Download CHEM 155: BASIC PHYSICAL CHEMISTRY I INTRODUCTION

CHEM 155: BASIC PHYSICAL CHEMISTRY I
INTRODUCTION
Physical chemistry is the study of the fundamental physical principles that govern the way that
atoms, molecules, and other chemical systems behave. Physical chemists study a wide array of
phenomena such as the rates of chemical reactions (kinetics), the way that light and matter
interact (spectroscopy), how electrons are arranged in atoms and molecules (quantum chemistry),
and the stabilities and reactivities of different compounds and processes (chemical
thermodynamics). Physical chemistry seeks to anchor the empirical rules of chemistry to the
laws of physics, and thus provide secure concepts to explain the trends seen in chemical
reactivity and molecular structure. Physical chemists use theoretical constructs and mathematical
computations to understand chemical properties and describe the behavior of molecular and
condensed matter.
A chemical system can be studied from either a microscopic or a macroscopic viewpoint.
The microscopic viewpoint is based on the concept of molecules. The macroscopic viewpoint
studies large-scale properties of matter without explicit use of the molecular concept.
Thermodynamics is a macroscopic science that studies the interrelationships of the various
equilibrium properties of a system and the changes in equilibrium properties in processes.
Molecules and the electrons and nuclei that compose them do not obey classical mechanics.
Instead, their motions are governed by the laws of quantum mechanics. Application of quantum
mechanics to atomic structure, molecular bonding and spectroscopy is the province of quantum
chemistry. The macroscopic science of thermodynamics is a consequence of what is happening
at the molecular (microscopic) level. The macroscopic and microscopic (molecular) levels are
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related to each other by the area of physical chemistry called statistical mechanics. Statistical
mechanics gives insight into why the laws of thermodynamics hold and allows calculations of
macroscopic thermodynamic properties from molecular properties. Kinetics is the study of rate
processes such as chemical reactions, diffusion and the flow of charge in an electrochemical cell.
The principles of physical chemistry provide a framework for all branches of chemistry.
Organic chemists use kinetics studies to figure out the mechanisms of reactions, use quantum
chemistry calculations to study the structures and stabilities of reaction intermediates, use
symmetry rules deduced from quantum chemistry to predict the course of chemical reactions and
use nuclear magnetic resonance (NMR) and infrared spectroscopy to help determine the structure
of compounds. Inorganic chemists use quantum chemistry and spectroscopy to study bonding.
Analytical chemists use spectroscopy to analyze samples. Biochemists use kinetics to study rates
of enzyme-catalyzed reactions; use thermodynamics to study biological energy transformations,
osmosis and membrane equilibrium and to determine molecular weights of biological molecules
and use spectroscopy to study processes at the molecular level. Environmental chemists use
thermodynamics to find the equilibrium composition of lakes and streams, use kinetics to study
the reactions of pollutants in the atmosphere and use physical kinetics to study the dispersion of
pollutants in the environment. Chemical engineers use thermodynamics to predict the
equilibrium composition of reaction mixtures, use kinetics to calculate how fast products will be
formed and use principles of thermodynamic phase equilibria to design separation procedures
such as fractional distillation. Geochemists use thermodynamic phase diagrams to understand
processes in the earth. Polymer chemists use thermodynamics, kinetics and statistical mechanics
to investigate the kinetics of polymerization, the molecular weights of polymers, the flow of
polymer solutions, and the distributions of conformations of a polymer molecule.
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LAWS, HYPOTHESES, THEORIES AND MODELS
A scientific law is a concise verbal or mathematical statement of a relation that expresses a
fundamental principle of science. It is a descriptive generalization about how some aspect of the
natural world behaves under stated circumstances. Simply put, a scientific law is a summary of
experience. A scientific law must always apply under the same conditions, and implies a causal
relationship between its elements. Examples are Boyle’s law of gases, Newton’s law of universal
gravitation and the law of conservation of mass.
A hypothesis is a tentative statement about the natural world leading to deductions that can be
tested. If the deductions are verified, it becomes more probable that the hypothesis is correct. If
the deductions are incorrect, the original hypothesis can be abandoned or modified. Simply put, a
hypothesis is an educated guess, based on observation. Hypotheses can be used to build more
complex inferences and explanations. Sometimes a hypothesis is developed that must wait for
new knowledge or technology to be testable. The concept of atoms was proposed by the ancient
Greeks, who had no means of testing it. Centuries later, when more knowledge became available,
the hypothesis gained support and was eventually proven, though it has had to be amended many
times over the years. Atoms are not indivisible, as the Greeks supposed.
A scientific theory is a well-substantiated explanation of some aspect of the natural world that
can incorporate facts, laws, inferences, and tested hypotheses. A theory must be internally
consistent and compatible with the evidence, firmly grounded in and based upon evidence, tested
against a wide range of phenomena, and demonstrably effective in problem-solving.
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A scientific model is a conceptual representation whose purpose is to explain and predict
observed phenomena. A model is an approximation or simulation of a real system that omits all
but the most essential variables of the system. The concept of an ideal gas is a model.
A scientific fact is an observation that has been repeatedly confirmed and for all practical
purposes is accepted as "true." Truth in science, however, is never final, and what is accepted as
a fact today may be modified or even discarded tomorrow.
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STATES OF MATTER
Matter is any material substance that constitutes the observable universe and, together with
energy, forms the basis of all objective phenomena.
Depending on the temperature and other conditions, matter may appear in several states. The
known states of matter are solid, liquid, gas, plasma and Bose-Einstein condensate (BEC).
A solid has a definite shape and takes up a fixed volume. A liquid has a fixed volume, but no
definite shape. A liquid will change its shape to fit its container. A gas has no definite shape and
no fixed volume. Like a liquid, a gas will change its shape to fit its container, but it will also
expand to fill the entire volume of the container.
Plasma is an ionized gas, a gas into which sufficient energy is provided to free electrons from
atoms or molecules and to allow both species, ions and electrons, to coexist. Examples of matter
in the plasma state are the sun and other stars, the solar wind, the interplanetary medium (space
between planets), the interstellar medium (space between star systems), the intergalactic medium
(space between galaxies) and interstellar nebulae. It is estimated that 99.9% of all observable
matter in the universe is in the plasma state.
Bose-Einstein condensate is a state of matter in which separate atoms or subatomic particles,
cooled to near absolute zero (0 K, -273.15 oC) coalesce into a single quantum mechanical entity
– that is, one that can be described by a wave function – on a near-macroscopic scale. This form
of matter was predicted in 1924 by Albert Einstein on the basis of the quantum mechanical
formulations of the Indian Physicist Satyendra Bose. Einstein demonstrated that cooling bosonic
atoms to a very low temperature would cause them to fall (or "condense") into the lowest
accessible quantum state, resulting in a new form of matter. This process is known as Bose–
Einstein condensation – the macroscopic ground-state accumulation of particles with integer
spin (bosons) at low temperature and high density. Although it had been predicted for decades,
the first atomic Bose-Einstein condensate was made only in 1995 when Eric Cornell and Carl
Wieman cooled a gas of rubidium atoms to 1.7 x 10-7 K above absolute zero. They succeeded in
merging 2,000 individual atoms into a “superatom”, a condensate large enough to observe with a
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microscope, that displayed distinct quantum properties. The first photon Bose-Einstein
condensate was observed in November, 2010.
GAS LAWS
Boyle's Law: The Pressure-Volume Law
Robert Boyle (1627-1691)
Boyle's law or the pressure-volume law states that the volume of a given amount of gas held at
constant temperature varies inversely with the applied pressure when the temperature and mass
are constant.
Vα
(at constant T, n)
PV = k
When pressure goes up, volume goes down. When volume goes up, pressure goes down.
From the equation above, this can be derived:
P1V1 = P2V2 = P3V3 etc.
This equation states that the product of the initial volume and pressure is equal to the product of
the volume and pressure after a change in one of them under constant temperature. P1V1 = P2V2
Boyle’s law is understandable from the picture of a gas as consisting of a huge number of
molecules moving essentially independently of one another (noninteracting point particles). The
pressure exerted by the gas is due to the impacts of the molecules on the walls. A decrease in the
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volume causes the molecules to hit the walls more often, thereby increasing the pressure. In
actuality, the molecules of a gas exert forces on one another, so Boyle’s law does not hold
exactly. In the limit of zero density (reached as the pressure goes to zero or as the temperature
goes to infinity), the gas molecules are infinitely far apart from one another, forces between
molecules become zero, and Boyle’s law is obeyed exactly. Boyle’s law is thus a limiting law: it
is not obeyed exactly by any actual gas, but describes a gas in the limit of zero pressure. A
hypothetical gas that obeys Boyle’s law at any pressure is called an ideal gas or a perfect gas.
Charles' Law: The Temperature-Volume Law
Jacques Charles (1746 - 1823)
This law states that the volume of a given amount of gas held at constant pressure is directly
proportional to the Kelvin temperature.
V α T (at constant P, n)
=k
As the volume goes up, the temperature also goes up, and vice-versa.
V1 / T1 = V2 / T2 = V3 / T3 etc.
The molecular explanation for Charles’ law lies in the fact that an increase in temperature means
the molecules are moving faster and hitting the walls harder and more often. Therefore, the
volume must increase if the pressure is to remain constant.
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Amontons' Law: The Pressure-Temperature Law
Guillaume Amontons (1663 – 1705)
This law states that the pressure of a given amount of gas held at constant volume is directly
proportional to the Kelvin temperature.
P α T (at constant V, n)
=k
As
the
pressure
goes
up,
the
temperature
also
goes
up,
and
vice-versa.
Also same as before, initial and final pressures and temperatures under constant volume can be
calculated.
P1 / T1 = P2 / T2 = P3 / T3 etc.
Avogadro's Principle: The Volume-Amount Relationship
Amedeo Avogadro (1776-1856)
Avogadro’s principle states that at a given temperature and pressure, equal volumes of gas
contain the same number of molecules. This gives the relationship between volume and amount
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when pressure and temperature are held constant. Since volume is one of the variables, that
means the container holding the gas is flexible in some way and can expand or contract.
If the amount of gas in a container is increased, the volume increases. If the amount of gas in a
container is decreased, the volume decreases.
V α n (at constant T, P)
=k
This means that the volume-amount fraction will always be the same value if the pressure and
temperature remain constant.
V1 / n1 = V2 / n2 = V3 / n3 etc.
Avogadro’s suggestion is a principle rather than a law, because it is not a direct summary of
experience (it is based on the model of a gas as a collection of molecules).
The molar volume of any substance is the volume it occupies per mole of molecules present in
the sample:
Vm =
Avogadro’s principle implies that the molar volume of a gas should be the same for all gases at
the same temperature and pressure.
The Combined Gas Law
Now we can combine everything we have into one proportion:
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The volume of a given amount of gas is proportional to the ratio of its Kelvin temperature and its
pressure.
Same as before, a constant can be put in:
PV / T = k
As
the
pressure
goes
up,
the
temperature
also
goes
up,
and
vice-versa.
Also same as before, initial and final volumes and temperatures under constant pressure can be
calculated.
P1V1 / T1 = P2V2 / T2 = P3V3 / T3 etc.
The Ideal Gas Law
The previous laws all assume that the gas being measured is an ideal gas, a gas that obeys them
all exactly.
Boyle's law:
Amontons' law:
Charles' law:
Avogadro's principle:
P α
Pα
Vα
V
1/V
T
T
n
(T and n constant)
(V and n constant)
(P and n constant)
(P and T constant)
Combining these relations into a single expression
PV α nT
Introducing a constant R gives
PV = nRT
This is called the ideal gas equation of state, where R is a constant called the universal gas
constant and is equal to approximately 0.082058 L atmK-1mol-1 or 8.3145 JK-1mol-1.
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MIXTURES OF GASES: DALTON’S LAW OF PARTIAL PRESSURES
In a mixture of gases, what contribution does each component of the mixture make to the total
pressure exerted by the gas? The answer to that lies in the Dalton’s law of partial pressures
which states that the pressure exerted by a mixture of perfect (ideal) gases is the sum of the
pressure exerted by the individual gases occupying the same volume alone.
The contribution that a gas makes to the total pressure is the partial pressure of that gas. The
partial pressure of each gas is equal to the pressure it would exert if it occupied the container
alone. Therefore Dalton’s law implies that the total pressure of a mixture of gases is the sum of
the partial pressures of all the gases present.
The partial pressure of each component in a mixture can be calculated from the perfect gas
equation of state once the amount of molecules of each gas is known. For example, if the mixture
consists of an amount nA of A molecules and nB of B molecules, then the partial pressures of A
and B are given by
=
=
According to Dalton’s law, the total pressure of the mixture is the sum of these partial pressures
=
+
=(
+
)
The mole fraction, χA, of a species A is the amount of A (in moles) expressed as a fraction of the
total amount of molecules (also in moles) present in the sample. In a sample that consists of an
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amount of nA of species A, and amount nB of a species B, an amount nC of species C etc, then the
mole fractions of A, B and C present in the mixture are
=
=
=
and
+
+
+
+
+
+
+
+
=1
The partial pressure of each component in a mixture is related to the total pressure by
PA =
xP
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THE KINETIC THEORY OF GASES
The kinetic theory of gases is the study of the microscopic behavior of gas molecules and the
interactions which lead to macroscopic relationships like the ideal gas law. The theory is an
attempt to explain macroscopic properties of gases, such as pressure, temperature, or volume, by
considering their molecular composition and motion. Essentially, the theory posits that pressure
is due not to static repulsion between molecules, as was Isaac Newton's conjecture, but due to
collisions between molecules moving at different velocities through Brownian motion.
The kinetic theory of gases is based on the following assumptions:
1. The gas is composed of a large number of identical molecules moving in random
directions, separated by distances that are large compared with their size;
2. The molecules undergo perfectly elastic collisions (no energy loss) with each other and
with the walls of the container, but otherwise do not interact;
3. The average kinetic energy of the gas particles depends only on the temperature of the
system.
4. The time during collision of molecule with the container's wall is negligible as compared
to the time between successive collisions.
The implications are that:
1. Statistical treatment can be applied since the number of molecules is so large.
2. Relativistic effects are negligible.
3. Quantum-mechanical effects are negligible. This means that the inter-particle distance is
much larger than the thermal de Broglie wavelength and the molecules are treated as
classical objects.
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THE PRESSURE OF A GAS FROM THE KINETIC THEORY
When a gas particle of mass m collides with the wall on the right its component of linear
momentum parallel to the x-axis changes from mυx (when it is travelling to the right) to –mυx
(when it is travelling to the left). Its momentum changes by 2mυx on each collision.
The number of collisions in an interval Δt is equal to the number of particles able to reach the
wall in that interval. A particle of speed υx in the x direction travels a distance υxΔt in an interval
Δt. Thus, all particles within a distance υ×Δt will strike the wall if travelling towards it. If the
wall has an area of A, then all particles in a volume AυxΔt will reach the wall if they are
travelling towards it.
Number density, N, is the number of particles per unit volume. Therefore the number of particles
in the volume is AυxΔt is NAυxΔt
On average, half the particles are moving to the right and half to the left. Therefore, the average
number of collision with the wall during the interval Δt is ½ NAυxΔt. The total momentum
change in that interval is the product of this number and the change 2mυ× that an individual
molecule experiences.
Momentum change = ½ NA υxΔt × 2mυx
= mNAυ2xΔt
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The rate of change of momentum is this change of momentum divided by the interval Δt during
which it occurs
Rate of change of momentum = mNAυ2x
Newton’s second law
F = mNAυ2x
Pressure = mNυ2x
The detected pressure, p, is the average of the quantity just calculated
p = mN<υ2x>
The root-mean-square speed, c, of the particles is
c = <υ2>1/2 = (<υ2x> + <υ2y> + <υ2z>)1/2
Because the particles are moving randomly the average of υ2x is the same as the average of y and
z, (<υ2x>, <υ2y> and <υ2z>) are all equal.
c = (3<υ2x>)1/2
υ2x = c2
p = Nmc2
(1)
The number density of molecules is the product of the amount n and Avogadro’s constant NA
divided by the volume (N = nNA/V); so (1) becomes
pV =
mNAmc2 = nMc2
pV = nMc2
where M is the molar mass and c is the root-mean-square speed of the molecules.
If we suppose that the root-mean-square speed of the molecules depends only on the
temperature, then at constant temperature and for a fixed amount of molecules, this equation
becomes
pV = constant
which is Boyle’s law (at constant temperature and for fixed amount of gas, pV = k). Thus the
kinetic theory has successfully accounted for Boyle’s law.
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The Speed of gas molecules
pV = nRT
pV = nMc2
Mc2 = RT
c=
Maxwell’s Distribution of speeds
The fraction of molecules that has a particular speed is called the distribution of molecular
speeds.
According to the Maxwell’s distribution of speeds, the fraction, f, of molecules that has a speed
in a narrow range s - s + Δs is
f = 4π
/
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Diffusion and Effusion
Graham’s law of effusion: at a given pressure and temperature, the rate of effusion of a gas is
inversely proportional to the square root of is molar mass.
Real Gases
All real gases are imperfect. Forces of attraction and repulsion come into play within range of
influence. If there were no forces of attraction, all matter would be gaseous, since there would
be nothing to bring the molecules together in the solid and liquid. The behavior of matter in
condensed phases is determined by the balance of the forces of attraction and repulsion.
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Compression factor:
From PV = nRT
=
=1
This is the compression factor. It must be 1 for ideal gases but is known to differ from 1 for real
gases. Thus, the compression factor is a measure of the deviation from ideality.
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Real Gas Equations of State
1. Van der Waal’s Equation of state:
P=
−
The presence of the repulsive interactions implies that two molecules cannot come closer
than some certain distance of each other. Therefore, instead of being free to travel anywhere
in a volume V, the actual volume in which the molecules can travel is reached by an amount
that is proportional to the number of molecules present in the sample and the volume they
each exclude. Thus, the volume available to the gas is V – nb
P=
The presence of attractive interactions between the molecules also reduce the pressure that
the gas exerts. The attraction experienced by a given molecule is proportional to the
concentration, n/V, of molecules in the container. Because the attractions slow down the
molecules, they strike the walls less frequently and with a lower impact. Because the pressure
is determined by the impact of molecules, and they are fewer and weaker, the reduction in
pressure is proportional to the square of the molar concentration.
Reduction in pressure ∝ (rate of impact) × (average strength of impact)
∝ ×
If the constant of proportionality is a, then
Reduction in pressure = a ×
Thus, P =
−a
Where a and b are empirical parameters
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2. Others
a. Virial:
b. Berthelot:
c. Clausius:
d. Redlich-Kwong:
e. Dieterici:
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THERMODYNAMICS
How can it be that dissolving one substance, such as NaOH, in water heats the liquid up, while
dissolving another, such as urea, cools it down? Why it is that mixing ethanol and butanol yields
a final volume greater than the combined volumes of the separated liquids, whereas mixing two
other substances, such as ethanol and water, gives a volume smaller than the total volume of the
initial substances? How can it be that compressing ice tends to melt it to water, whereas
compressing most other liquids tends to make them freeze? Why is it that when two objects are
placed in contact, heat flows naturally from the hotter object to the colder one, and not the other
way round? Why is it that as you use the books on your desk in a random way they tend to get
out of order rather than tending toward some special order, such as alphabetical?
These are a few of the diverse questions that are illuminated by thermodynamics, the branch of
physical chemistry that describes matter on the macroscopic level and the physical and chemical
changes that it undergoes. This description rests on a simplified representation, or model, of
physical reality, and on a small number of laws.
Thermodynamics deals with energy, its forms and transformations, and the interactions between
energy and matter. It is a central branch of science that has important applications in chemistry,
physics, biology, and engineering. Thermodynamics is a logical discipline that organizes the
information obtained from experiments performed on systems and enables us to draw
conclusions, without further experimentation, about other properties of the system. It allows us to
predict whether a reaction will proceed and what the maximum yield might be. As a macroscopic
science, thermodynamics deals with such properties as pressure, temperature, and volume. It
does not describe the microscopic constituents of matter and is not based on a specific model,
and is therefore unaffected by our changing concepts of atoms and molecules. Equations derived
from thermodynamics do not provide us with molecular interpretations of complex phenomena.
Furthermore, thermodynamics tells us nothing about the rate of a process except its likelihood.
THERMODYNAMIC TERMS: DEFINITIONS
A thermodynamic system is that part of the physical universe the properties of which are under
investigation. The system is confined to a definite place in space by the boundary which
separates it from the rest of the universe, the surroundings. A system is isolated when the
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boundary prevents any interaction with the surroundings. An isolated system produces no
observable effect or disturbance in its surroundings. A system is called open when mass passes
across the boundary, and closed when no mass passes the boundary.
Properties of a System: The properties of a system are those physical attributes that are
perceived by the senses, or are made perceptible by certain experimental methods of
investigation. Properties fall into two classes: (1) non-measurable, as the kinds of substances
composing a system and the states of aggregation of its parts ; and (2) measurable, as pressure
and volume, to which a numerical value can be assigned by a direct or indirect comparison with
a standard.
State of a System: A system is in a definite state when each of its properties has a definite value.
Specifying the state of a system means describing the condition of the system by giving the
values of a sufficient set of numerical variables.
There are two principal classes of macroscopic variables. Extensive variables are proportional to
the size of the system if temperature and pressure are constant, whereas intensive variables are
independent of the size of the system if temperature and pressure are constant. For example,
volume V, amount n and mass m of a system are extensive variables of the system, whereas
temperature T and pressure P are intensive variables. The quotient of two extensive variables is
an intensive variable. Examples: density (m/V), concentration (n/V).
A change in state is completely defined when the initial and the final states of the system are
specified. The path of a change in state is defined by giving the initial state, the sequence of
intermediate states arranged in the order traversed by the system, and the final state.
A process is the method of operation by means of which a change in state is effected. The
description of a process consists in stating some or all of the following: (1) the boundary; (2) the
change in state, the path, or the effects produced in the system during each stage of the process;
and (3) the effects produced in the surroundings during each stage of the process.
State Variable: A state variable is one that has a definite value when the state of a system is
specified.
HEAT AND WORK
Work is the transfer of energy between a system and its surrounding due to a macroscopic force
acting through a distance. Heat is the energy transfer between a system and its surrounding due
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to a temperature difference. Heat and work are forms of energy transfer rather than forms of
energy. Work is energy transfer due to the action of macroscopically observable forces. Heat is
energy transfer due to the action of forces at the molecular level. When bodies at different
temperatures are placed in contact, collision between molecules of the two bodies produce a net
transfer of energy to the colder body from the hotter, whose molecules have a greater average
kinetic than those in the colder body. Heat is work done at the molecular level.
Mechanical Work
The amount of work done on an object equals the force exerted on it times the distance it is
moved in the direction of the force. If a force Fz is exerted on an object in the z direction, the
work done on the object in an infinitesimal displacement dz in the z direction is
dw = Fzdz ( def i ni t i on of wor k)
where dw is the work done.
The Work Done on a Closed Fluid System
If a force is transmitted between a system and its surroundings and if the volume of the system
changes there is work done by the surroundings on the system or by the system on the
surroundings. Consider a gas confined in a cylinder with a piston as depicted below. Let an
external force Fext be exerted downward on the system by the piston. If the external force is
greater than the force exerted on the piston by the system, the piston will accelerate downward
and the surroundings perform work on the system. If the external force is smaller, the piston will
accelerate upward and the system performs work on the surroundings. If the height of the piston,
z, increases by an infinitesimal amount dz, the amount of work done on the surroundings is given
by
dws ur r= F( t r ans mi t t ed)
z d
where F(transmitted) is the force that is actually transmitted to the surroundings.
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At equilibrium, the force exerted on the piston by the gas is equal to
where P is the pressure of the gas and A is the area of the piston. We define the external pressure
Pext by
At equilibrium there is no tendency for the piston to move so that
where we define the transmitted pressure by
A reversible process is defined to be one that can be reversed in direction by an infinitesimal
change in the surroundings. In order for an infinitesimal change in Pext to change the direction
of motion of the piston, there can be no friction and there can be no more than an infinitesimal
difference between P, P(transmitted), and Pext. For an infinitesimal step of a reversible process
the work done on the surroundings is given by
where V is the volume of the system.
The work done on the surroundings as a negative amount of work done on the system, so
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NB: A positive value of w or dw corresponds to work being done on the system by the
surroundings. A negative value of w or of dw corresponds to work being done on the
surroundings by the system.
If a system can exchange work with its surroundings only by changing its volume, it is said to be
a simple system. This kind of work is sometimes called compression work or P-V work. A gas is
a simple system. A liquid is a simple system if the work required to create surface area by
changing the shape of the system can be neglected. A spring or a rubber band is not a simple
system because work is done to change its length. An electrochemical cell is also not a simple
system since work can be done by passing a current through it.
Work Done on an Ideal Gas in a Reversible Process
For a finite reversible change in volume of n moles of an ideal gas,
If the temperature is constant the process is called isothermal. For an isothermal process,
where V1 is the initial value of the volume and V2 is the final value.
THE LAWS OF THERMODYNAMICS
1. The Zeroth Law: Two systems that are each in thermal equilibrium with a third system
will be in thermal equilibrium with each other. The zeroth law of thermodynamics is an
observation. The law implies that thermal equilibrium between systems is a transitive
CHEM 155 BASIC PHYSICAL CHEMISTRY I NOTES – DR. RICHARD TIA
Page 25
relation, which affords the definition of an empirical parameter called temperature.
Objects in thermal equilibrium have the same temperature. The zeroth law permits the
construction of a thermometer to measure this property. We can calibrate the change in a
thermal property, such as the length of a column of mercury, by putting the thermometer
in thermal equilibrium with a known physical system at several reference points. Celsius
thermometers have the reference points fixed at the freezing and boiling point of pure
water. If we then bring the thermometer into thermal equilibrium with any other system,
such as the bottom of your tongue, we can determine the temperature of the other system
by noting the change in the thermal property.
2. The First Law: In addition to the macroscopic kinetic energy and potential energy of an
object, the object also has internal energy. This internal energy consists of: molecular
translational, rotational, vibrational and electronic energies; the relativistic rest-mass
energy mrestC2 of the electrons and nuclei, and potential energy of interaction between the
molecules.
Therefore the total energy of a body is
E=k+V+U
where K and V are the macroscopic (not molecular) kinetic and potential energies of the
body (due to motion of the body through space and the presence of fields that act on the
body) and U is the internal energy of the body (due to molecular motions and
intermolecular interactions. The first law asserts that for every process in a closed system
at rest,
ΔU = q + w
CHEM 155 BASIC PHYSICAL CHEMISTRY I NOTES – DR. RICHARD TIA
Page 26
where q is the heat flow into the system during the process and w is the work done on the
system during the process.
For an infinitesimal process,
dU = dq + dw
du is the infinitesimal change in system energy in a process with infinitesimal heat dq
flowing into the system and infinitesimal work dw done on the system.
Internal energy U is a function of the state of the system. For any process, ΔU depends
only on the final and initial states of the system and is independent of the path used to
bring the system from the initial state to the final state.
ΔU = U2 – U1 = Ufinal – Uinitial
ENTHALPY
The enthalpy, H, is defined as
H = U + PV …………. (1)
where P is the pressure of the system and V is its volume. Because U, P, and V are state
functions, the enthalpy is a state function.
The change in enthalpy is equal to the energy supplied as heat at constant pressure
(provided the system does no additional work).
dH = dq : heat transferred at constant pressure
For a measurable change
ΔH = qp
Proof:
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For a general infinitesimal change in the state of the system, U changes to U + dU, P to P
+ dP etc
so from (1)
H + dH = (U + dU) + (p + dP)(V+ dV)
= U + dU + PV + PdV + VdP + dPdV
The last term is the product of two infinitesimally small quantities and can therefore be
neglected.
H + dH = U + PV + dU + PdV + VdP
H + dH = H + dU + PdV + VdP
dH = dU + PdV + VdP
but since dU = dq + dw
dH = dq + dw + PdV + VdP
If the system is in mechanical equilibrium with its surrounding at a pressure P and does only
expansion work, then dw = -PdV
dH = dq + Vdp
At constant pressure, dP = 0
Therefore dH = dq
The enthalpy of a perfect gas is related to its internal energy by using PV = nRT in the definition
of H:
H = U + PV = U + nRT
CHEM 155 BASIC PHYSICAL CHEMISTRY I NOTES – DR. RICHARD TIA
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H = U + nRT
ΔH = ΔU + ΔnRT
where Δng is the change in the amount of gas molecules in the reaction.
Variation of Enthalpy with Temperature
The enthalpy of a substance increases as its temperature is raised.
The heat capacity at constant pressure, Cp,
=
=
∆
=
∆
HEAT CAPACITIES
The heat capacity, Cpr, of a closed system for an infinitesimal process pr is defined as
=
whose
and
in process.
/
are the heat flow into the system and the temperature change of the system
1. The heat capacity at constant pressure (or isobaric heat capacity), Cp,
≡
where
and
are the heat added to the system and the system’s temperature change in an
infinitesimal constant-pressure process.
2. Heat capacity at constant volume (or isochoric heat capacity), Cv, of a closed system is
≡
where
and
are the heat added to the system and the system’s temperature change in an
infinitesimal constant-volume process.
CHEM 155 BASIC PHYSICAL CHEMISTRY I NOTES – DR. RICHARD TIA
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=
⇒
and from
=
,
=
=
+
, at constant volume
=
closed system in equilibrium, P-V work only, V constant.
Cp and Cv give the rate of change of H and U with temperature.
CHEM 155 BASIC PHYSICAL CHEMISTRY I NOTES – DR. RICHARD TIA
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CHEMICAL KINETICS
Chemical kinetics is the study of the speed with which a chemical reaction occurs and the factors
that affect this speed. This information is especially useful for determining how a reaction
occurs. Chemistry is primarily concerned with the conversion of substances from one form into
another (i.e., chemical reactions). At the heart of this are two basic questions that must be
answered:
1. Does the reaction want to go? This is the subject of chemical thermodynamics.
2. If the reaction wants to go, how fast will it go? This is the subject of chemical kinetics.
Thermodynamics is the arrow of chemical reactions while chemical kinetics is the clock of
chemical reactions. Chemical kinetics is a subject of broad importance. It relates, for example, to
how quickly a medicine is able to work, to whether the formation and depletion of ozone in the
upper atmosphere are in balance, and to industrial problems such as the development of catalysts
to synthesize new materials. The speed (rate) of a reaction is the rate at which the concentrations
of reactants and products change. It is the rate of decrease of the concentration of a reactant or
the rate of increase of the concentration of a product.
E.g.
→
=
[ ]
or
=
[ ]
For a reaction such as:
+
→
+
Where a, b, c and d are the stoichiometric coefficients of the species A, B, C and D in the overall
balanced reaction equation.
=
[ ]
=
[ ]
=
[ ]
=
[ ]
CHEM 155 BASIC PHYSICAL CHEMISTRY I NOTES – DR. RICHARD TIA
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RATE LAWS
The measured rate of a reaction is after found to be proportional to the molar concentration of the
reactants raised to a power.
E.g. +
→
=
[ ] [ ] …………… (1)
A rate law is an equation that expresses the rate of reaction as a function of the concentration of
the species in the overall reaction (including the products).
Eqn (1) is a rate law.
The exponents x and y in the rate law are the orders of the reaction with respect to the species A
and B respectively.
The order of a reaction with respect to each species is the power to which the concentration of
that species is raised in the rate law.
Thus the reaction above is xth order with respect to A, yth order respect to B and (x + y)th order
overall. The overall order of a reaction is the sum of the orders of all the components.
(
=
[
)
+3
(
)
→2
(
)
+
(
)
][ ] → experimentally determined.
This reaction is first order in
, first order in
and second order overall. Thus, this
reaction is said to obey SECOND ORDER KINETICS.
INTEGRATED RATE LAWS
A. FIRST – ORDER REACTIONS –These are reactions in which the rate depends on the
reactant concentration raised to the first order.
E.g.
→ . Let us obtain an integrated rate law for this reaction, assuming it obeys first –order
kinetics.
CHEM 155 BASIC PHYSICAL CHEMISTRY I NOTES – DR. RICHARD TIA
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[ ]
=
Thus
[ ]
,
= [ ]
= [ ]
[ ]
=
[ ] [ ]
] [ ]
∫[
=− ∫
− l n[ ] + l n[ ] =
l n[ ] = l n[ ] −
⇒ln
[ ]
[ ]
=
Thus to test whether a given reaction obeys first ‒ order kinetics plot l n[ ] versus . If a straight
line results, the test is positive. The slope of that line should be − , from which the rate constant
can be calculated. The vertical intercept is l n[ ] .
Example: The conversion of cyclopropane to propene in the gas phases is a first order reaction
with a rate constant of 6.7×10-4 s-1 at 500℃.
(a). If the initial concentration of cyclopropane was 0.25M, what is the concentration after
8.8min-1 ?
(b). How long will it take for the concentration of cyclopropane to decrease from 0.25M to
0.15M?
(c). How long will it take to convert 74% of the starting materials?
Solution
(a). l n
ln
[ ]
[ ]
.
[ ]
=
= (6.7 × 10
) 8.8
×
[ ] = 0.18M
CHEM 155 BASIC PHYSICAL CHEMISTRY I NOTES – DR. RICHARD TIA
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(b). l n
[ ]
=
[ ]
.
ln
= (6.7 × 10
.
)
= 13
(c). Amount left after time is (100 − 74)% = 26%
= (6.7 × 10
Hence
)
= 33
B. SECOND – ORDER REACTIONS –These are reactions in which the rate depends on the
concentration of one reactant raised to the second power or on the concentration of two
different reactants, each raised to the first order.
→ . Let us obtain an integrated rate law for this reaction if it obeys second – order
E.g.
kinetics.
[ ]
=
Thus
[ ]
[ ] [ ]
] [ ]
∫[
⟹
[ ]
=
[ ] =
[ ]
[ ]
:∫
=
+
=−
[ ] [ ]
]
=
= − ∫
[ ]
=[
,
+
…………………….. (∗)
[ ]
[ ]
CHEM 155 BASIC PHYSICAL CHEMISTRY I NOTES – DR. RICHARD TIA
Page 34
Equation (∗) shows that to test for second‒order reaction, we should plot 1 [ ] against and
expect a straight line. If it is straight, then the reaction is second ‒ order in A and the slope of the
line is equal to the rate constant.
THE ARRHENIUS EQUATION
The dependence of the rate constant of a reaction on temperature can be expressed by the
following equation, known as the Arrhenius equation:
=
⟹ln =ln −
where
is called the pre-exponential factor and
is called the activation energy. Collectively,
the two parameters are called the Arrhenius parameters of the reaction.
The Arrhenius parameters of a reaction can be determined by the plotting l n versus1
gives a straight line. The slope is −
⁄
, from which
can be calculated since
, which
is a known
constant. The vertical intercept is l n .
An equation relating the rate constant
and
at temperature
and
can be used to calculate
the activation energy or find the rate constant at another temperature if activation energy is
known. To derive such an equation, proceed as follows.
ln
=ln −
…………… (1)
ln
=ln −
…………… (2)
Subtracting (2) from (1)
ln
−l n
ln =
=
−
−
CHEM 155 BASIC PHYSICAL CHEMISTRY I NOTES – DR. RICHARD TIA
Page 35
ln =
.
E.g. The rate constant of a first – order reaction is 3.46×10−2 s−1 at 298K. What is the rate
constant at 350 K if the activation energy for the reaction is 50.2 kJmol−1?
Solution
K1 = 3.46×10−2s−1
K2 =?
T1 = 298K
T2 = 350K
Substituting this into the equation above gives
ln
.
×
=
. ×
(
.
)(
)
= 0.702
REACTION MECHANISMS
Elementary reactions – these are a series of simple reactions that represent the progress of the
overall reaction at molecular level.
Reaction mechanism is the sequence of elementary steps that leads to product formation.
Consider this reaction:
2
( )
+
( )
→2
( )
The products are not formed directly from the collision of two NO molecules with an O2
molecule as predicted by the overall balanced equation above. The reaction takes place in a
series of steps:
Step 1: 2
( )
⇌
Step 2:
( )
+
( )
( )
→2
( )
CHEM 155 BASIC PHYSICAL CHEMISTRY I NOTES – DR. RICHARD TIA
Page 36
Steps 1 and 2 are the elementary reactions. Together they represent the mechanism of the
reaction. The net chemical equation, which is represents the overall change is given by the sum
of the elementary steps.
Intermediates are species that appear in the mechanism of the reaction (that is, the elementary
steps) but not the overall balanced equation. An intermediate is always formed in an elementary
step and consumed in a later elementary step. N2O2 in the example above is an intermediate.
The molecularity of a reaction is the number of molecules reacting in an elementary step.
Unimolecular – an elementary step in which only one reacting molecule participates.
Bimolecular – an elementary step that involves two molecules.
THE FORMULATION OF RATE LAWS: THE STEADY – STATE ASSUMPTIONS
Consider the following elementary steps of the overall reaction
+
Step 1:
→
:
k1
A
Step 2:
k2
+
B
→
From step 1;
Rate of formation of
[ ] ...................... (1)
=
Rate of consumption of
=
[ ] ................. (2)
=
[ ][ ] ........... (3)
From step 2;
Rate of consumption of
Rate of formation of
=
[ ][ ] ................ (4)
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Page 37
Eqn (4) expresses the rate of the overall reaction. However, this expression is not an acceptable
overall rate law because it is expressed in terms of the intermediate B: an acceptable rate law for
an overall reaction is expressed SOLELY in terms of the species that appear in the overall
reaction. Therefore, we need to find an expression for the concentration of . To do this, we
consider the net rate of the formation of the intermediate, the difference between its rates of
formation and consumption.
Net rate of formation of
where
⟹
=
i ns t ep1 −
is the rate of formation and
[ ]
=
[ ]−
[ ]
=
[ ]−(
[ ]−
+
n s t ep1 −
i ns t ep2.
is the rate of consumption
[ ][ ]
[ ])[ ]................. (5)
At this stage, we introduce the steady – state assumption:
In the steady – state assumption, it is supposed that the concentration of all the intermediates
remain constant throughout the reaction (except right at the beginning and right at the end).
Employing the assumption, NET rate of formation of
⟹ Rat eof f or mat i on
of
=0
= Rat eof cons umpt i onof
It follows from this assumption that eqn (5) becomes
[ ]−(
+
[ ])[ ] = 0
Making [ ] the subject;
[ ]=
[ ]
[ ]
Substituting this into eqn (4), we obtain
Rate of formation of
=
[ ][ ]
[ ]
………… (6)
CHEM 155 BASIC PHYSICAL CHEMISTRY I NOTES – DR. RICHARD TIA
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This rate law is complex. It can conform to first – order or second – order kinetics depending on
the relative rates of consumption of
If the rate of consumption of
in steps 1 and 2.
in step 1 is much greater than its rate of consumption in step 2;
then
[ ]≫
⟹
∴
[ ][ ]
[ ]
≫
+
(NB:
>>
=
+
≃ )
[ ]≃
Putting this into (6) gives
Rate of formation of
Let
=
[ ][ ]
=
Rate of formation of
= [ ][ ]. This follows second – order kinetics.
However, if the rate of consumption of
in step 2 is much greater than its rate of consumption
in step 1; then
[ ][ ] ≫
⟹
∴
[ ]
[ ]≫
[ ]+
≃
[ ]
Putting this into (6) gives
Rate of formation of
=
Rate of formation of
=
[ ][ ]
[ ]
[ ]. This follows first – order kinetics.
CHEM 155 BASIC PHYSICAL CHEMISTRY I NOTES – DR. RICHARD TIA
Page 39
Set up the rate expression for the following mechanism:
k1
A
Step 1:
+
Step 2:
B
k2
→
Applying the steady – state approximation. Hence show that this reaction may follow the first –
order equation at high pressure and the second – order at low pressure.
Solution
NB: this is the same problem dealt with above.
[ ]
=
[ ][ ]
[ ]
=
[ ]−(
[ ]
[ ])[ ] = 0 ⟹ Steady – state approximation.
[ ]
[ ]=
⟹
+
[ ]
[ ][ ] =
=
[ ]
[ ]
At high pressure
[ ][ ] ≫
[ ]≫
[ ]
⟹
[ ]
=
[ ]. 1st – order kinetics
At low pressure
[ ]≫
≫
[ ][ ]
[ ]⟹
[ ]
=
[ ][ ]
. 2nd – order kinetics.
A certain reaction follows the following differential
CHEM 155 BASIC PHYSICAL CHEMISTRY I NOTES – DR. RICHARD TIA
Page 40
− [ ]⁄
= [ ]
a. Integrate the above equation if initial concentration is [
].
b. How would you check graphically that rate law follow 3 2 order reaction?
c. Calculate the half – life.
Solution
[ ]
a.
= [ ]
[ ] [ ]
][ ]
− ∫[
=
∫
⟹ 2
[ ]
=
−
[
=
]
⟹
[ ]
=
+
[
]
Hence the integrated rate law is
[ ]
=
[
+
]
…………. (∗)
b. Rat e= [ ]
⟹ l ogRat e= l og + l og[ ]
Plot a graph of l og(Rat e) against l og[ ] and expect a straight line whose slope is .
c. At =
=
,
2
Substitute these into (∗)
=
√
[ ]
2
=
[
]
[
]
√
−
[ ]
[
=
+
]
√
=
[
+
]
=
=
( .
)
[
]
.
[
]
CHEM 155 BASIC PHYSICAL CHEMISTRY I NOTES – DR. RICHARD TIA
Page 41
ENZYME KINETICS
Enzyme catalysis is the catalysis of (bio)chemical reactions by enzymes.
Enzymes are biological catalysts that increase the rates of biochemical reactions. In some cases,
the enzyme-catalyzed reaction is nearly 1015 times faster than the uncatalyzed reaction. Enzymes
are proteins (in some cases RNA) with very specific functions and are active under mild
conditions.
The mechanisms of enzyme catalysis is similar in principle to other types of chemical catalysis.
By providing an alternative reaction route, enzymes lower the activation energy of the desired
reaction. The reduction in activation energy increases the number of reactant molecules with
enough energy to reach the activation energy and form the product.
Enzymes differ from ordinary catalysts in their reaction rates, their action under milder reaction
conditions, their greater reaction specificities, and their capacity for regulation.
Enzymes are highly specific for their substrates and reaction products. Hence the enzyme and its
substrate(s) must have geometric, electronic and stereospecific complementarity. Enzymes, for
example, yeast alcohol dehydrogenase, can distinguish between prochiral groups.
Many enzymes require cofactors for activity. Cofactors may be metal ions or organic molecules
known as coenzymes. Many vitamins are coenzymes. Coenzymes may be cosubstrates, which
must be regenerated in a separate reaction, or prosthetic groups, which are permanently
associated with the enzyme. An enzyme without its cofactor(s) is called an apoenzyme and is
inactive, and an enzyme with its cofactor(s) is a holoenzyme and is active.
Michaelis-Menten (steady-state) Kinetics of Enzyme Catalysis
The Michaelis-Menten model for enzyme kinetics presumes a simple 2-step reaction:
Step 1: Binding – the substrate binds to the enzyme
Step 2: Catalysis – the substrate is converted to product and released
CHEM 155 BASIC PHYSICAL CHEMISTRY I NOTES – DR. RICHARD TIA
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Several simplifying assumptions allow for the derivation of the Michaelis-Menten equation:
1. The binding step (E + S ⇌ ES) is fast, allowing the reaction to quickly reach equilibrium
ratios of [E], [S], and [ES]. The catalytic step (ES⇌ E + P) is slower, and thus ratelimiting.
2. At early time points, where initial velocity (VO) is measured, [P] ≈ 0.
3. ES immediately comes to steady state, so [ES] is constant (throughout the measured
portion of the reaction).
4. [S] >> [ET], so the fraction of S that binds to E (to form ES) is negligible, and [S] is
constant at early time points.
5. The enzyme exists in only two forms: free (E), and substrate-bound (ES). Thus, the total
enzyme concentration (ET) is the sum of the free and substrate-bound concentrations:
[ET] = [E] + [ES].
From (1), the overall rate of the reaction is determined by the rate of the catalytic step:
Vo = k2[ES] – k-2[E][P]
From (2), the second term equals zero, so
Vo = k2[ES]
Since ES is an intermediate, [ES] is not easy to measure. It is better to describe Vo in measurable
quantities. Since [S] is known, express [ES] in terms of [S] making use of assumption (3) –
steady-state assumption:
Rate of formation of ES = Rate of breakdown of ES
k1[ E] [ S] +
-2[kE] [ P] =
-1[kES] +2[kES]
k1[ E] [ S] =
-1[kES] +2[kES]
Factoring out [ES] and grouping the rate constants
CHEM 155 BASIC PHYSICAL CHEMISTRY I NOTES – DR. RICHARD TIA
Page 43
k1[ E] [ S] = [ ES]
{k k2}
-1 +
[ E] [ S] = [{ES]
}
Thi s r at i o of r at e cons t ant s i s def i ned as t he Mimchael
:
i s Cons t ant , K
Km = {
}
Subs t i t ut i ng mi fnorK t he r-cons
at e t ant r at i o gi ves
[ E] [ S] [=ES] Km
Fr om as s umpt i on ( 5) , [ E]
[ ES]
[E
T] – =
{[ ET]– [ ES] }[ S] = [ ES]
m K
Obt ai ni ng an expr es s i on f or [ ES] i n t er ms of meas ur abl e quant i t i es ,
[ ET] [ S]– [ ES] [ S] = [ ES]
m K
[ ET] [ S] = [ ES]
K [ ES] [ S]
m+
[ ET] [ S] = [ ES]m {K
+ [ S] }
[
][ ]
[ ]
= [ES]
Thus VO = k2[ES] =
2
[ ]
+[ ]
At hi gh [ S] ( when [ S] m>>>K
) , near l y al l enz yme wi l l have s ubs t r at e bound, and [ ES]
appr oaches T[]E. Thi s i s when
Voaches max
V . Si nceo =V k2[ ES]
o appr
Then: Vmax = k2[ ET]
Or mat hemat i cal l y, whe n [ S] m>>>
, mK i sK negl i gi bl e, and t he equat i on s i mpl i f i es t o:
Vmax =
[
][ ]
[ ]
= k2 [ET]
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Page 44
Substituting the Vmax in to the rate equation gives the Michaelis-Menten equation:
VO =
[ ]
[ ]
At very low substrate concentration, when [S] is much less than KM, V0 = (Vmax/KM)[S]; that is,
the rate is directly proportional to the substrate concentration. At high substrate concentration,
when [S] is much greater than KM, V0 = Vmax; that is, the rate is maximal, independent of
substrate concentration.
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