Download First Law of Thermodynamics

Concept of Internal Energy, U
Internal energy is the sum of the kinetic and potential
energies of the particles that make up the system.
First Law of Thermodynamics
• translational energy of the molecules.
• energy in the form of molecular vibrations and rotations.
• potential energy of the constituents of the system due to
the environmental effects (intra).
• energy stored in the form of chemical bonds that
can be released through a chemical reaction.
• potential energy of interaction between molecules (inter).
Chapter 2
Conservation of Energy
All types of energy of molecules, except translational
energy are quantized.
Translational
whole atom or molecule changes its location in
three dimensional space
Rotational
Motion of
whole
molecule
whole molecule spins around an axis in three
dimensional space
Vibrational
At molecular level, contributors to the internal energy, U
are;
 Motion within molecule
motion that changes the shape of the molecule – stretching,
bending, and rotation of bonds
KE 
1
m v2
2
The first law of thermodynamics
Changing U;
The first law of thermodynamics states that energy can be
neither created nor be destroyed, if the energies of both the
system and the surroundings are taken into account.
In a closed system in which no chemical reactions or phase
changes occur; heat, work, or a combination of both is the
means to change U.
U Total  U sys  U Surr
U sys  U Surr
In such a system the flow of heat, q, and/or work, w,
across the boundary between the system and surroundings
during a process which change U of the system ;
The internal energy U of an isolated system is constant, i.e.
Usys = 0.
Heat into the system, q > 0
Work done on the system, w > 0
Simple processes:
Work: w (isothermal process) on the system (+ve):
The simplest processes; one of P, V or T remains constant.
xf
w
A constant temperature process is referred to as isothermal.
A constant pressure process is referred to as isobaric.
A constant temperature process is referred to as isochoric.
 F  dx
xi
xf
Vf
xi
Vi
w    Pext Adx    Pext dV
Vf
Work: mechanical work (P-V) w :
P-V work is the energy transfer across the boundary
between the system and the surroundings due to a force
acting through a distance (from expansions (-ve) and
contractions (+ve) of system volume).
Work is performed by other ways too, e.g. electrical,
sonar,..
w  
Vi
nRT
dV
V
xi
xf
x=0
Vf
1
dV
V
Vi
w   nRT 
w   nRT ln
Vf
Vi
Vf
w    PdV
Vi
w  0, on system-contraction
w  0, by system-expansion
Heat:
Molecular Level Perspective;
Heat is the energy transfer across the boundary between
the system and the surroundings (flows from high
temperature to low temperature) due the temperature
difference between them.
In the final analysis matter has to be viewed in terms of
their existence of entities as atoms, molecules,
macromolecules…
Surroundings:
At molecular level the energy an entity can acquire is
quantized. Allowed energies of entities are well defined, 1,
2, 3, 4, …
Everything other than the system is surroundings.
As a practical matter for a given situation only the
immediate region that can interact with the system is the
(effective) surrounding.
The relative probability of a molecule in allowed
energy states 1 and 2 (> 1 ) is given by;
2 > 1
n2
e
n1
 (  2  1 )
kbT
as ( 2  1 )  ,
All the molecules in a system do not have the same energy.
There is a distribution of energies among them at a given
temperature.
(2 - 1)=Const.
n2

n1
n2
e
n1
 (  2  1 )
kbT
as T  ,
n2

n1
T = const.
Smaller energy gaps
Bigger energy gaps
Low T
High T
Heat Capacity, C
Flow of heat in/out of matter (system) results in a
temperature change of the matter in the system. The
amount of heat (strictly speaking energy) required C, to
change the temperature is defined as the heat capacity
q
C  lim
T  0 T  T
f
i
C
dq
dT
Heat capacity depends on the experimental conditions as
well. Constant P heat capacity, CP, is different from
constant V heat capacity, CV.
Molecular view of C
Heat exchanging with matter changes the temperature,
and therefore the populations in the energy levels change.
The energy levels in a molecule in general is the sum of
different (energetic degrees of freedom) types of energies;
Etot = Etr + Erot + Evib + Eelec + …
K-1.
C depends on the material - SI unit; J
per mole of material, Cm - SI unit; J K-1 mol-1.
Energy levels – comparison
Molecules can gain/lose energy from.to other molecules via
molecular collisions.
Equipartition Theorem
The law of equipartition of energy states: that each
quadratic term in the classical expression for the energy
contributes ½RT (J mol-1 K-1) the average energy.
For instance, the translational motion of an atom or
molecule has three degrees of freedom (number of ways
of absorbing energy), corresponding to the x, y and z
components of its momentum.
# degrees of freedom for each energy types: for an entity
with n atoms, Tr = 3, Rot. = 3 or 2, Vib. = (3n - 6) or (3n – 5).
Each degree of freedom has its own energy levels.
Since these components of momenta appear quadratically
in the kinetic energy, every atom has an average kinetic
energy of (3/2) RT in thermal equilibrium.
http://chemwiki.ucdavis.edu/Physical_Chemistry/Statistical_Mechanics/Equipartition_Theorem
The contribution to heat capacity CVm for a gas at a
temperature of T not much lower than 300 K is R/2 for
each translation and rotational degree of freedom, where
R is the ideal gas constant.
Each vibrational degree of freedom for which the relation
E/kT < 0.1 (is active) contributes R to CVm.
If E/kT > 10 (is inactive) such degree of freedom does
not contribute to CVm.
For 10 > E/kT > 0.1, the degree of freedom contributes
partially to CVm.
Each active degree of freedom contribute to the heat capacity.
CVm for gases :
Monoatomic gases:
Polyatomic gases with rotations active: Etot = Etr + Erot
CVm = 3(R/2)+2(R/2) - linear
CVm = 3(R/2)+3(R/2) - non linear
Polyatomic gases with rotations and vibrations active
(upper limit):
2 quadratics terms !!
Etot = Etr + Erot +Evib
CVm = 3(R/2)+2(R/2)+2(3n-5)(R/2) - linear
CVm = 3(R/2)+3(R/2)+ 2(3n-6)(R/2) - non linear
CV 
Calculation of qV;
300K
CVm = 3(R/2) J mol-1 K-1
Etot = Etr
dqV
dT
Tsys , f
 qV 

CV dT
Tsys ,i
Heat for constant pressure processes (gases):
22.78 + 4 normal modes
22.78 + 1 normal mode
12.47
Under constant pressure changing temperature would
change volume of the gas. This involves pushing or
pulling of ‘surrounding’, thus involves mechanical work.
The energy (heat) required to change the temperature, CPm,
for constant pressure processes are different from CVm.
Tsys , f
qP 

Tsys ,i
Tsur , f
CPsys dT  

Tsur ,i
CPsur dT
For gases CP > CV because energy is needed to effect
volume changes.
CP  CV  nR
i.e. CPm  CVm  R
:Borrowed Chapter 3
Tsys , f
qP 

Tsys ,i
Tsur , f
CPsys dT  

CPsur dT
Tsur ,i
A state function describes the current state of a system.
State functions and path functions:
U  q  w
dU   q   w
state
path
Any quantity that does not depend on the path it takes to
move from stage 1 to stage 2 is a state function. All
thermodynamic quantities are state functions. State of a
single phase of fixed composition is characterized by any
two of P, V and T.
Any quantity that depend on the path it takes to move from
stage 1 to stage 2 is a path function; q and w are
path functions.
The overall change for a cyclic process of a state function
is zero.
f
How the system came to be in that particular state is of no
consequence.
The following are state functions:
Pressure, P
Volume, V
Temperature, T
Mass, m
Quantity, n
Internal Energy, U
Enthalpy, H
Entropy, S
Gibbs Energy, G
U   dU  U f  U i
i
 dU  0
:cyclic path
Thermodynamics applies to systems in internal equilibrium.
Reversible process:
It implies any change done must be performed giving
sufficient time to achieve equilibrium internally and with the
surroundings; rate of change - slow – very slow;
quasi-static process.
Therefore the no gradients, currents or Eddys exist.
To prevent all in-homogeneities, a reversible process must
be carried out infinitely slowly.
We deal with quasi-static processes which are reversible.
Reversible process: A reversible process is a process where
the effects of following a thermodynamic path can be
undone be exactly reversing the path. It is a process that is
always at equilibrium even when undergoing a change.
Ideally the composition throughout the system must be
homogeneous.
A truly reversible processes is non-existent. However,
many systems are approximately reversible. Assuming
reversible processes facilitates calculations of various
thermodynamic state functions.
Fact: Maximum work is achieved from the system during a
reversible expansion and vice versa.
Pi ,Vi ,Ti
P-V isotherms do not cross.
P
nRT
V
:isothermal process
Pf ,Vf ,Tf
All points on the surface correspond to all possible
P, V and T values (equilibrium state) of 1mol of an ideal gas.
Reversible P-V changes of an ideal gas (isothermal);
Reversible P-V changes of an ideal gas (isothermal);
Pi ,Vi ,T
Pi ,Vi ,T
PV  nRT  constant :isothermal process
PV  nRT  constant :isothermal process
Vf
Vf
Vi
Vi
w    Pext dV   
Pf ,Vf ,T
Pf ,Vf ,T
w   nRT ln
Vf
Vi
wexp ansion   nRT ln
indicator diagram
nRT
dV
V
V2
V1
indicator diagram
Maximum work is involved during a reversible expansion (or
compression).
w  P  V indicator diagram plot area
For the irreversible process, P-V work
For the cyclic reversible process(isothermal);
P1 ,V1 ,T
w   Pext V
PV  nRT  constant :isothermal process
Reversible process follows the IGE.
Vf
P2 ,V2 ,T
Vf
w    Pext dV   
Vi
w   nRT ln
Vi
Vf
nRT
1
dV   nRT  dV
V
V
Vi
w  P  V plot area
P1 ,V1 ,T
w1
w    Pext V
w2
P2 ,V2 ,T
steps
w1  0
w2   P2 (V2  V1 )
Vf
Vi
wexp ansion   nRT ln
V2
V1
w   Pext V
w  P  V indicator diagram plot area
wtotal  0; qtotal  0
For the cyclic irreversible process, P-V work
w   Pext V
P1 ,V1 ,T
w1  0
w  P  V plot area
w3  0
P2 ,V2 ,T
w    Pext V
cycle
w2
U  qtotal  wtotal  0
w4
:U function of state
 qtotal   wtotal ; qtotal  0 :because wtotal  0
Example Problem 2.4 p.32
For P-V work wirrev < wrev.
4.50L, 25.0bar
Use Boyle’s Law !!
11.25L, 4.5bar
25.0L, 4.5bar
Use Boyle’s Law, easier !!
The magnitude of the work is greater for the two-step process than
for the single-step process, but less than that for the reversible
process.
For P-V work |wirrev|< |wrev|.
Compression
Expansion
2
2
5
5
10
10
Smaller steps (slower) process closer to reversible.
Maximum work is involved during a reversible expansion (or
compression).
Smaller steps (slow) process closer to reversible.
Enthalpy:
Determination of U
U  q  w  q   Pexternal dV
if dV  0;
Processes performed under constant pressure dU would be;
dU  dqP  Pexternal dV
U  q  qV
U  qV
 dU  q   P dV  q
P
P
 P  dV
U f  U i  qP  Pf (V f  Vi )
Perform the process under constant volume, and
measure the heat flow, qV associated.
and Pf  Pi
qP  U f  U i  Pf V f  PV
i i
H  U  PV
Enthalpy
q P  H f  H i  H
H  q P
Calculation of q, w, U and H for ideal gases:
Needed: equation of state, initial state, final state and
the path taken.
U  qV   Cv dT
Isothermal process;
H  U  0
U function of T
Adiabatic process:
H  U (T )   ( PV )  U (T )   (nRT ) H function of T
 q P  C P T
H  q P  C P  T
w    Pexternal dV   
Note:
nRT
dV
V
q=0
P-V work, constant volume:
definition
w=0
Reversible Adiabatic Expansion/Compression: Ideal Gas
Consider the adiabatic expansion of an ideal gas. Because
q = 0. the first law takes the form;
CV ln
CV ln
U  w    Pexternal dV   CV dT
dV
CV dT   nRT
V
dT
dV
CV
  nR
T
V
Tf
V
 CV ln
  nR ln f
Ti
Vi
Two systems containing 1 mol
of N2 have the same P and V
values at 1 atm.
Isothermal:
P >1  q < 0 to keep T const.
P <1  q > 0
Adiabatic: q = 0
P >1  T increase, Pad>Piso
P <1  T decrease, Pad<Piso
PV   const
PV  const :isothermal
ln
Tf
Ti
Tf
Ti
Tf
Ti
  nR ln
Vf
ln
Vi
 (CP  CV ) ln

(CP  CV )
ln
CV
Vi
Vf
 (  1) ln
where  
Vf
Vi
CP
CV
Vf
Vi
 Vf 
 ln  
Ti
 Vi 
Tf
 Vf 
 
Ti  Vi 
Tf
Tf
Ti

(1 )
(1 )
 Vf 
 
PV
i i
 Vi 
Pf V f
(1 )
Pf V f  Vi 
 
PV
i i  Vf 

Pf V f   PV
i i
1
(1 )