Download THERMODYNAMICS Ideal Gases. Also for gases we concentrate on

UNIVERSITY OF PERUGIA
1ST LEVEL DEGREE IN BIOTECHNOLOGIES - PHYSICS FUNDAMENTAL
THERMODYNAMICS
Ideal Gases. Also for gases we concentrate on the simple case of no viscosity (and no internal
forces among particles), thus defining what is called an ideal gas.
Various experiments, and the kinetic theory at the end of the XIX century established that all
materials (gases in particular) are formed by small particles (atoms, molecules) rapidly moving in
an empty space and continuously hitting each other, so that while their individual parameters are
essentially unknown, their average and distribution can be established by statistical methods. An
average measure of their kinetic energy (apart from constants) is provided by the Temperature
of a gas, in the absolute scale (called the Kelvin scale) that has its zero at -273.16 centigrade
degrees.
A measure of the 'degree' can be established by dividing into 100 equal parts the interval (in a
linear pipe) over which a given fluid expands, when put in contact first with melting ice, then
with boiling water (in external conditions typical of the sea level). In general, the temperature is
defined operationally by its measurement, made using the expansion properties of fluids (e.g.
alchool or Hg).
Heat. Another concept difficult to be specified unless a measurement is done is that of 'heat'. We
say we are providing heat to a system if we see its temperature increasing. After two centuries
spent in trying to identify this elusive 'fluid', Mayer in Germany and Joule in United Kingdom
established in the XIX century that it is a form of energy, and the kinetic theory of gases says that
heating a system means increasing the kinetic energy (temperature) of its particles, i.e. letting
them move faster.
Thermal capacity and specific heat. One calls 'thermal capacity' C the amount of heat (∆Q)
necessary to increase by one degree the temperature of a body. This is normally defined in some
special conditions (e.g. for water between 13.5 and 14.5 degrees Celsius). For a unit mass (e.g. 1
kg) , the thermal capacity is called specific heat, c. Hence:
C = lim
(t is here the temperature).
∆Q dQ
=
dt
∆t
c=
1 dQ
m dt
1st level degree in Biotechnologies - Physics Fundamental
Thermodynamics
Equivalence of heat and work. The equivalence between heat and work (or energy) was proven
by Joule in an experiment dissipating all the mechanical work done in a liquid into heat. Before
that, heat was already measured in calories, 1 calorie being the heat necessary to increase the
temperature of 1 g of water by 1 degree. Joule's experiments show that 4.185 calories = 1 N . m
(Newton times meter): this last unit is called Joule.
Avogadro's law and number. Avogadro established at the end of the XVIII century that equal
volumes of different gases contain a same number of particles (Avogadro's law). If we also define
a mole as the weight in grams equal to the molecular weight, it turns out that 2 g of H2 (mol.
weight = 2) contain the same number of molecules as 32 g of 02 (mol. weight 32). This number is
called Avogadro's number and is equal to 6.02.1023.
Laws of ideal gases. Boyle found that ideal gases (or real gases at sufficiently low density)
follow the law:
pV = const
when they expand or contract. This 'constant' in fact turns out to be dependent on the temperature
and proportional to it:
P V = nRT
(for n moles)
The above is called ‘equation of state’ of a perfect gas. R is called the constant of gases [R=8.317
Joule/(mole.K)]. If we refer to 1 mole of H and divide by its 'molar volume' (22.4 liters in
standard conditions) we get
p=
R
R M
k
⋅T = ⋅
T=
ρT
V
V N A mH
mH
where k = R N A = 1.38 ⋅10−16 erg/K, NA is the Avogadro’s number and ρ is the density.
Internal energy and degrees of freedom. It can be shown that the equivalence between the
concept of temperature and that of kinetic energy of molecules can be expressed quantitatively in
terms of the internal energy U, by:
U=
1
3
N A ⋅ m ⋅ v 2 = RT
2
2
which means that the average kinetic energy is
1
3
m ⋅ v 2 = kT
2
2
2
1st level degree in Biotechnologies - Physics Fundamental
One also says that the internal energy is
Thermodynamics
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kT for each 'degree of freedom'.
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State variables and conservation of energy. Saying that heat is a form of energy allows to
express a more general form of conservation of energy. Even in presence of viscosity the part of
energy transformed into heat remains mechanical energy, only it is 'confused', passing at the
microscopic level where particle motions are chaotic. Energy is thus preserved, but degraded at a
level less usable to produce new work. The heat necessary to produce a certain work cannot
therefore be uniquely specified, as it depends on how it is given: transformation of it into work
can be more or less efficient depending, on conditions. For a given transformation of a gas (e,g.
an expansion) one cannot (in general!) uniquely reconstruct which heating was necessary to
obtain the result. One says that 'heat is NOT a state variable'.
Contrary to heat, the internal energy of a system is specified perfectly (e.g. by its temperature)
and is unique: it is a state variable. When a system varies the final internal energy is not
dependent on the particular way in which the final state was obtained (see analogy with
conservative force fields!!).
Principles of Thermodynamics
First principle (or conservation of energy):
dU = δQ - pdV
Giving a heat amount to a system we vary the internal (kinetic) energy and/or we perform a
macroscopic work (in this case an expansion by a volume dV). The symbol δ is used instead of 'd'
to remember that Q is not a state variable (so that we cannot construct 'exact' differential
quantities). This principle also states that one cannot produce work without consuming energy
(impossibility of the first specie of perpetuum motion).
Second principle (first form)
i) 'It is impossible to make a thermodynamic transformation whose only result is converting into
work heat coming from a single source' (impossibility of the second specie of perpetuum
motion).
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1st level degree in Biotechnologies - Physics Fundamental
Thermodynamics
Gas transformations
Isothermal We already know that when a gas expends or contracts at constant T it follows the
law pV = RT , with T = const .
Adiabatic If the expansion or contraction occurs without energy (heat) exchanges with the
environment it is called adiabatic. One can prove that for adiabatic transformations:
γ
pV = const
or
Tp
−
γ −1
γ
= const
Here γ is the ratio between the specific heat at constant pressure and that at constant volume:
cV =
3
R;
2
5
R;
2
cp =
cp
cV
=
5
=γ
3
Alternative forms of the second principle
ii) 'It is impossible to perform a spontaneous transformation whose unique result is to let heat
pass from a cooler to a warmer body'.
Entropy. The quantity Q/T of a system that acquires or gives energy (heat) at a temperature T is
called Entropy. Entropy is a state variable and its variation between two states depends only on
the initial and final conditions. This is expressed mathematically by:
dS = δQ/T
For a cyclic ideal transformation
v∫
δQ
T
=0
Using the entropy concept, another form of the second principle is:
iii) 'In real transformations the entropy is always increasing'.
An increase means that the Q increases more than T or that T decreases: in both cases energy is
degraded at a lower efficiency. We know that the Universe as a whole is expanding and cooling:
its entropy is increasing. Entropy is also a measure of the disorder in a system and its increase
means a decrease of information and an increase of chaos.
Thermal machines and Carnot’s cycle
Whenever we use heat to produce work we say we are using a thermal machine. Original thermal
engines, like a steam train, are very inefficient. They take heat from a cauldron, use it to produce
steam from water (thus dissipating a lot of heat on containers, pipes etc..) and then eject both
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1st level degree in Biotechnologies - Physics Fundamental
Thermodynamics
steam itself through valves and hot exhaust gases of the combustion.
The French physicist Carnot defined the ideal thermal machine, having the highest possible
efficiency: it is an ideal case not physically possible, though simulations approaching the
theoretical limit in efficiency have been invented.
The ideal machine would exchange heat with two isothermal sources (taking Q1 from the hotter
one at T=T1 and leaking Q2 at temperature T2 to the cooler one). then all other passages are
adiabatic: an adiabatic expansion from T1 to T2 and an adiabatic compression from T2 to T1. the
system performs therefore a cycle whose typical representation in the p,V plane is a closed curve.
The efficiency of a thermal machine exchanging heat with two sources is always:
η=
Q1 − Q2
Q1
η=
T1 − T2
T1
For an ideal (Carnot’s) machine it is also:
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