HEAT CAPACITY
When heat flows into a system, provided no phase change occurs, its temperature will
increase. The increment dT by which T increases is proportional to the amount of heat flow
dq.
 dq 
The ratio 
 is called the heat capacity [Units J/K].
 dT 
Definition: The heat capacity (denoted by C) is the increment in heat dq required to
increase the temperature by an amount dT.
C is an extensive quantity but molar heat capacity Cm = C/n is an intensive quantity
(property). Usually subscript m is omitted. The quoted units then define whether C or Cm is
intended. The term specific heat capacity, i.e. heat capacity per gram, is also used.
The value of the heat capacity however depends on the circumstances under which the
system is heated. So in order to have a defined heat capacity one must specify the
condition.
1). Isochoric (constant volume), heat capacity is abbreviated as “CV”. If heat is supplied at
constant volume (dV = 0), the system can do no PV work (assuming the system does no
other kind of work). Then, from first law of thermodynamics
dq V
 dU
 q 
 U 
By definition  CV     

 T V  T V
dU
 U 
From an ideal gas 
 
 T V dT
CV 
dU
 dU  CV dT
dT
2). Isobaric (constant pressure) heat capacity is abbreviated as “Cp”. If the heat is supplied
at constant pressure, usually the sample expands doing work against the external
pressure. Thus in addition to raising the temperature some extra heat is required for the
expansion work. Hence, more heat will be required to raise the system temperature
under constant pressure condition than under constant volume condition. Thus CP > CV
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From 1st law of thermodynamics dU = dq + dw and dw= -PdV, then
dU  dq  P dV
dq  dU  P dV
dq U  P dV  H 

CP  p  


  
dT 
T
P T P
For an ideal gas:
dH
 H 

 
 T  P dT
 CP 
dH
 dH  C P dT
dT
In order to obtain U when an ideal gas is heated from temperature T1 to T2 at constant
volume, the integrated form of dU  CV dT is to be used.
T2
U   CV dT
where CV is the heat capacity at constant volume.
T1
Similarly the enthalpy change H when an ideal gas is heated from temperature T1 to T2 at
T2
constant pressure is given by H   C P dT .
T1
where CP is the heat capacity at constant pressure.
Note if CV and Cp are given as molar heat capacities then;
T2
T2
T1
T1

U   nCV dT and H  nC P dT .
For an ideal gas it can be shown that Cp – Cv = R
HEAT CAPACITIES OF GASES
The heat capacity of a gas at constant pressure Cp exceeds the heat capacity of that gas at
constant volume, Cv, because the gas performs work on its surroundings as it expands at
constant pressure. If the volume is held constant such expansion work by the gas is
impossible.
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From the kinetic molecular theory of gases, the energy, U, of one mol of ideal monatomic
gas as a function of temperature is given as:
U  32 RT
U 
Therefore Cv     32 R
T v
and Cp = Cv, + R =
5
2
R since, the enthalpy of an ideal gas is greater than its internal energy
by PV (= RT). So that H =
3
2 RT
5
+ RT = 2 RT
H 
C p     52 R
T p
The heat capacity of a perfect gas is independent of temperature.
Predicting Heat Capacities of gases: The equipartition theorem
We can predict the heat capacities of monoatomic, diatomic or polyatomic gases based on
equipartition theorem, simply by counting the degrees of freedom. The mean energy per
degree of freedom per mol of monatomic ideal gas is
1
2
RT. Note however that the
classical equipartition theorem holds in the high-temperature limit.
The internal energy of 1 mol of monatomic ideal gas is
possess, on average,
3
2 k BT
3
2 RT.
This means that each particle
units of energy. Monatomic particles have only three
translational degrees of freedom, corresponding to their motion in three dimensions. They
possess no internal rotational or vibrational degrees of freedom. Thus, the mean energy per
degree of freedom per mol of monatomic ideal gas is
1
2
RT.
For rigid (non-vibrating) linear molecules and rigid diatomic molecules, the internal energy
U=
3
2 RT
+ RT =
5
2 RT;
where RT is the contribution of rotational degrees of freedom
(rotational energy).
U 
Therefore giving Cv     52 R for rigid diatomic molecules.
T v
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P = linear momentum = mv
J = angular momentum = Iω
I = moment of Inertia = Σmiri2
ω = angular velocity
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Real Gases: Temperature Dependence of Heat Capacities:
In figure 1 the dependence of constant pressure molar heat capacity, Cp, on temperature is
shown for a number of gases. In general the more complex the molecule, the greater is its
molar heat capacity and the greater is the increase of heat capacity with rising temperature.
It has been customary to express the molar heat capacity of a gas empirically as a power
series in the temperature, either as:
CP    T  T 2  ...
or
CP  a  bT  cT 2  ...
The value of constant volume molar heat capacity, Cv, can then be found from Cp at any
temperature by using the relation: Cp = Cv + R.
C2H6
110
CP,mean / J mol-1 K-1
90
C2H4
SO3
70
C2H2
SO2
50
H2S
Cl2
NH3
HCl
30
300
500
700
900
1100
1300
1500
T/K
Fig. 1: Variation of molar heat capacities with temperature for some gases.
Example
1. One mol of methane gas originally at 298 K and 1 atm is heated at constant pressure
until the volume is doubled. Assuming the gas behaves ideally, calculate U and H
for the process. The molar heat capacity of methane at constant pressure is Cp = 22.34 +
48.1 x 10-3T J mol-1 K-1.
Answers:
10587 and 13065 J.
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