Specific heat of ideal gases and the
equipartition theorem
Specific heats revisited
The specific heat of a material will be different depending
on whether the measurement is made at constant volume or
constant pressure.
 Molar specific heat at constant volume
cV  (1/n) dQ/dT|V
But if dV = 0 then dW = 0 and by the first law of
thermodynamics,
dEint = dQ ,

cV = (1/n) Eint/T |V
true for all materials.
Specializing to ideal gases, we know Eint(T)
cV = (1/n) dEint/dT

dEint = ncVdT
or
true for all ideal gases.
 Molar specific heat at constant pressure
cP  (1/n) dQ/dT|P
and using the first law,
cP = (1/n) [dEint+dW]/dT|P = (1/n)dEint/dT|P + (1/n)dW/dT|P
For an ideal gas, Eint(T) and
dW|P = d(PV)|P = P dV|P = P d(nRT/P)|P = nRdT

cP = c V + R
true for all ideal gases.
Equipartition Theorem
Each degree of freedom of a system has an energy ½ kBT.
A degree of freedom is an independent mode of motion:
translation, rotation, and vibration. Let the number of
degrees of freedom be denoted f. Neglecting vibration,
for ideal gases:
N=nNA and R=NAkB
System
1/ndE/dT
cV
cV+R
cP
f
Eint(T)
3N
(trans.)
3N ½ kBT
3/2 nRT
3/2 R
5/2 R
d.i.g.
5N
5N ½ kBT
dumbbell(trans.+rot.) 5/2 nRT
5/2 R
7/2 R
p.i.g.
6/2 R
4R
m.i.g.
6N
6N ½ kBT
(trans.+rot.) 6/2 nRT
Ideal Gas Toolkit
Ideal gas law
PV = nRT
First Law of Thermo.
dEint = dQin - dWout
Processes
isothermal, adiabatic, etc.
Work by gas
Wout = ∫p dV
Internal energy of gas
Eint(T) = f ½ kBT
EXAMPLES[in class]