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. We neglected to mention this before, now we fix it.

 

· Molar specific heat at constant volume =

cV º (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 or

dEint = ncVdT , true for all ideal gases

 

 

 

· Molar specific heat at constant pressure =

cP º (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 = P dV|P = P d(nRT/P)|P = nRdT

 

cP = cV + 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 N molecules of an ideal gas:



System......f.........E(T).................cv...........cp

m.i.g. ......3N......3NkT/2............3R/2......5R/2


d.i.g. .......5N......5NkT/2............5R/2......7R/2


p.i.g. .......6N......6NkT/2............6R/2......8R/2



EXAMPLES{in class}