Heat Capacities

Since we have looked at both Cv and Cp already, now is a good time to compare these heat capacities.

Heat Capacity at constant Volume, Cv

The equipartition theory states that translational and rotational degrees of freedom each contribute RT/2 JK-1 to the constant volume heat capacity, whilst vibrational degrees each contribute RT JK-1.

Mode (Type) of Freedom Number possessed by a molecule
Translational 3
Rotational (a) linear
(b) non-linear
Vibrational (a) linear
(b) non-linear
3n – 5
3n – 6
Rotational and Vibrational degrees of freedom can only be accessed at higher temperatures (as these energy levels are separated by a greater amount than translational levels), resulting in a graph somewhat like this one. ie: when the system becomes hot enough, rotational degrees may be accessed and at even higher temperatures, vibrational degrees. This produces the steps on the graph – a particular mode will only contribute to the heat capacity at a temperature high enough for it to be fully excited.

The more ‘perfect’ the gas is, the more it will adhere to these guidelines. So argon shows good similarities. The more molecular interactions there are, the more the theory falls apart.

We can see then, how Cv changes with temperature. What about Cp?

Heat Capacity at constant Pressure, Cp

You’ll be relieved to hear that Cp changes with temperature in a more simple fashion.

We can model it with the equation Cp,m = a + bT + c/T2. a, b and c are constants and independent of temperature, and at the lowest temperatures Cp,m = aT3. Once again, a is a constant independent of temperature.

So, how do we use Cp?1) If we take a system and heat it up, then the enthalpy changes, that’s the definition.
\dH = CpδT (constant pressure)
So, over a changing temperature range,
H1H2dH = ∫T1T2CpdT
which reduces to
ΔH = [aT + bT2/2 – C/T]T2T12) If we know the enthalpy change for a given reaction at T1, we can estimate the enthalpy change at T2 using Cp.
ΔHT2 – ΔHT1 = ∫T1T2ΔCpdT
where ΔCp is the difference in heat capacity between reactants and products. Over limited temperature ranges you can assume that Cp is independent of temperature. This equation is known as Kirchhoff’s Law.

Finally, there is an important relationship between Cv and Cp for perfect gases:

Let us consider Cp – Cv.

Cp – Cv = (dH/dT)p – (dU/dT)p
and remembering that H = U + nRT
\ dH/dT = dU/dT + nR
\ Cp – Cv = (dU/dT)p + nR – (dU/dT)p = nR

Cp – Cv = nR