# Theoretical Calculation of Specific Heat of a Gas

I've read that the theoretical specific heat of a monatomic gas (like dissociated hydrogen or oxygen) is $20.8\, \mathrm{\dfrac{J}{mol\cdot K}}$ at constant pressure and $12.5\, \mathrm{\dfrac{J}{mol\cdot K}}$ at constant volume.

1. Is there a way to calculate theoretical values for the specific heats of polyatomic gases?
2. Is there a way to calculate the temperature dependence of specific heats for monatomic species?

I don't know much about quantum mechanics, but maybe this is one of the "tricks" you can use it for? The specific heats of everything else I've seen (experimental data) depends on temperature, so I assume monatomic gases are the same. I think the theoretical values I have are for $298\,\textrm{K}$, $1$ atm.

I'm working on some code that simulates the elementary reactions going on in the combustion of several fuels and in order to accurately calculate the Gibbs free energy change to determine whether or not each reaction will occur, I need specific heats that are as accurate as possible, especially in the 2000-4000K range. It's hard to find experimental data for less common species like OH, HO2, etc.

• In the energy range you're asking about the vibrational modes of atoms contribute substantially to the heat capacity, and makes the calculation much less trivial than it can be for simple molecules at low temperatures. – dmckee --- ex-moderator kitten Jul 19 '16 at 20:55

## 3 Answers

Leaving this here for now... Will update with more information and references later.

Einstein and Debye showed that specific heat is a function of temperature, but is asymptotic at high* temperatures. Here is a simple explanation why:

Heat, with regard to everyday applications, is simply a measure of the motion of atoms and molecules. Let's start with gases. Classical theory tells us there are three types of motion for an atom: translational, rotational, and vibrational. Here's where quantum theory comes in. Quantum Mechanics tells us that there is a "temperature" threshold where each of these types of motion can begin to occur. We know from thermodynamics that there is a distribution of temperatures with regard to the atoms in a gas, so some atoms may have breached this temperature threshold and are in motion in more ways than others--let's call this having more degrees of freedom. This is where the temperature depends comes in. At low temperatures, there is a significant variance in degrees of freedom between atoms, and atoms continuously climb and fall below these thresholds. However, at high temperatures, most atoms have attained enough energy to gain all possible degrees of freedom. Therefore, there is little temperature dependence at high temperature for gases. In solids and liquids, magnetism must be considered, so they can exhibit different behavior at high temperatures.

The above discussion completely ignores pressure and volume, which are far more likely to have an impact on your calculations than temperature. Classical theory says polyatomic gases should have a constant specific heat at high temperatures, but I doubt it is that simple. I will do some reading on these issues and get back to you.

*Varies depending on the compound. Most gases have stable specific heats around room temperature, but behavior can vary wildly. See Einstein Temperature or Debye Temperature, specifically for Diamond.

In the temperature range you are talking about, and assuming we are talking about pressure that are not close to vacuum, then a monotomic gas (and I'd prefer talking about Ar, or He, and not a monotomic O, or H)

Cp/Cv=5/3 (billiard ball atoms - no vibration/rotation) Cp-Cv =R (R is gas constant, and there is a universal gas constant) and from these, I believe you can get Cp and Cv (you also need the molecular weight to get the specific R for the gas you are interested in).

For diatomic molecules, and below the point where vibration rears its wobbly head, Cp/Cv=7/5, and again Cp-Cv =R.

The temperature range you are talking about is, however, where the vibration will start to occur, and then we need experiment, but you should be able to find a lot on the NIST site. At least that used to be the case.

For references, see Vincenti and Kruger "Introduction to Physical Gas Dynamics".

However: you are looking for Gibb's free energy.You need a lot more than Cp and Cv, and again, I'd look to NIST. At this point you need entropy in an absolute sense.

The way you calculate (classically) the heat capacities for gases is by comparing the expressions for the internal energy given by thermodynamics and kinetic theory.

The Equipartition Theorem says that at thermal equilibrium at temperature $T$ each quadratic term in the (mechanical) energy of the molecule contributes with $kT/2$. For a gas with $N$ molecules whose each of them has $q$ quadratic terms in its energy the total internal energy is $$U=Nq\frac{kT}{2},$$ which implies (for a closed system) $$dU=Nq\frac{kdT}{2}.$$

On the other hand, thermodynamics empirically have the relation $$dU=nC_vdT,$$ where $C_v$ is the molar heat capacity at constant volume. Comparing the equations above and using $nR=Nk$ one gets $$C_v=\frac{qR}{2}.$$

For a monoatomic gas, $q=3$, corresponding to the kinetic energy $\frac{1}{2}(v_x^2+v_y^2+v_z^2)$, and therefore $$C_v=\frac{3R}{2}\approx 12.47\, \mathrm{\frac{J}{mol\cdot K}}.$$ The heat capacity at constant pressure is simply $C_p=C_v+R$, which for the monoatomic gas gives $$C_p\approx 20.79\, \mathrm{\frac{J}{mol\cdot K}}.$$