Ideal Gas Equation

At low density (caused by either high temperature or low pressure), the separation between gases is large enough that any intermolecular interactions can be ignored and the gas can be assumed to be “ideal”.

The equation of state for such a gas is called the Ideal Gas Equation. It was first stated by Emile Clapeyron in 1834 by combining Boyle’s Law, Charle’s Law, Avogadro’s Law and Gay-Lussac’s Law. For $n$ moles of an ideal gas, the equation is: $pV = nRT$ where R is the Universal Gas Constant. For, $N$ particles, however, the equation becomes $pV = NkT$ in which, $k$ is the Boltzmann constant. $R = N_A k$. $N_A$ is Avogadro’s Number. The meaning of the other terms in both the equations are as conventionally known and they have also been stated later.

It turns out, the Ideal Gas Equation can be obtained theoretically by invoking the entropy of a lattice model (described below). And this is exactly what I will do in this article.

Background

Let’s first have a quick introduction to the concepts of entropy and multiplicity for those unfamiliar. You can skip to the next section otherwise.

Concepts

Entropy, $S$ is the measure of the degree of randomness in a system. A state with higher number of degenerate arrangements, or higher multiplicity, has higher entropy. $S$ is assumed to be a function of $U$, $V$, $N$ where $V$ is the volume and $N$ is the number of particles. So, $S = S(U, V, N)$. Differentiating:

$$ dS = \left(\frac{\partial S}{\partial U}\right)_{V,N} dU + \left(\frac{\partial S}{\partial V}\right)_{U,N}dV \newline + \sum_{j=1}^M \left(\frac{\partial S}{\partial N_j}\right)_{U,V,N_{i \neq j}}dN_j $$

Now, $(\partial S/\partial U)_{V, N} = 1/T $, $ (\partial S/\partial V)_{U, N} = p/T $, and $ (\partial S/\partial N)_{U, V, N_{i \neq j} } = -\mu_j/T $ , where $T$ is the temperature, $p$ is the pressure and $\mu_j$ is the chemical potential. These are measureable properties. So,

$$ dS = \frac{1}{T}dU + \frac{p}{T}dV - \sum_{j=1}^M \frac{\mu_j}{T}dN_j $$

Multiplicity is represented by $W$. It is the count of the possible number of arrangements while satisfying specific conditions. For example, the multiplicity of a system with $M$ boxes and $N$ particles, given that each box can have only one particle, can be calculated as: $$ W = {M \choose N} = \frac{M!}{N!(M-N)!} $$

Entropy is given by $S = k\ln{(W)}$ where $k$ is the Boltzmann constant.

Stirling’s Approximation is used to approximate factorials. It is given as, $$n! \sim \sqrt{2\pi n}\left(\frac{n}{e}\right)^n$$ Inside logarithms, this yields $(1/2)\ln{(2\pi n)} + n\ln{\left(n/e\right)}$. For large enough values of $\bf{n}$, the first term can be neglected. So, $$\ln{(n!)} \sim n\ln{(n/e)}$$ $$n! \sim \left(\frac{n}{e}\right)^n$$

Model

To derive the ideal gas equation, we consider a gas with $N$ particles, encased in a container of volume V. The container has $M$ sites where the $N$ particles can go, each site accomodating only one particle at max. The position of one particle is not influenced by the other particles other than making a few potential sites unavailable for it to occupy. We also assume that $N \ll M$ which is reasonable given the density of most gases are very low.

Derivation

Let’s start by calculating the multiplicity of the model described above.

Calculating multiplicity

If $N$ particles are to be arranged in $M$ lattice sites, the multiplicity $$ W = \frac{M!}{N!(M - N)!} $$ Using Stirling’s approximation, $$ W = \frac{M^M}{N^N (M - N)^{(M - N)}} $$

Invoking entropy

Entropy $S$ is, $$S = k\ln{(W)}$$ $$\frac{S}{k} = \ln{\left(\frac{M^M}{N^N (M - N)^{(M - N)}}\right)}$$ $$ \frac{S}{k} = M\ln{(M)} - N\ln{(N)} - (M - N)\ln{(M - N)} $$

From the definition of $dS$, $$ \frac{p}{T} = \left(\frac{\partial S}{\partial V}\right) $$ $$ \frac{p}{T} = \left(\frac{\partial S}{\partial M}\right) \left(\frac{\partial V}{\partial M}\right) $$

Now, $$ \left(\frac{\partial S}{\partial M}\right) = k\left[ 1 + \ln{(M)} - 1 - \ln{(M - N)} \right] $$ $$ \left(\frac{\partial S}{\partial M}\right) = k\ln{\left(\frac{M}{M - N}\right)} $$ $$ \left(\frac{\partial S}{\partial M}\right) = - k\ln{\left(1 - \frac{N}{M}\right)} $$

Also, $(\partial M / \partial V) = 1/v_0 = (M / V)$ where $v_0$ is the volume per lattice.

Combining everything, $$ \frac{p}{T} = k \left(\frac{M}{V}\right)\ln{\left(1 - \frac{N}{M}\right)} $$

Using the expansion of $\ln{(1 - x)} = - x/1 - x^2/2 - …$ $$ \frac{p}{T} = - k \left(\frac{M}{V}\right)\left( - \frac{N}{M} - \frac{N}{2 M} - … \right) $$ For gases, $N \ll M$, so the higher order terms are negligible. Thus, $$ \frac{p}{T} = - k \left(\frac{M}{V}\right)\left( - \frac{N}{M}\right) $$ $$ \frac{p}{T} = k \left(\frac{M}{V}\right)\left(\frac{N}{M}\right) $$

Thus, we have $\boxed{pV = NkT}$ which is the ideal gas equation for $N$ molecules of gas.