Entropy of Mixing

When two different ideal gases mix, entropy increases even without any heat exchange. This is called the entropy of mixing and arises purely from the increased number of accessible microstates.

Formula

For a mixture of $k$ ideal gas components:

$\Delta S_{\text{mix}} = -nR\sum_{i} x_i \ln x_i$

where $x_i = n_i / n_{\text{total}}$ is the mole fraction of species $i$ and $n = \sum_i n_i$ is the total number of moles. For two gases:

$\Delta S_{\text{mix}} = -R\!\left(n_1 \ln x_1 + n_2 \ln x_2\right)$

Since $0 < x_i < 1$, each $\ln x_i < 0$, so $\Delta S_{\text{mix}} > 0$ always — mixing is always spontaneous for ideal gases.

Maximum Mixing Entropy

$\Delta S_{\text{mix}}$ is maximised when all mole fractions are equal ($x_i = 1/k$). For two gases this means $x_1 = x_2 = 0.5$.

Boltzmann's Entropy Formula

Ludwig Boltzmann connected macroscopic entropy to the microscopic world:

$S = k_B \ln \Omega$

where $k_B = 1.381 \times 10^{-23}\,\text{J/K}$ is the Boltzmann constant and $\Omega$ is the number of microstates consistent with the macroscopic state. More microstates $\Rightarrow$ higher entropy.

Your Task

Implement the two functions below.