Question 15.7.3: The mixing of gases is always accompanied by an increase in ......

For a binary mixture the entropy per mole of the mixture formed is given by

For entropy of mixing to be maximum, the first derivative \frac{\delta\left(\Delta \mathrm{S}_{\text {mixing }}\right)}{\delta \mathrm{X}_{1}} should be zero and the second derivative should be negative. Differentiating \Delta S_{\text {mixing }} with respect to X_1 we get

or   \ln \frac{\mathrm{X}_{1}}{1-\mathrm{X}_{1}}=0 \quad  \text{or} \quad \mathrm{X}_{1}=1-\mathrm{X}_{1}\\

And hence \mathrm{X}_{2}=1-0.5=0.5.