Question

a) Derive a relation between partial pressure and mole fraction of gas.
b) Write the Vander Walls equation for 1 mole.

Answer 1

Solution:-

• (a) Ideal gas equation for individual gases-
  

$PV=nRT.....\left(1\right)$

Ideal gas equation for the mixture-

$P_{\text{total}}V=n_{\text{total}}RT.....\left(1\right)$

Dividing equation $\left(1\right)$ by $\left(2\right)$, we have

$\frac{PV}{P_{\text{total}}V}=\frac{nRT}{n_{\text{total}}RT}$

$\Rightarrow \frac{P}{P_{\text{total}}}=\frac{n}{n_{\text{total}}}.....\left(3\right)$

As we know that molar fraction of a gas is-

$X=\frac{n}{n_{\text{total}}}.....\left(4\right)$

From equation $\left(3\right)\&\left(4\right)$, we have

$P=X{P}_{\text{total}}\Rightarrow$ Required relation

Hence the relation between partial pressure and mole fraction of a gas is-

$P=X{P}_{\text{total}}$

• (b) Vander Wall's equation-
  

$(P+\frac{a{n}^2}{V^2})(V-nb)=nRT.....\left(1\right)$

for $1$ mole, i.e.,

$n=1$

Substituting the value of $n$ in equation $\left(1\right)$, we get

$(P+\frac{a}{V^2})(V-b)=RT$

Hene Vander Waal's equation for $1$ mole of a gas is-

$(P+\frac{a}{V^2})(V-b)=RT$