Consider a system consisting of a pure liquid phase in equilibrium with a gaseous phase, composed of a mixture of the vapor of the liquid and an inert gas that is insoluble in the liquid. Suppose that the inert gas (sometimes called the foreign gas) can flow into or out of the gaseous phase, so that the total pressure can be varied at will.
(a) How many components are there, and what is the variance?
(b) Assuming the gaseous phase to be a mixture of ideal gases, show that
$$
g^{\prime \prime}=R T(\phi+\ln p)
$$
where $g^{\prime \prime}$ is the molar Gibbs function of the liquid, and $\phi$ and $p$ refer to the vapor.
(c) Suppose that a little more foreign gas is added, thus increasing the pressure from $P$ to $P+d P$, at constant temperature. Show that
$$
v^{\prime \prime} d P=R T \frac{d p}{P}
$$
where $\nu^{\prime \prime}$ is the molar volume of the liquid, which is practically constant.
(d) Integrating at constant temperature from an initial state, where there is no foreign gas, to a final state, where the total pressure is $P$ and the partial vapor pressure is $p$, show that
$$
\ln \frac{P}{P_{0}}=\frac{v^{\prime \prime}}{R T}\left(P-P_{0}\right) \quad \text { (Gibbs' equation), }
$$
where $P_{0}$ is the vapor pressure when no foreign gas is present.
(e) In the case of water at $0^{\circ} \mathrm{C}$, at which $P_{0}=4.58 \mathrm{~mm} \mathrm{Hg}$, show that, when there is sufficient air above the water to make the total pressure equal to $10 \mathrm{~atm}$, $p=4.62 \mathrm{~mm} \mathrm{Hg}$.