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Chemistry LibreTexts

24.6: Vapor Pressures of Volatile Binary Solutions

Gibbs-Duhem and Henry's law

What happens is Raoult does not hold over the whole range? Recall that in a gas:

\[μ_j = μ_j^o + RT \ln  \dfrac{P_j}{P^o} \label{B}\]

or

\[μ_j = μ_j^o + RT \ln P_j \]

after dropping \(P^o=1\; bar\) out of the notation. Note that numerically this does not matter, since \(P_j\) is now assumed to be dimensionless.

Let's consider \(dμ_1\) at constant temperature:

\[dμ_1 = RT\left(\dfrac{\partial \ln P_1}{ \partial x_1}\right)dx_1\]

likewise:

\[dμ_2 = RT\left(\dfrac{\partial \ln P_2}{ \partial x_2}\right)dx_2\]

If we substitute into the Gibbs-Duhem expression we get:

\[x_1 \left(\dfrac{∂\ln P_1}{ ∂x_1}\right) dx_1+x_2 \left(\dfrac{∂\ln P_2}{∂x_2} \right) dx_2=0\]

Because \(dx_1= -dx_2\):

\[x_ 1 \left( \dfrac{∂\ln P_1}{ ∂x_1} \right) =x_2 \left( \dfrac{∂\ln P_2}{∂x_2} \right)\]

(This is an alternative way of writing Gibbs-Duhem).

If in the limit for \(x_1 \rightarrow 1\) Raoult Law holds then

\[P_1 \rightarrow x_1P^*_1\]

Thus:

\[ \dfrac{∂ \ln P_1}{∂x_1} = \dfrac{1}{x_1}\]

and

\[\dfrac{x_1}{x_1}=x_2 \dfrac{∂ \ln P_2}{∂x_2}\]

\[1=x_2 \dfrac{∂ \ln P_2}{ ∂x_2} \]

\[\dfrac{1}{x_2}= \dfrac{∂ \ln P_2}{∂x_2} \label{EqA12}\]

We can integrate Equation \(\ref{EqA12}\) to form a logarithmic impression, but it will have an integration constant:

\[\ln P_2 =\ln x_2 + constant\]

This constant of integration can be folded into the logarithm as a multiplicative constant, \(K\)

\[\ln P_2 = \ln \left(K x_2 \right)\]

So for \(x_1 \rightarrow 1\) (i.e., \(x_2 \rightarrow 0\)), we get that

\[P_2=K x_2\]

where \(K\) is some constant, but not necessarily \(P^*\). What this shows is that when one component follows Raoult the other must follow Henry and vice versa. (Note that the ideal case is a subset of this case, in that the value of \(K\) then becomes \(P^*\) and the linearity must hold over the whole range.)

Margules Functions

Of course a big drawback of the Henry law is that it only describes what happens at the two extremes of the phase diagram and not in the middle. In cases of moderate non-ideality, it is possible to describe the whole range (at least in good approximation) using a Margules function:

\[P_1= \left(x_1P^*_1 \right)f_{Mar}\]

The function \(f_{Mar}\) has the shape:

\[f_{Mar}= \text{exp} \left[ αx_2^2+βx_2^3+δx_2^3 + .... \right]\]

Notice that the Margules function involves the mole fraction of the opposite component. It is an exponential with a series expansion. with the constant and linear term missing. As you can see the function has a number of parameters \(α\), \(β\), \(δ\) etc. that need to be determined by experiment. In general, the more the system diverges from ideality, the more parameters you need. Using Gibbs-Duhem is is possible to translate the expression for \(P_1\) into the corresponding one for \(P_2\).