Application to Mixtures of Ideal Gases

It has been shown that if several phases are in equilibrium with each other, then the chemical potential of any chemical species will be the same in each phase. Therefore, if we can calculate the chemical potential in some phase we know its value in any other phase that is in equilibrium with it. A clever approach would be to determine the chemical potential in the phase where it is most simple to calculate. It is particularly simple to calculate the chemical potential in an ideal gas mixture. In this manner, the determination of chemical potential in any phase can be determined by finding the ideal gas mixture that is equilibrium with it.

We begin by applying the condition that applies for equilibrium to a a single phase mixture of ideal gases at constant pressure and temperature.

To apply equilibrium condition:

$$\input{equations/14-1A}$$ (20-4)

to an ideal gas, an expression for the Gibbs free energy associated with each component is calculated. For an ideal gas, $U$ is a function of $T$ only. And since $H = U + PV = U + NRT$, $H$ is a function of $T$ only. $\mu_i$ is the Gibbs free energy per mole of species $i$ (this will be shown to be true later), so we need to find an expression for:

$$\input{equations/molarG}$$ (20-5)

for each deal gaseous species $i$ that comprises and ideal gas mixture at constant pressure, $P$ and temperature, $T$.

To find a model for the chemical potential in an ideal gas mixture², one might imagine that the system is in contact with $C$ pistons and each of the pistons only interacts with one gas. The total pressure is the sum of the partial pressures on each of the pistons. Or, one might associate a sub-volume $V_i$ with each of the $C$ components.

For an ideal gas,

$$\input{equations/tds-idealgas}$$ (20-6)

Because the individual volumes associated with each gas species are the ``internal degrees of freedom'' which may adjust themselves so that the system attains equilibrium, an expression for the the molar entropy can be calculated for any ideal gas species occupying volume $V_i$ at some temperature $T$ by integrating that volume from some reference state.

It will be useful to introduce a particular reference state, $\ensuremath{\overline{\ensuremath{{V_i}^{\mbox{ref}}}}}$, which is the volume that a mole of isolated gas would occupy at some temperature $T$ and some reference pressure $\ensuremath{{P}^{\mbox{ref}}}$³

Therefore changing the state of each of species $i$ from its reference state to any molar volume $\ensuremath{\overline{V_i}}$ at constant temperature can be calculated by integrating Equation 20-6 along an isotherm:

$$\input{equations/S-idg-molar}$$ (20-7)

And because

$$\input{equations/ideal-gas-pv}$$ (20-8)

therefore

$$\input{equations/ideal-gas-g}$$ (20-9)

Can be written by changing variables from $V$ to $P$:

$$\input{equations/G-idg}$$ (20-10)

where in an ideal gas, the ratio of partial pressure to the total pressure $P_i/\ensuremath{{P}^{\mbox{total}}} = X_i$.

if we take $\ensuremath{{P}^{\mbox{ref}}}$ to be some convenient standard state (i.e. STP) then $P_i/\ensuremath{{P}^{\mbox{ref}}} = P_i/1\mbox{atm} \equiv \ensuremath{{P_i}^{\mbox{atm}}}$ is the partial pressure of the gaseous species $i$ with respect to one atmosphere. $\ensuremath{{P_i}^{\mbox{atm}}}$ is the numerical value of the partial pressure of the i$^{th}$ species, but it is really unitless.

Typically, a standard is adopted where all pressure measurements are assumed to be in atmospheres and thus the $\log P_i/\ensuremath{{P}^{\mbox{ref}}}$ term is written as $\log P_i$ where it is understood that $P_i$ is exactly the numerical value of the unitless $\ensuremath{{P_i}^{\mbox{atm}}}$ and there is no difficulty in carrying around units like log(atmospheres). We will adopt this standard below.

From these considerations derives an expression for the chemical potentials of ideal gases in a mixture:

$$\input{equations/mu-idg}$$ (20-11)

Compare this to $\ensuremath{\overline{G}} = \mu = \ensuremath{\overline{H}} - T \ensuremath{\overline{S}}$

$$\input{equations/ideal-entropy}$$ (20-12)

for an ideal gas mixture.