Up to this point, we have focused primarily on systems whose composition remains constant. In many real systems, however, substances are mixed together, causing the composition of the system to change.
To understand the thermodynamics of mixtures, consider two ideal gases initially separated by a partition.
Both gases are at the same temperature and pressure. When the partition is removed, the gases spontaneously mix until each occupies the entire volume of the container.
Since the gases mix without any external intervention, we expect the process to be spontaneous.
Assume that the gases behave ideally so that interactions between molecules are unimportant.
Each gas undergoes an isothermal expansion into the volume previously occupied by the other gas.
For an ideal gas, the enthalpy depends only on temperature:
\[ \Delta H = nC_P\Delta T \]
Because the mixing process is isothermal,
\[ \Delta T = 0 \]
and therefore
\[ \boxed{ \Delta H_{\mathrm{mix}} = 0 } \]
This result is characteristic of an ideal mixture. No energy is released or absorbed during mixing because the interactions between unlike molecules are assumed to be the same as those between like molecules.
The entropy change can be calculated by viewing the process as two separate isothermal expansions.
For component \(i\),
\[ \Delta S_i = n_iR \ln\left( \frac{V_f}{V_i} \right) \]
Summing the contributions of all components yields
\[ \Delta S_{\mathrm{mix}} = -R \sum_i n_i \ln x_i \]
where \(x_i\) is the mole fraction of component \(i\).
Since
\[ 0 < x_i \le 1 \]
it follows that
\[ \ln x_i < 0 \]
for all components present in the mixture.
Therefore,
\[ \boxed{ \Delta S_{\mathrm{mix}} > 0 } \]
Mixing increases entropy because the molecules become distributed over a larger number of possible arrangements.
At constant temperature and pressure,
\[ \Delta G = \Delta H - T\Delta S \]
For an ideal mixture,
\[ \Delta H_{\mathrm{mix}} = 0 \]
so
\[ \Delta G_{\mathrm{mix}} = -T\Delta S_{\mathrm{mix}} \]
or
\[ \boxed{ \Delta G_{\mathrm{mix}} = RT \sum_i n_i \ln x_i } \]
Since \(\ln x_i\) is always negative,
\[ \boxed{ \Delta G_{\mathrm{mix}} < 0 } \]
for all ideal mixtures.
The negative Gibbs function change confirms that mixing is a spontaneous process.
The thermodynamics of ideal mixing are driven entirely by entropy.
Because there is no enthalpy change,
\[ \Delta H_{\mathrm{mix}} = 0 \]
the spontaneity of mixing comes solely from the increase in entropy caused by distributing the molecules among a larger number of possible arrangements.
Real mixtures may deviate from this behavior because intermolecular interactions can make mixing either more favorable or less favorable than predicted by the ideal model.
Big picture: For an ideal mixture, mixing produces no enthalpy change, increases entropy, and lowers the Gibbs function. The tendency of gases to mix spontaneously is therefore a direct consequence of the Second Law of Thermodynamics.
Consider \(1.00\ \mathrm{mol}\) of gas A at \(1.00\ \mathrm{atm}\) mixing with \(2.00\ \mathrm{mol}\) of gas B at \(2.00\ \mathrm{atm}\). Both gases are initially at \(298\ \mathrm{K}\). Calculate the initial volumes, the final volume after mixing, \(\Delta H_{\mathrm{mix}}\), \(\Delta S_{\mathrm{mix}}\), and \(\Delta G_{\mathrm{mix}}\).
\[ V_A=\frac{n_ART}{p_A} = \frac{(1.00\ mol)(0.08206\frac{atm\ L}{mol\ K})(298\ K)}{1.00\ atm} = 24.\underline{4}54\ \mathrm{L} \]
\[ V_B=\frac{n_BRT}{p_B} = \frac{(2.00\ mol)(0.08206\frac{atm\ L}{mol| K})(298\ K)}{2.00\ atm} = 24.\underline{4}54\ \mathrm{L} \]
Therefore, the total volume after mixing is
\[ V_{\mathrm{tot}} = V_A+V_B = 48.\underline{9}1\ \mathrm{L} \]
For ideal gases, enthalpy depends only on temperature. Since the mixing is isothermal,
\[ \boxed{ \Delta H_{\mathrm{mix}}=0 } \]
Each gas expands from its original volume into the total volume:
\[ \Delta S_{\mathrm{mix}} = n_AR\ln\left(\frac{V_{\mathrm{tot}}}{V_A}\right) + n_BR\ln\left(\frac{V_{\mathrm{tot}}}{V_B}\right) \]
\[ \Delta S_{\mathrm{mix}} = (1.00\ mol)R\ln\left(\frac{48.9\ L}{24.5\ L}\right) + (2.00\ mol)R\ln\left(\frac{48.9\ L}{24.5\ L}\right) \]
\[ \Delta S_{\mathrm{mix}} = \left( 3.00\ mol \right) \left( 8.314 \frac{J}{mol\ K} \right)\ln(2) = 17.3\ \mathrm{J\,K^{-1}} \]
Since \(\Delta H_{\mathrm{mix}}=0\),
\[ \Delta G_{\mathrm{mix}} = \Delta H_{\mathrm{mix}} - T\Delta S_{\mathrm{mix}} \]
\[ \Delta G_{\mathrm{mix}} = 0 - (298)(17.3) \]
\[ \boxed{ \Delta G_{\mathrm{mix}} = -5.15\ \mathrm{kJ} } \]
Interpretation: The gases mix spontaneously because the entropy of the system increases. Since the ideal gases do not interact energetically, \(\Delta H_{\mathrm{mix}}=0\), and the decrease in \(G\) is entirely entropy-driven.
A pair of ideal gases will be generated at random. Calculate the initial volumes, total volume after mixing, \(\Delta H_{\mathrm{mix}}\), \(\Delta S_{\mathrm{mix}}\), and \(\Delta G_{\mathrm{mix}}\).
| Quantity | Your answer |
|---|---|
| \(V_A\) | |
| \(V_B\) | |
| \(V_{\mathrm{tot}}\) | |
| \(\Delta H_{\mathrm{mix}}\) | |
| \(\Delta S_{\mathrm{mix}}\) | |
| \(\Delta G_{\mathrm{mix}}\) |
Big picture: The mixing of ideal gases is a spontaneous process because it increases the entropy of the system without changing its enthalpy. As a result, the Gibbs function decreases during mixing, providing a thermodynamic explanation for why gases naturally diffuse and mix with one another.