Consider a system containing liquid water and water vapor in a sealed container. At first glance, it may seem surprising that both phases can exist simultaneously. Why doesn't all of the liquid evaporate? Why doesn't all of the vapor condense?
The answer lies in the concept of chemical potential. Recall that the chemical potential measures the change in Gibbs energy associated with adding a small amount of a substance to a system. As with other thermodynamic potentials, systems tend to evolve toward states of lower chemical potential.
Suppose the chemical potential of a substance in the liquid phase is greater than its chemical potential in the vapor phase:
\[ \mu_{\text{liquid}} > \mu_{\text{vapor}} \]In this situation, molecules can lower the Gibbs energy of the system by moving from the liquid phase into the vapor phase. Evaporation will therefore occur spontaneously.
Conversely, if
\[ \mu_{\text{liquid}} < \mu_{\text{vapor}} \]molecules can lower the Gibbs energy of the system by moving from the vapor phase into the liquid phase, and condensation will occur spontaneously.
Equilibrium is reached only when neither process is thermodynamically favored.
\[ \mu_{\text{liquid}} = \mu_{\text{vapor}} \]At this point, evaporation and condensation may still occur, but they occur at equal rates and there is no net transfer of material between the phases.
The same idea applies to any pair of phases. If a substance is present in two phases, designated \(\alpha\) and \(\beta\), equilibrium requires
\[ \mu^{\alpha} = \mu^{\beta} \]If three phases coexist, such as at a triple point, the chemical potential must be the same in all three phases:
\[ \mu^{\text{solid}} = \mu^{\text{liquid}} = \mu^{\text{vapor}} \]More generally, every phase present in a system at equilibrium must have the same chemical potential for a given component.
This criterion is one of the most important ideas in thermodynamics because it allows us to determine the conditions under which phases can coexist.
The condition of equal chemical potential is the foundation of phase equilibrium. In the remainder of this chapter, we will use it to develop the Gibbs Phase Rule, the Clapeyron equation, and the Clausius-Clapeyron equation. We will also use it to understand the structure of phase diagrams and the behavior of mixtures.
Every phase boundary on a phase diagram corresponds to a set of temperatures and pressures for which the chemical potentials of two phases are equal.
Big picture: Phase equilibrium is governed by the same principle that governs all spontaneous processes: systems move toward lower Gibbs energy. Equilibrium between phases is reached when there is no longer a thermodynamic driving force for material to move from one phase to another, which occurs when the chemical potential is the same in every phase present.
Consider a closed container containing liquid water and water vapor in equilibrium. Predict how each of the following changes will affect the equilibrium.
The chemical potential of a phase depends on temperature and pressure. For a gas, the dependence on pressure is especially strong because gases are highly compressible. Equilibrium requires
\[ \mu_{\text{liquid}} = \mu_{\text{vapor}} \]Any change that disrupts this equality will cause material to move from one phase to the other until equilibrium is restored.
The chemical potential decreases with increasing temperature according to
\[ \left(\frac{\partial \mu}{\partial T}\right)_p = -\bar S \]Since the molar entropy of the vapor is much larger than that of the liquid,
\[ \bar S_{\text{vapor}} > \bar S_{\text{liquid}} \]increasing the temperature causes the chemical potential of the vapor to decrease more rapidly than that of the liquid.
As a result,
\[ \mu_{\text{vapor}} < \mu_{\text{liquid}} \]and additional liquid evaporates until equilibrium is re-established.
Prediction: Increasing the temperature favors the vapor phase.
The pressure dependence of chemical potential is
\[ \left(\frac{\partial \mu}{\partial p}\right)_T = \bar V \]Since the molar volume of the vapor is much larger than that of the liquid,
\[ \bar V_{\text{vapor}} \gg \bar V_{\text{liquid}} \]increasing the pressure raises the chemical potential of the vapor much more than that of the liquid.
Consequently,
\[ \mu_{\text{vapor}} > \mu_{\text{liquid}} \]and vapor condenses until equilibrium is restored.
Prediction: Increasing the pressure favors the liquid phase.
Increasing the volume decreases the pressure of the vapor phase.
Since the chemical potential of a gas decreases as its pressure decreases,
\[ \mu_{\text{vapor}} < \mu_{\text{liquid}} \]and additional liquid evaporates to restore equilibrium.
Prediction: Increasing the volume favors the vapor phase.
Physical interpretation: Phase equilibrium is controlled by the relative chemical potentials of the phases. Increasing temperature favors the phase with the larger entropy, while increasing pressure favors the phase with the smaller molar volume. These simple ideas provide a powerful way to predict how phase boundaries will shift when external conditions change.
| Question 1 |
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A sealed container contains liquid water and water vapor. Which condition must be satisfied
when the two phases are at equilibrium?
A. The liquid and vapor must have the same density. B. The liquid and vapor must contain the same number of molecules. C. The chemical potentials of the liquid and vapor must be equal. D. The vapor pressure must be zero. |
Show AnswerAnswer: CPhase equilibrium occurs when there is no thermodynamic driving force for material to move from one phase to another. This requires \[ \mu_{\text{liquid}} = \mu_{\text{vapor}} \] |
| Question 2 |
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A liquid and its vapor are initially at equilibrium. The pressure is then increased while the
temperature remains constant. Which phase will be favored?
A. The vapor phase B. The liquid phase C. Neither phase; nothing changes D. The effect cannot be predicted |
Show AnswerAnswer: BIncreasing pressure raises the chemical potential of the phase with the larger molar volume most strongly. Since the vapor phase has a much larger molar volume than the liquid phase, compression favors condensation and therefore favors the liquid phase. |
| Question 3 |
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A liquid and its vapor are initially at equilibrium. The temperature is increased while the
pressure is held constant. Which phase will generally be favored?
A. The liquid phase B. The vapor phase C. Both phases equally D. The effect depends only on the molar mass |
Show AnswerAnswer: BThe vapor phase has a much larger molar entropy than the liquid phase. As temperature increases, the chemical potential of the vapor decreases more rapidly than that of the liquid, making the vapor phase more thermodynamically favorable. This is why increasing temperature generally promotes evaporation and shifts phase equilibrium toward the gas phase. |
Big picture: Chemical potential is the driving force behind phase changes. Phase equilibrium is achieved when the chemical potential of a component is the same in every phase present, eliminating any thermodynamic incentive for material to move from one phase to another. This simple idea forms the basis for the entire thermodynamic treatment of phase diagrams and phase transitions.