Equality of chemical potentials in the presence of surface...

Equality of chemical potentials in the presence of surface effects. We proved this using entropy in Section 24.1. Obtain the same result from the Gibbs function, as follows. Consider a system consisting of a drop of liquid in the presence of its vapour. Ascribe to the surface of the drop internal energy $U_s$ and entropy $S_s$, but no volume or particles. Define $G \equiv U+$ $p_v V-T S$, where $p_v$ is the pressure of the vapour. Consider a change in which $p_v, T$ are constant but material moves from one phase to the other. By considering $\mathrm{d} U, \mathrm{~d} S$, and $\mathrm{d} V$ show that $$ \begin{aligned} \mathrm{d} G= & \left(u_l+p_v v_l-T s_l\right) \mathrm{d} N_l \\ & +\left(u_v+p_v v_v-T s_v\right) \mathrm{d} N_v+\sigma \mathrm{d} A, \end{aligned} $$ if the liquid is assumed to be incompressible. Relate $v_l \mathrm{~d} N_l$ to $\mathrm{d} A$ and hence show that $\mathrm{d} G=\mathrm{d} N_l g_l+\mathrm{d} N_v g_v$, where $g_l$ is evaluated at the pressure of the liquid and $g_v$ at the pressure of the vapour. Hence show that $g_v=g_l$ in phase equilibrium.

Equality of chemical potentials in the presence of surface effects. We proved this using entropy in Section 24.1. Obtain the same result from the Gibbs function, as follows. Consider a system consisting of a drop of liquid in the presence of its vapour. Ascribe to the surface of the drop internal energy $U_s$ and entropy $S_s$, but no volume or particles. Define $G \equiv U+$ $p_v V-T S$, where $p_v$ is the pressure of the vapour. Consider a change in which $p_v, T$ are constant but material moves from one phase to the other. By considering $\mathrm{d} U, \mathrm{~d} S$, and $\mathrm{d} V$ show that
$$
\begin{aligned}
\mathrm{d} G= & \left(u_l+p_v v_l-T s_l\right) \mathrm{d} N_l \\
& +\left(u_v+p_v v_v-T s_v\right) \mathrm{d} N_v+\sigma \mathrm{d} A,
\end{aligned}
$$
if the liquid is assumed to be incompressible. Relate $v_l \mathrm{~d} N_l$ to $\mathrm{d} A$ and hence show that $\mathrm{d} G=\mathrm{d} N_l g_l+\mathrm{d} N_v g_v$, where $g_l$ is evaluated at the pressure of the liquid and $g_v$ at the pressure of the vapour. Hence show that $g_v=g_l$ in phase equilibrium.

Key Concepts

Gibbs Free Energy

The Gibbs free energy is a thermodynamic potential defined as G = U + pV ? TS. It represents the maximum reversible work obtainable from a closed system at constant temperature and pressure. In analyzing phase changes and chemical equilibria, the Gibbs free energy is minimized at equilibrium, making it a central tool for determining the conditions under which different phases coexist.

Chemical Potential

The chemical potential is the partial molar Gibbs free energy, signifying the change in Gibbs free energy when an additional amount of substance is introduced to the system. It is a key factor in phase transitions because, at equilibrium, the chemical potentials of the species in different phases must be equal, ensuring no net transfer of matter between the phases.

Phase Equilibrium

Phase equilibrium occurs when multiple phases of a substance exist together without net mass exchange. This equilibrium is achieved when the chemical potentials of the substance in each phase are equal. The derivation using the Gibbs free energy confirms that, under constant temperature and pressure, the equality of the chemical potentials is a necessary condition for phase equilibrium.

Surface Effects

Surface effects become significant especially in systems like small droplets where the surface-to-volume ratio is large. The interfacial region contributes additional energy (and entropy) that is quantified by the surface tension and the change in surface area. Incorporating surface effects into the thermodynamic analysis allows for a more complete description of phase transitions and equilibrium conditions in heterogeneous systems.

Thermodynamic Differential Relations

Thermodynamic differential relations, such as dG = dU + pdV ? TdS, are fundamental in relating changes in thermodynamic potentials to changes in other state variables. They provide a systematic method to account for contributions from different parts of the system, including bulk phases and interfaces, and are essential for deriving conditions like the equality of chemical potentials at equilibrium.

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