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An isolated system has two phases, denoted by A and B, each...

An isolated system has two phases, denoted by $\mathrm{A}$ and B, each of which consists of the same two substances, denoted by 1 and $2 .$ Show that necessary conditions for equilibrium are 1. the temperature of each phase is the same, $T_{\mathrm{A}}=T_{\mathrm{B}}$. 2. the pressure of each phase is the same, $p_{\mathrm{A}}=p_{\mathrm{B}}$. 3. the chemical potential of each component has the same value in each phase, $\mu_{1}^{\mathrm{A}}=\mu_{1}^{\mathrm{B}}, \mu_{2}^{\mathrm{A}}=\mu_{2}^{\mathrm{B}}$.

An isolated system has two phases, denoted by $\mathrm{A}$ and B, each of which consists of the same two substances, denoted by 1 and $2 .$ Show that necessary conditions for equilibrium are
1. the temperature of each phase is the same, $T_{\mathrm{A}}=T_{\mathrm{B}}$.
2. the pressure of each phase is the same, $p_{\mathrm{A}}=p_{\mathrm{B}}$.
3. the chemical potential of each component has the same value in each phase, $\mu_{1}^{\mathrm{A}}=\mu_{1}^{\mathrm{B}}, \mu_{2}^{\mathrm{A}}=\mu_{2}^{\mathrm{B}}$.

Key Concepts

Thermal Equilibrium

Thermal equilibrium refers to the condition under which all parts of a system reach the same temperature, ensuring no net flow of heat between different regions. In the context of phase equilibrium, the equality of temperature in all phases is necessary for stability, as any temperature gradient would drive heat transfer until a uniform temperature is achieved.

Mechanical Equilibrium

Mechanical equilibrium occurs when a system exhibits uniform pressure across all its parts, meaning there is no net force driving the movement of matter from one region to another. In a multiphase system, equal pressure in each phase prevents any mechanical work being done on or by the system, thereby maintaining overall stability.

Chemical Equilibrium

Chemical equilibrium is the state in which the chemical potentials of each component are equal across all phases of a system. This ensures that there is no net transfer of substances between phases, as any difference in chemical potential would lead to a shift in the composition of the phases until equilibrium is restored.

Phase Equilibrium

Phase equilibrium is the overall state wherein multiple distinct phases coexist within a system without any net change over time. Achieving phase equilibrium requires the conditions of thermal, mechanical, and chemical equilibrium to be met simultaneously, ensuring that the system is stable and no spontaneous changes in the state of the matter occur.

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