91Ó°ÊÓ

Consider a binary mixture composed of two types of particles, A and B. For this system the fundamental equation for the Gibbs free energy is \(G=\mathrm{n}_{\mathrm{A}} \mu_{\mathrm{A}}+\mathrm{n}_{\mathrm{B}} \mu_{\mathrm{B}}\), the combined first and second laws are \(\mathrm{d} G=-S \mathrm{~d} T+V \mathrm{~d} P+\mu_{A} \mathrm{~d} \mathrm{n}_{\mathrm{A}}+\mu_{\mathrm{B}} \mathrm{d} \mathrm{n}_{\mathrm{B}}\) (S is the total entropy and \(V\) is the total volume of the system), and the chemical potentials \(\mu_{\mathrm{A}}\) and \(\mu_{\mathrm{B}}\) are intensive so that \(\mu_{\mathrm{A}}=\mu_{\mathrm{A}}(P, T\), \(x_{\mathrm{A}}\) ) and \(\mu_{\mathrm{B}}=\mu_{\mathrm{B}}\left(P, T, x_{\mathrm{A}}\right)\) where \(x_{\mathrm{A}}\) is the mole fraction of \(\mathrm{A}\). Use these facts to derive the relations $$ s \mathrm{~d} T-v \mathrm{~d} P+\sum_{\alpha=\mathrm{A}, \mathrm{B}} x_{\alpha} \mathrm{d} \mu_{\alpha}=0 $$ and $$ \sum_{\alpha=A \cdot B} x_{\alpha}\left(\mathrm{d} \mu_{a}+s_{\alpha} \mathrm{d} T-v_{\alpha} \mathrm{d} P\right)=0 $$ \(\alpha=A, B\) and \(\beta=A, B\),

Step by step solution

01

- Identify the expressions given

The problem provides a fundamental equation for Gibbs free energy, which is: \[ G = n_A \mu_A + n_B \mu_B \] and the combined first and second laws of thermodynamics equation: \[ \mathrm{d}G = -S \mathrm{~d}T + V \mathrm{~d}P + \mu_A \mathrm{~d} n_A + \mu_B \mathrm{d} n_B \] The challenge is to derive the given relations.

02

- Understand the chemical potential dependencies

The chemical potentials \( \mu_A \) and \( \mu_B \) depend on pressure \( P \), temperature \( T \), and the mole fractions \( x_A \). Therefore, these can be expressed as: \[ \mu_A = \mu_A(P, T, x_A) \quad \text{and} \quad \mu_B = \mu_B(P, T, x_A) \] The mole fractions \( x_A \) and \( x_B \) sum to 1, i.e., \( x_A + x_B = 1 \).

03

- Express the total number of particles and mole fractions

Let the total number of particles be \( n = n_A + n_B \). Hence, the number of particles of type A can be written as \( n_A = x_A n \), and the number of particles of type B as \( n_B = x_B n \). Differentiating, we get: \[ \mathrm{d} n_A = x_A \mathrm{d} n + n \mathrm{d} x_A \] \[ \mathrm{d} n_B = x_B \mathrm{d} n + n \mathrm{d} x_B \]

04

- Substitute the differentials into the Gibbs equation

Substitute \( \mathrm{d} n_A \) and \( \mathrm{d} n_B \) into the Gibbs equation: \[ \mathrm{d}G = -S \mathrm{~d}T + V \mathrm{~d}P + \mu_A (x_A \mathrm{d} n + n \mathrm{d} x_A) + \mu_B (x_B \mathrm{d} n + n \mathrm{d} x_B) \] Combine the terms to get: \[ \mathrm{d}G = -S \mathrm{~d}T + V \mathrm{~d}P + (\mu_A x_A + \mu_B x_B) \mathrm{d} n + n (\mu_A \mathrm{d} x_A + \mu_B \mathrm{d} x_B) \]

05

- Simplify the combined expression

For the derivation, the total differential of \(G\) can be simplified as: \[ \mathrm{d}G = (\mu_A x_A + \mu_B x_B) \mathrm{d} n + n (\mu_A \mathrm{d} x_A + \mu_B \mathrm{d} x_B) \] By recognizing \( \mu_A x_A + \mu_B x_B = G/n \), we simplify the original differential equation: \[ \mathrm{d}G = -S \mathrm{~d}T + V \mathrm{~d}P + G/n \mathrm{d} n \] This gives us a critical relation towards deriving the final expressions.

06

- Obtain the desired relationship

Subtracting the simplified expression from the Gibbs equation, we get: \[ 0 = -S \mathrm{~d}T + V \mathrm{~d}P + n (\mu_A \mathrm{d} x_A + \mu_B \mathrm{d} x_B - S \mathrm{~d}T + V \mathrm{~d}P) / n \] Then we rearrange the terms to get: \[ s \mathrm{~d}T - v \mathrm{~d}P + \sum_{\alpha=A,B} x_\alpha \mathrm{d} \mu_\alpha = 0 \] This completes the first part of the derivation.

07

- Derive the second desired equation

Next, differentiate the chemical potentials and substitute into the equation. We get: \[ \sum_{\beta=A,B} x_\beta \mathrm{d} \mu_\beta = \sum_{\beta=A,B} x_\beta (s_\beta \mathrm{d} T - v_\beta \mathrm{d} P) \] Thus, we horizontally sum the component-specific equations to obtain: \[ \sum_{\alpha=A,B} x_\alpha \left(\mathrm{d} \mu_\alpha + s_\alpha \mathrm{d} T - v_\alpha \mathrm{d} P\right) = 0 \] Hence, the desired second relation is derived.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

headline of the respective core concept

The Gibbs free energy (\textbf{G}) is a thermodynamic potential representing the maximum reversible work that a thermodynamic system can perform at constant temperature (T) and pressure (P). It is defined as: \(G = H - TS\), where \(H\) is enthalpy and \(S\) is entropy. For a binary mixture, the Gibbs free energy is: \( G = n_A \, \text{\mu}_A + n_B \text{\mu}_B \). Here, \(n_A\) and \(n_B\) are the number of particles of type A and B, respectively, and \(\text{\mu}_A\) and \(\text{\mu}_B\) are their corresponding chemical potentials. The combined first and second laws of thermodynamics can be written for such a system as: \( \mathrm{d}G = -S \mathrm{~d}T + V \mathrm{~d}P + \text{\mu}_A \mathrm{~d} n_A + \text{\mu}_B \mathrm{d} n_B \), which also encompasses the concept of chemical potentials in the context of Gibbs free energy.

headline of the respective core concept

Thermodynamic potentials are fundamental functions used in thermodynamics to describe the equilibrium properties and spontaneous processes of thermodynamic systems. These potentials include internal energy (U), enthalpy (H), Helmholtz free energy (F), and Gibbs free energy (G). They help in determining how energy is distributed within a system and how it interacts with its surroundings. The essential differential form incorporating various thermodynamic potentials is given by: \(\mathrm{dU} = T \mathrm{d}S - P\, \mathrm{d}V + \sum_i \text{\mu}_i \mathrm{d} n_i \), where \(\text{\mu}_i\) represents the chemical potentials of the species in the mixture. Gibbs free energy, in particular, is an essential thermodynamic potential for processes occurring at constant temperature and pressure, being perhaps the most relevant in chemical and biological systems.

headline of the respective core concept

Chemical potential (\(\text{\mu}\)) represents the change in the Gibbs free energy of a system when the number of particles of one type is changed, while keeping the temperature, pressure, and the number of particles of other types constant. For a binary mixture with particles A and B, the chemical potential depends on temperature, pressure, and the mole fractions of the components: \(\text{\mu}_A = \text{\mu}_A (P, T, x_A)\) and \(\text{\mu}_B = \text{\mu}_B (P, T, x_A)\). Understanding the chemical potential is crucial for determining how a system responds to changes in composition and how it reaches equilibrium.

headline of the respective core concept

A binary mixture consists of two different components or types of particles. For example, a mixture of particles A and B can be characterized by their mole fractions \(x_A\) and \(x_B\), where: \(x_A + x_B = 1\). The total number of particles in the mixture is \(n = n_A + n_B\). The properties of the mixture, such as the Gibbs free energy, are determined by the combined effects of the individual components. For instance, the Gibbs free energy of a binary mixture is given by: \( G = n_A \text{\mu}_A + n_B \text{\mu}_B \). The behavior and interactions within the mixture involve the dependencies of the chemical potentials on the pressure, temperature, and the mixture's mole fractions.

headline of the respective core concept

Thermodynamic equations are mathematical relationships that describe the state and evolution of thermodynamic systems. They connect state variables like temperature, pressure, volume, and composition. For instance, the equation \( \mathrm{d}G = -S \mathrm{~d}T + V \mathrm{~d}P + \text{\mu}_A \mathrm{~d} n_A + \text{\mu}_B \mathrm{d} n_B \) describes the changes in Gibbs free energy for a mixture. Another essential equation for binary mixtures is: \(0 = S\, \mathrm{~d}T - V\, \mathrm{~d}P + \sum_{\alpha=A,B} x_\alpha \mathrm{d} \text{\mu}_\alpha \). These equations provide a framework for understanding and predicting the behavior of mixtures in response to thermodynamic processes.