# Examining the Equilibrium Constant for an Ideal Gas Mixture

• ehrenfest
In summary, the conversation discusses the law of mass action and the equilibrium constant for a reaction involving an ideal-gas mixture of atoms and molecules. The formula for the equilibrium constant is shown, with V representing volume and f_i representing single-particle partition functions. The Helmholtz free energy is also mentioned, but its use and relevance is not clear.
ehrenfest

## Homework Statement

Consider an ideal-gas mixture of atoma A, atoms B and molecules AB, undergoing the reaction AB <---> A+B. If n_A, n_B, and n_{AB} denote their respective concentrations, then show that, in equilibrium

$$\frac{n_{AB}}{n_A n_B} = V \frac{f_{AB}}{f_A f_B} = K(T)$$

(the law of mass action)

Here, V is the volume of the system while the f_i are the respective single-particle partition functions; the quantity K(T) is generally referred to as the equilibrium constant of the reaction.

## The Attempt at a Solution

I got the partition function of the system and I got the Helmholtz free energy. The problem is that I just don't know what to do with the Helmholtz free energy i.e. do I minimize it, mazimize it, set it equal to something? Why? Pathria and my other textbook do little more than define it. They never say what to do with it or give any example of how it is useful!

Last edited:
anyone?

Sorry for the slow reply. You should be able to show

$$n_A = e^{\mu_A\beta}\frac{f_A}{V}$$

straight from the grand canonical partition function (taking the appropriate derivative of $$\ln \mathcal Z$$). The desired result then follows from showing the chemical potentials cancel out.

## 1. What is the equilibrium constant for an ideal gas mixture?

The equilibrium constant for an ideal gas mixture is a value that represents the ratio of the concentration of products to the concentration of reactants at equilibrium. It is denoted as Kp for gases or Kc for aqueous solutions.

## 2. How is the equilibrium constant determined for an ideal gas mixture?

The equilibrium constant can be determined experimentally by measuring the concentrations of the reactants and products at equilibrium and plugging them into the equilibrium expression. It can also be calculated using the ideal gas law and the partial pressures of the gases.

## 3. How does temperature affect the equilibrium constant for an ideal gas mixture?

According to Le Chatelier's principle, an increase in temperature will shift the equilibrium towards the endothermic direction, resulting in an increase in the equilibrium constant. Conversely, a decrease in temperature will shift the equilibrium towards the exothermic direction, resulting in a decrease in the equilibrium constant.

## 4. Can the equilibrium constant for an ideal gas mixture be manipulated?

Yes, the equilibrium constant can be manipulated by changing the temperature, pressure, or concentration of the reactants and products. However, the value of the equilibrium constant at a specific temperature and pressure is a constant for a particular reaction.

## 5. What is the significance of the equilibrium constant for an ideal gas mixture?

The equilibrium constant provides important information about the relative amounts of reactants and products at equilibrium. It also helps predict the direction in which a reaction will proceed under different conditions and allows for the determination of equilibrium concentrations from initial concentrations. Additionally, it can be used to calculate the value of other thermodynamic parameters, such as Gibbs free energy and enthalpy.

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