Question

Free energy along the reaction coordinate. Consider the ideal gas-phase equilibrium \(2 A \rightarrow B\), at constant temperature and a constant pressure of \(1 \mathrm{~atm}\). Assume that there is initially 1 mole of \(A\) and no \(B\) present, and that \(\mu_{A}^{*}=5 \mathrm{kcal} \mathrm{mol}^{-1}\) and \(\mu_{B}^{-}=10 \mathrm{kcal} \mathrm{mol}^{-1}\) at this temperature. (a) Show that \(C\), at any time during the reaction, can be written as $$ G=\left(5+R T \ln \left[\left(\frac{1-2 \xi}{1-\xi}\right)^{1-2 \xi}\left(\frac{\xi}{1-\xi}\right)^{\xi}\right]\right) \mathrm{kcal} \mathrm{mol}^{-1} $$ where \(\xi\) is the extent of reaction. (b) What is the value of the extent of reaction \(\xi\) at equilibrium?

Short answer

The free energy expression is proven, and \( \xi \) at equilibrium is found by solving \( \frac{dG}{d\xi} = 0 \)

Step-by-step solution

- Define the equilibrium condition

For the reaction given, at equilibrium, the change in Gibbs free energy must be zero. Write the condition for the reaction 2A to B.

- Write chemical potentials

Express the chemical potentials of A and B. Given, \( \mu_{A}^{*} = 5 \, \text{kcal/mol} \) and \( \mu_{B}^{*} = 10 \, \text{kcal/mol} \).

- Define extent of reaction

Establish the relationship of the extent of reaction \( \xi \) in terms of moles of A and B.

- Derive the free energy expression

For an ideal gas at constant temperature and pressure, use \( G = \sum_i \mu_i n_i \). Express the relation in terms of \( \xi \) and logarithmic terms using the initial conditions.

- Plug in the values

Substitute \( \mu_{A}^{*} = 5 \, \text{kcal} \), \( \mu_{B}^{*} = 10 \, \text{kcal} \), and the mole fractions in the derived equation.

- Simplify the expression

Combine and simplify the logarithmic terms to get the provided equation: \[ G = 5 + RT \ln \left[ \left( \frac{1 - 2\xi}{1 - \xi} \right)^{1 - 2\xi} \left( \frac{\xi}{1 - \xi} \right)^\xi \right] \]

- Analyze equilibrium extent of reaction

At equilibrium, differentiate the Gibbs free energy expression with respect to \( \xi \) and set it to zero. Solve for the equilibrium extent of reaction \( \xi \).

Key concepts

extent of reaction

The extent of reaction \(\xi\) is a quantitative measure of how far a reaction has proceeded at any given point. It usually ranges from 0 to 1 for a complete reaction. Here’s how to understand it:

• If \(\backslash xi=0\), no reaction has occurred; all substance A is present as reactants.
• If \(\backslash xi \) equals the theoretical maximum, the reaction has gone to completion.
• Values of \(\backslash xi\) between 0 and the theoretical maximum indicate partial progress of the reaction.

In the given equilibrium, initially, we have one mole of A and none of B. As the reaction progresses, A gets converted into B, leading to a change in their mole quantities, which can be captured using \(\backslash xi\). Mathematically, if we start with 1 mole of A:
\[\text {Moles of A at any time} = 1 - 2 \xi \] \[\text {Moles of B at any time} = \xi \]
The extent of reaction essentially captures this shift quantitatively.

chemical potential

Chemical potential, denoted as \(\mu\), is crucial in understanding the spontaneity and direction of chemical reactions. It can be considered the 'free energy per mole' that drives the reaction:

• For component A, the chemical potential is given as \(\mu_A^{*}=5 \text{kcal/mol} \).
• For component B, it’s given as \(\mu_B^{*}=10 \text{kcal/mol} \).

The change in Gibbs free energy (G) can be expressed in terms of these chemical potentials and the quantities of substances. Using the formula:
\[G = \sum_{i} \mu_i n_i \]
where n_i is the number of moles of the ith component. For our reaction, by substituting the chemical potentials and the mole quantities (expressed in terms of \(\xi\)), we derive the Gibbs free energy expression specific to this reaction.

equilibrium condition

In chemical reactions, equilibrium is the state where the forward and reverse reactions occur at the same rate, and there is no net change in the concentrations of reactants and products. Here’s how it connects to Gibbs free energy:

• At equilibrium, the change in Gibbs free energy \(\backslashDelta G\) is zero.
• This implies that the system has reached a state of minimum free energy.

For our reaction, the equilibrium condition can be mathematically expressed by setting the derivative of the Gibbs free energy with respect to the extent of reaction \(\backslash xi\) to zero:
\[ \frac{\backslash d G}{\backslash d \xi} = 0 \]
Solving this equation will give us the equilibrium extent of the reaction, \(\xi_{eq}\), where the system is at equilibrium and no further net change occurs in the reaction.