Question

At \(1500 \mathrm{K},\) the process $$\begin{aligned} &\mathbf{I}_{2}(g) \longrightarrow 2 \mathbf{I}(g)\\\ &10 \mathrm{atm} \quad 10 \mathrm{atm} \end{aligned}$$ is not spontaneous. However, the process $$\begin{aligned} &\mathbf{I}_{2}(g) \longrightarrow 2 \mathbf{I}(g)\\\ &0.10 \mathrm{atm} \quad 0.10 \mathrm{atm} \end{aligned}$$ is spontaneous at \(1500 \mathrm{K}\). Explain.

Short answer

At 1500 K, the process \(I_{2}(g) \longrightarrow 2I(g)\) is spontaneous at 0.1 atm but non-spontaneous at 10 atm due to the effect of pressure on the Gibbs free energy change (∆G). As pressure decreases, the chemical potential difference between products and reactants (∆µ) decreases, resulting in a more negative ∆G, which makes the process spontaneous. In contrast, at higher pressures, the change in Gibbs free energy (∆G) becomes less negative (or even positive) causing the process to be non-spontaneous.

Step-by-step solution

Write Down the Reaction Equation

The given reaction is: \( I_{2}(g) \longrightarrow 2I(g) \) This reaction involves dissociation of an iodine molecule into two iodine atoms at two different pressures and the same temperature, 1500 K.

Write Down the Equation for Gibbs Free Energy

The equation for Gibbs free energy is: ∆G = ∆H - T∆S

Consider the Effects of Pressure on the Reaction Spontaneity

In this specific problem, both ∆H and ∆S do not change with pressure. Therefore, to analyze the effect of pressure on reaction spontaneity, we must turn our attention to the parameter of Gibbs free energy that's related to pressure: chemical potential. The equation for chemical potential, µ, is: µ = µ° + RT ln(P) Where µ° is the standard chemical potential, R is the gas constant, T is the temperature in Kelvin, and P is the pressure.

Analyze the Effects of Pressure on the Chemical Potential

As pressure decreases, the natural logarithm term in the chemical potential equation (ln(P)) also decreases, making the chemical potential for products lower. Consequently, the difference in the chemical potential between products and reactants (∆µ) will be smaller at lower pressures as compared to higher pressures.

Relate the Chemical Potential Difference to Gibbs Free Energy Change

Because the change in Gibbs free energy is proportional to the difference in chemical potential between products and reactants (∆G = -n∆µ), a smaller ∆µ will lead to a smaller ∆G. Therefore, at lower pressures, the change in Gibbs free energy for the given reaction will be more negative compared to the one at the higher pressure.

Explain the Spontaneity at Two Different Pressures

At lower pressure (0.1 atm), the change in Gibbs free energy (∆G) becomes more negative as compared to the reaction at a higher pressure (10 atm), leading to a spontaneous process. On the other hand, at higher pressure (10 atm), the Gibbs free energy change (∆G) is less negative (or even positive), making the process non-spontaneous.

Key concepts

Chemical Potential

Chemical potential is a fundamental concept in thermodynamics that bears a strong influence on reaction behavior. It is a measure of the free energy per particle, capturing the tendency of particles to change their state or phase. At a given temperature, the chemical potential, denoted as \( \mu \), determines how particles favorably rearrange themselves.

The equation for chemical potential is written as:

• \( \mu = \mu^\circ + RT \ln(P) \)

Here, \( \mu^\circ \) is the standard chemical potential, \( R \) is the gas constant, \( T \) is the temperature in Kelvin, and \( P \) is the pressure.

The natural logarithm term, \( \ln(P) \), indicates how pressure influences chemical potential. At lower pressures, this term becomes less positive, meaning the chemical potential decreases. Lower chemical potential in the reaction products relative to reactants implies a more favorable condition for the reaction to proceed.

Therefore, in reactions where pressure plays a role, reducing the pressure can lower the chemical potential, promoting reaction processes.

Reaction Spontaneity

Understanding reaction spontaneity involves assessing whether a reaction will proceed without external intervention. A spontaneous reaction corresponds to a decrease in Gibbs Free Energy, \( \Delta G \). This is given by the equation:

• \( \Delta G = \Delta H - T\Delta S \)

Where \( \Delta H \) is the enthalpy change, \( T \) is the temperature, and \( \Delta S \) is the entropy change.

A reaction is spontaneous if \( \Delta G \) is negative, equilibrating the effects of both enthalpy and entropy. For example, the dissociation of iodine from \( I_2(g) \) to \( 2I(g) \) may become spontaneous at lower pressures where the \( \Delta G \) becomes more negative.

Reaction spontaneity indicates that the system moves towards equilibrium in a natural manner. In thermodynamic terms, this often involves balancing the energy fronts (enthalpic and entropic effects) to reach a configuration where the system can minimize energy use overall. As such, reactions tend to be spontaneous under conditions where \( \Delta G < 0 \).

Effect of Pressure on Reactions

Pressure significantly affects gaseous reactions, influencing how molecules interact and form products. For the iodine dissociation reaction, lowering the pressure impacts how the reaction energy states are configured.

According to the equation for chemical potential \( \mu = \mu^\circ + RT\ln(P) \), lower pressure reduces the logarithmic term, affecting the chemical potential and thus the Gibbs Free Energy \( \Delta G \). When pressure decreases, the Gibbs Free Energy change becomes more negative.

This means that decreasing pressure can drive a reaction to become more spontaneous, as it shifts the energy balance favorably towards products. However, at higher pressures, the reaction’s tendency to progress decreases, making it less favorable to spontaneously proceed.

In conclusion, pressure changes the landscape of the reaction's energy profile, altering the likelihood of the reaction occurring spontaneously. For gas reactions, as pressure decreases, the tendency for the reaction becomes more spontaneous as the lowest energy pathway is preferred.