Question

Steam undergoes decomposition at high temperature as per the reaction \(\mathrm{H}_{2} \mathrm{O}(\mathrm{g}) \rightleftharpoons \mathrm{H}_{2}(\mathrm{~g})+\frac{1}{2} \mathrm{O}_{2}(\mathrm{~g})\) \(\Delta \mathrm{H}^{\circ}=200 \mathrm{~kJ} \mathrm{~mol}^{-1}, \Delta \mathrm{S}^{\circ}=40 \mathrm{~kJ} \mathrm{~mol}^{-1}\). The temperature at which equilibrium constant is unity is : (1) 3000 Kelvin (2) 5000 Kelvin (3) 5333 Kelvin (4) 5 Kelvin

Short answer

The temperature at which the equilibrium constant is unity is 5000 Kelvin.

Step-by-step solution

Identify the Given Information

The problem provides the following data: - A chemical reaction: \[\text{H}_{2} \text{O} (\text{g}) \rightleftharpoons \text{H}_2 (\text{g}) + \frac{1}{2} \text{O}_2 (\text{g})\] - Change in enthalpy, \(\Delta \text{H}^\circ = 200 \text{ kJ mol}^{-1}\) - Change in entropy, \(\Delta \text{S}^\circ = 40 \text{ kJ mol}^{-1}\)

Determine the Relationship Between Enthalpy, Entropy, and Temperature

The equilibrium constant can be expressed in terms of the Gibbs free energy change (\(\Delta G^\circ\)). At equilibrium, \(\Delta G^\circ = 0\) when the equilibrium constant \(K = 1\). The relationship is given by the equation: \[\Delta G^\circ = \Delta H^\circ - T \Delta S^\circ\]

Set the Gibbs Energy Change to Zero

Since \(K = 1\) at equilibrium, set \(\Delta G^\circ\) to 0: \[0 = \Delta H^\circ - T \Delta S^\circ\] Rearrange to find the temperature \(T\): \[T = \frac{\Delta H^\circ}{\Delta S^\circ}\]

Plug in the Given Values

Substitute the values for \(\Delta H^\circ\) and \(\Delta S^\circ\) into the equation: \[T = \frac{200 \text{ kJ mol}^{-1}}{40 \text{ kJ mol}^{-1}} = 5 \text{ kJ/K} \] Note that \(kJ/K\) can be converted to Kelvin by direct comparison since the units are consistent.

Convert to Temperature in Kelvin

The calculation gives: \[T = 5 \times 10^3 \text{ K} = 5000 \text{ K}\]

Key concepts

Gibbs Free Energy

Gibbs free energy (\text{G}) is a thermodynamic potential that measures the maximum reversible work a thermodynamic system can perform at constant temperature and pressure. It is used to predict the direction of chemical reactions and determine equilibrium positions. The change in Gibbs free energy (\text{ΔG}) for a reaction is given by the equation:

\[\text{ΔG} = \text{ΔH} - T\text{ΔS}\]
Here, \text{ΔH} represents the change in enthalpy, \text{ΔS} is the change in entropy, and T is the temperature in Kelvin.

When \text{ΔG} is negative, the reaction proceeds spontaneously in the forward direction. When \text{ΔG} is zero, the system is at equilibrium. Lastly, a positive \text{ΔG} means the reaction is non-spontaneous in the forward direction.

Equilibrium Constant

The equilibrium constant (K) provides insight into the proportions of reactants and products when a chemical reaction reaches equilibrium. For the reaction:

\[\text{H}_{2}\text{O}(\text{g}) \rightleftharpoons \text{H}_{2}(\text{g}) + \frac{1}{2}\text{O}_{2}(\text{g})\]
When K equals 1, the concentrations of reactants and products are equal under standard conditions. The link between Gibb's free energy and the equilibrium constant is expressed as:

\[\text{ΔG}^{\text{°}} = -RT\text{ln}K\]
At equilibrium, where \text{K} is 1, \text{ln}1 equals 0, leading to \text{ΔG} = 0, which further simplifies to:

\[\text{ΔH}^{\text{°}} = T\text{ΔS}^{\text{°}}\]
This equation can be used to calculate the temperature at which the equilibrium constant is unity, as we did in the provided solution.

Enthalpy and Entropy

Enthalpy (\text{H}) represents the total heat content of a system. It includes both the internal energy of the system and the product of its pressure and volume. The change in enthalpy (\text{ΔH}) is an important factor in predicting reaction behavior:

• A negative \text{ΔH} indicates an exothermic reaction, releasing heat to the surroundings.
• A positive \text{ΔH} signifies an endothermic reaction, absorbing heat.

Entropy (\text{S}), on the other hand, is a measure of the disorder or randomness of a system. The change in entropy (\text{ΔS}) reflects how much the disorder increases or decreases during a reaction:

• A positive \text{ΔS} suggests an increase in disorder.
• A negative \text{ΔS} indicates a decrease in disorder.

Both \text{ΔH} and \text{ΔS} are crucial in determining the feasibility and spontaneity of a reaction through the Gibbs free energy equation:

\[\text{ΔG} = \text{ΔH} - T\text{ΔS}\]
In summary, understanding \text{ΔH} and \text{ΔS} allows us to predict how a chemical reaction will behave under different temperature conditions.