Question

A certain constant-pressure reaction is barely nonspontaneous at \(45^{\circ} \mathrm{C}\) . The entropy change for the reaction is 72 \(\mathrm{J} / \mathrm{K} .\) Estimate \(\Delta H .\)

Short answer

The change in enthalpy (ΔH) for the constant-pressure reaction can be estimated by using the Gibbs Free Energy equation: ΔG = ΔH - TΔS, where T is the temperature in Kelvin and ΔS is the entropy change. Given that the reaction is barely nonspontaneous, we can assume ΔG ≈ 0. After converting the temperature to Kelvin (T(K) = 45°C + 273.15 = 318.15 K), we can substitute the values into the equation: ΔH ≈ (318.15 K)(72 J/K) ≈ 22874.8 J.

Step-by-step solution

Convert temperature to Kelvin

To convert temperature from Celsius to Kelvin, we use the following formula: T(K) = T(°C) + 273.15 Given T(°C) = 45°C, we have: T(K) = 45 + 273.15 = 318.15 K

Calculate ΔG

Since the reaction is barely nonspontaneous, we can assume that the Gibbs Free Energy (ΔG) is approximately equal to 0. Thus, we have: ΔG ≈ 0

Use the Gibbs Free Energy equation to estimate ΔH

We can use the following equation to estimate ΔH: ΔG = ΔH - TΔS Substituting the values of ΔG, T, and ΔS, we get: 0 ≈ ΔH - (318.15 K)(72 J/K)

Solve for ΔH

Rearrange the equation in Step 3 to solve for ΔH: ΔH ≈ (318.15 K)(72 J/K) Calculating the value of ΔH, we get: ΔH ≈ 22874.8 J Thus, the change in enthalpy (ΔH) for the reaction is approximately 22874.8 J.

Key concepts

Gibbs Free Energy

Gibbs Free Energy (G) is a thermodynamic property that can determine the spontaneity of a chemical reaction at constant pressure and temperature. A negative value for Gibbs Free Energy indicates a spontaneous process, while a positive value suggests that the reaction is nonspontaneous.

In the context of a barely nonspontaneous reaction, the Gibbs Free Energy is very close to zero. As shown in the exercise, for a reaction occurring at 318.15 K, a Gibbs Free Energy (\(\Delta G\)) value approximating zero suggests that the system is at equilibrium or very close to it. This means that the forward and reverse reactions occur at almost the same rate.

The relationship between Gibbs Free Energy and enthalpy (\(\Delta H\)), entropy (\(\Delta S\)), and temperature (T) of a system is given by the equation:
\(\Delta G = \Delta H - T\Delta S\).

This equation signifies that the change in Gibbs Free Energy for a process is influenced by the enthalpy change (\(\Delta H\)) which is the heat absorbed or released, the temperature of the system (T), and the entropy change (\(\Delta S\)), which represents the disorder or randomness in the system. Our exercise used this very relationship to estimate the enthalpy change for a chemical reaction at a specific temperature.

Entropy Change

Entropy (\(S\)) is a measure of the disorder or randomness in a system. An increase in entropy corresponds to an increase in the number of ways energy can be distributed within a system, and hence more disorder. For a reaction to be spontaneous, the total entropy change of the system and its surroundings generally should increase.

In a reaction with an entropy change of 72 J/K, as in the given exercise, the system experiences an increase in disorder as it proceeds. Thermodynamically, this would contribute favorably to the spontaneity of the reaction. However, it's important to remember that entropy is only one part of the spontaneity puzzle. The influence of entropy on the system's free energy is tempered by the temperature, which is why reactions that absorb heat (\(\Delta H > 0\)) can still be spontaneous if they produce enough disorder (\(T\Delta S > \Delta H\)).

In the given problem, the entropy change functions as a crucial part of the Gibbs Free Energy equation, interacting with temperature to influence the overallsystem stability and the possibility of the reaction proceeding on its own.

Enthalpy Change

The enthalpy change (\(\Delta H\)) of a reaction is the heat absorbed or released under constant pressure. It's a central component in energy exchange for chemical reactions. Endothermic reactions absorb heat (\(\Delta H > 0\)), while exothermic reactions release heat (\(\Delta H < 0\)).

In our exercise, the goal was to estimate the enthalpy change. Since our reaction is barely nonspontaneous at a certain temperature, its Gibbs Free Energy is approximately zero, which allows us to rearrange the Gibbs Free Energy equation to solve for \(\Delta H\). The estimated enthalpy change turns out to be positive, indicating an endothermic process at this temperature. It's worth noting that this thermodynamic quantity plays a pivotal role in determining the favorability of a reaction, together with entropy changes, under different temperature conditions.

Understanding how \(\Delta H\) relates to chemical bonding and reactions can provide deeper insight into the potential energy landscape of a reaction. In the context of reaction spontaneity, the Gibbs Free Energy equation integrates both changes in enthalpy and entropy to assess the energy profile and predict the behavior of the reaction.