Question

Consider the reaction: $$\mathrm{H}_{2} \mathrm{S}(g)+\mathrm{SO}_{2}(g) \longrightarrow 3 \mathrm{S}(g)+2 \mathrm{H}_{2} \mathrm{O}(l)$$ for which \(\Delta H\) is \(-233 \mathrm{kJ}\) and \(\Delta S\) is \(-424 \mathrm{J} / \mathrm{K}\) . a. Calculate the free energy change for the reaction \((\Delta G)\) at 393 \(\mathrm{K} .\) b. Assuming \(\Delta H\) and \(\Delta S\) do not depend on temperature, at what temperatures is this reaction spontaneous?

Short answer

The free energy change for the reaction at 393 K is ΔG = -66,368 J. The reaction is spontaneous (ΔG < 0) at temperatures above 549.06 K.

Step-by-step solution

Recall the Gibbs free energy equation

We will use the Gibbs free energy equation to calculate the free energy change for the reaction and to find the conditions for spontaneity. The equation is given by: ΔG = ΔH - TΔS where ΔG is the change in free energy, ΔH is the change in enthalpy, T is the temperature in Kelvin, and ΔS is the change in entropy.

Calculate the free energy change at 393 K

We are given ΔH = -233 kJ and ΔS = -424 J/K. The temperature for part a is 393 K. We first need to convert ΔH to J since ΔS is in J/K. ΔH = -233 kJ × (1000 J/1 kJ) = -233,000 J Now we can plug in the values into the Gibbs free energy equation: ΔG = ΔH - TΔS = -233,000 J - (393 K × -424 J/K) = -233,000 J + 166,632 J = -66,368 J So, the free energy change for the reaction at 393 K is ΔG = -66,368 J.

Determine the conditions for spontaneity

To determine the temperatures at which this reaction is spontaneous, we need to analyze the criteria for spontaneity. A reaction is spontaneous when ΔG < 0. Therefore, we have: ΔG = ΔH - TΔS < 0 Now, we need to find the temperature boundaries that make this inequality true. We can rearrange the inequality as follows: T > ΔH / ΔS Note that the signs of both ΔH and ΔS are negative. So, we have: T > (-233,000 J) / (-424 J/K) = 549.06 K

Conclusion

The free energy change for the reaction at 393 K is ΔG = -66,368 J. The reaction is spontaneous (ΔG < 0) at temperatures above 549.06 K.

Key concepts

Spontaneity and Gibbs Free Energy

In thermodynamics, the spontaneity of a reaction tells us whether it will proceed without any external energy input. A spontaneous reaction occurs naturally and favors the formation of products under given conditions. To determine if a reaction is spontaneous, we use Gibbs free energy (\(\Delta G\)).

- If \(\Delta G < 0\), the reaction is spontaneous.- If \(\Delta G > 0\), the reaction is non-spontaneous and requires energy input.- If \(\Delta G = 0\), the system is at equilibrium.

For the given chemical reaction, we calculated \(\Delta G = -66,368 \text{ J}\) at 393 K, indicating that the reaction is spontaneous at this temperature. This calculation uses the equation:\[\Delta G = \Delta H - T\Delta S\]where \(\Delta H\) is the change in enthalpy, \(T\) is the temperature in Kelvin, and \(\Delta S\) is the change in entropy.

Keep in mind that spontaneity can depend on the temperature. For example, if temperatures exceed 549.06 K in this situation, \(\Delta G\) remains negative, supporting spontaneity across those conditions. This understanding helps us predict under which circumstances a reaction will naturally occur.

Understanding Enthalpy

Enthalpy (\(\Delta H\)) represents the total heat content of a system. It reflects the energy absorbed or released during a reaction under constant pressure.

- A positive \(\Delta H\) indicates an endothermic process, where the system absorbs heat.- A negative \(\Delta H\) indicates an exothermic process, where the system releases heat.

For the given reaction, \(\Delta H = -233 \text{kJ}\) signifies that the process is exothermic, releasing a substantial amount of heat. Typically, exothermic reactions are often spontaneous because releasing heat tends to favor the formation of products at lower temperatures in many practical situations.

Enthalpy is a crucial factor to consider when evaluating \(\Delta G\) and understanding whether the direction of a chemical reaction will favor products or reactants. Knowing the heat relationship helps in making predictions about reaction dynamics and potential energy changes.

Role of Entropy

Entropy (\(\Delta S\)) measures the degree of randomness or disorder within a system. Entropy change impacts the spontaneity of reactions significantly.

- \(\Delta S > 0\) means an increase in disorder, which is often a favorable condition for spontaneity.- \(\Delta S < 0\) implies a decrease in disorder, often less favorable without energy compensation.

In the exercise, \(\Delta S = -424 \text{J/K}\) means the reaction results in a decrease in randomness. This might suggest that the reaction would be non-spontaneous since the system's order increases. However, because \(\Delta H\) is significantly negative, the loss in entropy is compensated by the energy release, allowing the reaction to be spontaneous at certain temperatures.

Entropy highlights the balance needed between energy changes and disorder to understand fully why some reactions naturally proceed while others do not. In many chemical processes, especially those involving gas to liquid or solid phases, entropy changes play a pivotal role in determining feasibility and directionality.