Question

For a particular reaction, \(\Delta H=-32 \mathrm{~kJ}\) and \(\Delta S=-98 \mathrm{~J} / \mathrm{K}\). Assume that \(\Delta H\) and \(\Delta S\) do not vary with temperature. (a) At what temperature will the reaction have \(\Delta G=0\) ? (b) If \(T\) is increased from that in part (a), will the reaction be spontaneous or nonspontaneous?

Short answer

The reaction will have \(\Delta G=0\) at approximately 326.53 K. If the temperature is increased from 326.53 K, the reaction will be non-spontaneous.

Step-by-step solution

Write down the given values and equation

We are given: \(\Delta H = -32 \mathrm{~kJ} = -32000 \mathrm{~J}\) \(\Delta S = -98 \mathrm{~J/K}\) We are going to use the Gibbs-Helmholtz equation: \(\Delta G = \Delta H - T\Delta S\)

Calculate the temperature for \(\Delta G=0\)

Set \(\Delta G = 0\) in the equation and solve for temperature: \(0 = -32000\mathrm{~J} - T(-98 \mathrm{~J/K})\) Rearrange the equation to isolate T: \(T = \frac{-32000 \mathrm{~J}}{-98 \mathrm{~J/K}}\)

Solve for temperature

Now, we'll simplify the equation: \(T = \frac{32000}{98} \approx 326.53\mathrm{~K}\) So the reaction will have \(\Delta G=0\) at approximately 326.53 K.

Determine spontaneity when temperature is increased

We know that when the reaction is spontaneous, \(\Delta G < 0\). If the temperature is increased, we can analyze the Gibbs-Helmholtz equation to determine the effect on \(\Delta G\): \(\Delta G = \Delta H - T\Delta S\) Given \(\Delta H < 0\) and \(\Delta S < 0\), increasing the temperature makes the term \(T\Delta S\) less negative (since a negative multiplied by a positive is negative, and increasing the positive value decreases the absolute value of the result). Since both terms have the same sign (negative), decreasing the absolute value of \(T\Delta S\) will result in a less negative \(\Delta G\). As \(\Delta G\) approaches zero or becomes positive, the reaction becomes non-spontaneous. Therefore, if the temperature is increased from 326.53 K, the reaction will be non-spontaneous.

Key concepts

Enthalpy

Enthalpy is a measure of the total energy in a thermodynamic system. It combines the system's internal energy with the work required to make room for it in the environment, often visualized as the energy needed to maintain pressure and volume. Here are some key points about enthalpy:

• Enthalpy, denoted as \( \Delta H \), reflects the heat change at constant pressure.
• If \( \Delta H \) is negative, the process releases heat, known as an exothermic reaction. This means the system loses energy to its surroundings, like in our exercise where \( \Delta H = -32 \mathrm{~kJ} \).
• Conversely, if \( \Delta H \) is positive, heat is absorbed from the surroundings, marking an endothermic process.

Understanding enthalpy helps in evaluating the energy changes during reactions, providing insight into whether they are energy consuming or releasing.

Entropy

Entropy is a measure of randomness or disorder in a system. The second law of thermodynamics tells us that entropy tends to increase in isolated systems. Here’s what you need to know about entropy:

• Entropy is denoted by \( \Delta S \). In our exercise, \( \Delta S = -98 \mathrm{~J/K} \), indicating a reduction in disorder during the reaction.
• A positive \( \Delta S \) suggests increased disorder or chaos, while a negative \( \Delta S \) implies a more ordered system.
• Entropy helps predict the direction of spontaneous processes: more disorder naturally occurs, so processes that increase entropy are more likely to be spontaneous.

Entropy indicates the balance between system order and randomness, and it plays a crucial role in determining whether reactions favor certain directions.

Reaction Spontaneity

Reaction spontaneity describes whether a chemical reaction will proceed on its own without any input of external energy. This is where Gibbs Free Energy (\( \Delta G \)) comes into play:

• Gibbs Free Energy combines enthalpy, temperature, and entropy effects to predict the spontaneity of a reaction.
• If \( \Delta G < 0 \), the reaction is spontaneous, occurring without added energy input.
• If \( \Delta G > 0 \), the reaction is non-spontaneous, requiring external energy to proceed.
• At \( \Delta G = 0 \), the system is in equilibrium, meaning no net change happens over time, as seen with the reaction temperature at about 326.53 K in our exercise before increasing the temperature even more.

Understanding these conditions helps predict and control chemical reactions, essential for many practical applications.