Problem 65 - Chemistry: The Central Science

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?

Step-by-Step Solution

Answer

The temperature at which the reaction has ΔG=0 is approximately 326.53 K. At temperatures higher than this value, the reaction becomes nonspontaneous.

Step 1: Determine the given values

We are given the following values: ΔH = -32 kJ = -32000 J (Converting kJ to J) ΔS = -98 J/K

Step 2: Use the Gibbs free energy equation to find the temperature at which ΔG = 0

The equation for Gibbs free energy change is: ΔG = ΔH - TΔS We need to find the temperature (T) at which ΔG = 0. So, we can set up the equation as follows: 0 = -32000 J - T(-98 J/K)

Step 3: Solve for the temperature T

Now, we can solve for T: 0 = -32000 J + 98T 32000 J = 98T T = 32000 J / 98 J/K T = 326.53 K

Step 4: Determine the spontaneity of the reaction at temperatures higher than the one found in step 3

We have found that ΔG = 0 at T = 326.53 K. Now, let's analyze the reaction's spontaneity at temperatures higher than this value. For a reaction to be spontaneous, ΔG must be negative. If ΔG is positive, the reaction is nonspontaneous. If we increase the temperature T from 326.53 K, we can analyze the sign of ΔG using the expression: ΔG = ΔH - TΔS We know that both ΔH and ΔS are negative in this case. When multiplying a negative value for ΔS by a temperature higher than 326.53 K, the value of TΔS will become more positive. As a result, the entire expression on the right-hand side of the equation (ΔH - TΔS) will become more positive. Thus, ΔG will become more positive at temperatures higher than 326.53 K.

Step 5: Conclude whether the reaction becomes spontaneous or nonspontaneous at higher temperatures

Since the sign of ΔG becomes more positive as the temperature increases beyond 326.53 K, the reaction becomes nonspontaneous at higher temperatures.

Key Concepts

Chemical Thermodynamics

Chemical thermodynamics is a branch of physical chemistry that deals with the relationship between heat, work, and energy changes in chemical reactions. Understanding thermodynamics is essential for predicting whether a chemical reaction is energetic enough to proceed on its own, which is referred to as spontaneity.

At the heart of chemical thermodynamics is the first law, which states that energy cannot be created or destroyed, only transformed. This principle ensures that when chemicals react, energy is conserved. The total amount of energy in the universe remains constant, but during a reaction, energy can change forms, such as transforming from potential energy within chemical bonds into thermal energy (heat).

Another crucial point is the second law of thermodynamics, which introduces the concept of entropy and states that the total entropy of an isolated system can never decrease over time. Entropy is often described as the measure of disorder or randomness within a system. In chemical reactions, entropy can determine the spontaneity of the process, as reactions tend to evolve towards a state of higher entropy.

Spontaneity of Reactions

The spontaneity of chemical reactions is a fundamental concept in thermodynamics and refers to the tendency of a process to occur without continuous external intervention. A reaction that happens spontaneously releases free energy and is capable of performing work.

To determine if a reaction is spontaneous, scientists consider the Gibbs free energy change, denoted as \(\Delta G\). The Gibbs free energy equation, \(\Delta G = \Delta H - T\Delta S\), combines the system's enthalpy (\(\Delta H\)), entropy (\(\Delta S\)), and temperature (T). Spontaneity is favored when \(\Delta G\) is negative. If it is positive, the reaction is nonspontaneous and requires energy input.

The Importance of Temperature

Temperature plays a key role in spontaneity. For the reaction mentioned in the exercise, \(\Delta G\) becomes zero at a specific temperature, which means at that exact temperature, the reaction is at equilibrium. Above this temperature, if \(\Delta S\) is negative, the reaction becomes less spontaneous because the term \(T\Delta S\) reduces the negative impact of \(\Delta H\), leading to a positive \(\Delta G\) and thus, a nonspontaneous reaction.

Entropy and Enthalpy

Entropy (\(\Delta S\)) and enthalpy (\(\Delta H\)) are two concepts that describe different aspects of a chemical reaction's energy changes.

Enthalpy is a measurement of the total heat content of a system and reflects the strength of chemical bonds. A negative \(\Delta H\) indicates that energy is released during the reaction, known as an exothermic process. Conversely, a positive \(\Delta H\) corresponds to an endothermic reaction, which absorbs energy.

Entropy, on the other hand, measures the disorder within a system, and increasing entropy often corresponds to greater randomness or chaos. A positive \(\Delta S\) indicates the disorder is increasing during the reaction, while a negative \(\Delta S\) suggests a decrease in disorder - the system is becoming more ordered.

Interplay Between Enthalpy and Entropy

These two properties, enthalpy and entropy, work together to determine the spontaneity of a reaction. If the loss of enthalpy (\(\Delta H\)) is high enough to overcome the decrease in entropy (\(\Delta S\)), the reaction will be spontaneous at certain temperatures. However, as the temperature increases, the entropy change becomes more significant, and if \(\Delta S\) is negative, it can drive \(\Delta G\) to be positive, which results in nonspontaneity. It is this delicate balance and interplay that guides the direction and feasibility of chemical reactions.