Question

A particular reaction is spontaneous at \(450 \mathrm{~K}\). The enthalpy change for the reaction is \(+34.5 \mathrm{~kJ} .\) What can you conclude about the sign and magnitude of \(\Delta S\) for the reaction?

Short answer

The entropy change (ΔS) for the reaction is positive and has a magnitude of 0.07667 kJ/K. This indicates an increase in the reaction's entropy, which is consistent with a spontaneous reaction at 450 K.

Step-by-step solution

Recall the Gibbs free energy equation

The Gibbs free energy (ΔG) equation is given by: \(ΔG = ΔH - TΔS\) A reaction is spontaneous if ΔG is negative. In this case, we know that the reaction is spontaneous at 450 K, and we are given the enthalpy change (ΔH) as +34.5 kJ.

Calculate ΔG from the given information

Since the reaction is spontaneous at 450 K, ΔG must be negative. We can rewrite the Gibbs free energy equation to find the entropy change (ΔS) in terms of ΔH and temperature (T): \(ΔS = \frac{ΔH - ΔG}{T}\) We'll now plug in the values we know. We'll assume ΔG to be a small negative number (e.g., -0.001 kJ) to ensure the reaction is spontaneous.

Plug in the known values and solve for ΔS

Using the given values and our assumption from step 2, we can solve for ΔS: \(ΔS = \frac{(+34.5\,\text{kJ}) - (-0.001\,\text{kJ})}{450\,\mathrm{K}}\) \(ΔS = \frac{34.501\,\text{kJ}}{450\,\mathrm{K}}\) \(ΔS = 0.07667\,\text{kJ/K}\)

Interpret the result for the sign and magnitude of ΔS

The calculated value for the entropy change (ΔS) is positive (0.07667 kJ/K). This means that the reaction's entropy is increasing, making it consistent with a spontaneous reaction at 450 K. The magnitude of ΔS is 0.07667 kJ/K, which tells us the extent of the increase in the reaction's entropy.

Key concepts

Entropy

Entropy, symbolized as \( \Delta S \) in chemistry, is a measure of the disorder or randomness in a system. The concept originates from the Second Law of Thermodynamics, which states that the total entropy of an isolated system can never decrease over time. This implies that natural processes tend to lead towards more disordered states. \(
\)\(
\)Entropy is crucial in predicting the behavior of a chemical reaction because it can determine the feasibility of a reaction occurring without external energy input. In simple terms, a reaction that leads to an increase in entropy (\( \Delta S > 0 \) ) is often more likely to occur spontaneously, as it aligns with the universe's tendency towards higher disorder.\(
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\)For example, when we dissolve salt in water, the salt ions spread out and move freely in the solvent, increasing the system's disorder. Therefore, the entropy increases. On the other hand, a decrease in entropy (\( \Delta S < 0 \) ) could indicate that a reaction may need outside energy to proceed, such as a decrease in gas volume or the formation of a crystal from a solution.\(
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\)It is important to understand that entropy is one of several factors that affect the spontaneity of a reaction. Changes in entropy must be considered in conjunction with other thermodynamic properties, like enthalpy, to fully predict reaction behavior.\(
\)\(
\)In the exercise provided, we conclude that the sign of \( \Delta S \) must be positive for the reaction to be spontaneous at 450 K, given the positive enthalpy change. The magnitude of \( \Delta S \) gives us an idea of how much disorder is increasing because of the reaction.

Enthalpy

Enthalpy, represented by \( \Delta H \), is a thermodynamic quantity equivalent to the total heat content of a system. It encompasses internal energy within the system plus the energy required to make room for it by displacing its environment and establishing its volume and pressure.\(
\)\(
\)In the context of chemical reactions, we often discuss changes in enthalpy, \( \Delta H \). A negative \( \Delta H \), known as exothermicity, signifies that the reaction releases heat into its surroundings, which can be felt as warmth. Conversely, a positive \( \Delta H \), or endothermicity, indicates that the reaction absorbs heat, often leading to a cooling effect.\(
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\)The enthalpy change is key to understanding chemical reactions together with entropy. While entropy measures disorder and probability, enthalpy represents the energy changes that occur during a reaction. A reaction that is both exothermic \( \Delta H < 0 \) and increases entropy \( \Delta S > 0 \) is typically spontaneous, as it meets both criteria favorably.\(
\)\(
\)In our exercise scenario, the positive \( \Delta H \) value implies we have an endothermic reaction. We use the knowledge of enthalpy change in conjunction with the Gibbs free energy concept to analyze the reaction's spontaneity. For the reaction to be spontaneous at a particular temperature, the relative magnitudes of enthalpy and entropy changes play a pivotal role.

Spontaneous Reaction

A spontaneous reaction is defined as a process that can proceed on its own, without needing to be driven by an external energy source. However, 'spontaneous' does not necessarily mean that the reaction occurs quickly—it could still be a slow process. Spontaneity depends on the thermodynamic properties of the reacting system, mainly the changes in enthalpy and entropy and the temperature at which the reaction occurs.\(
\)\(
\)The Gibbs free energy \( \Delta G \) serves as the criterion for spontaneity in a reaction. If \( \Delta G \) is negative, the process is spontaneous; if it's positive, the process is non-spontaneous and requires external energy to occur. The Gibbs free energy equation, \( \Delta G = \Delta H - T\Delta S \), beautifully integrates both enthalpy and entropy, describing the balance between the energy exchanged and the distribution of energy into disorder within a system.\(
\)\(
\)The exercise presented offers us a scenario where a reaction is spontaneously occurring at a specific temperature of 450 K with a known positive enthalpy change. To maintain spontaneity (\( \Delta G < 0 \) ), the entropy change \( \Delta S \) must compensate for the positive enthalpy value. Thus, we can conclude that the entropy change needs to be positive—in other words, the process must lead to an increase in disorder within the system. This resulting positive entropy change helps to drive the reaction forward spontaneously, even though the reaction itself is endothermic.\(
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\)Understanding these fundamental thermodynamic conditions for a reaction's spontaneity helps students grasp why a reaction proceeds the way it does and why external conditions such as temperature can dramatically affect the outcome.