Question

Calculate \(\Delta S^{\circ}\) for the reaction \(3 \mathrm{H}_{2}(g)+\mathrm{N}_{2}(g) \rightarrow\) \(2 \mathrm{NH}_{3}(g)\) at \(725 \mathrm{K} .\) Omit terms in the temperature-dependent heat capacities higher than \(T^{2} / \mathrm{K}^{2}\)

Short answer

The standard entropy change, \(\Delta S^\circ\), for the reaction \(3\mathrm{H}_{2}(g)+\mathrm{N}_{2}(g) \rightarrow 2 \mathrm{NH}_{3}(g)\) at 725K is -198.40 J/mol·K.

Step-by-step solution

Find the standard entropy values for the reactants and products.

In order to solve this problem, first, we need to gather the standard entropy values (\(\mathrm{S}_{\text{molar}}^\circ\)) of all reactants and products involved in the reaction. In this case, we have H₂(g), N₂(g), and NH₃(g). The values are as follows (given in textbook or reference tables): Sₘ⁰ (H₂) = 130.60 J/mol·K Sₘ⁰ (N₂) = 191.60 J/mol·K Sₘ⁰ (NH₃) = 192.50 J/mol·K

Apply the equation and compute the entropy change.

Now, we can apply the following equation to determine the standard entropy change, \(\Delta S^\circ\): \[\Delta S^\circ = \sum \text{ (moles of products * Sₘ⁰ of products)} - \sum \text{ (moles of reactants * Sₘ⁰ of reactants)}\] For this reaction, the equation will look like this: \[\Delta S^\circ = [2(\text{moles of NH₃}) * Sₘ⁰ (\text{NH₃})] - [ 3(\text{moles of H₂}) * Sₘ⁰ (\text{H₂}) + (\text{moles of N₂}) * Sₘ⁰ (\text{N₂}) ]\] Now plug in the values: \[\Delta S^\circ = [2(192.50 \ \mathrm{J/mol \cdot K})] - [ 3(130.60 \ \mathrm{J/mol \cdot K}) + (191.60 \ \mathrm{J/mol \cdot K}) ]\]

Calculate the value of ΔS⁰.

Now we can calculate the value of ΔS⁰ by performing the arithmetic operations. \[\Delta S^\circ = (385.00 \ \mathrm{J/mol \cdot K}) - (391.80 + 191.60 \ \mathrm{J/mol \cdot K})\] \[\Delta S^\circ = (385.00 - 583.40) \ \mathrm{J/mol \cdot K}\] \[\Delta S^\circ = -198.40 \ \mathrm{J/mol \cdot K}\] Therefore, the standard entropy change, \(\Delta S^\circ\), for the reaction at 725K is -198.40 J/mol·K.

Key concepts

Entropy

Entropy is a fundamental concept in thermodynamics, which describes the degree of disorder or randomness in a system. Entropy, represented by the symbol S, plays a vital role in determining the direction and feasibility of chemical reactions.
It provides a measure of the number of ways energy can be distributed among the molecules of a system, indicating how spread out or dispersed the energy is. For students, it’s helpful to think of entropy as how messy the system is becoming.
A key point to remember: an increase in entropy generally means more disorder.

• If the total entropy of a system and its surroundings increases, the process can occur spontaneously.
• Entropy is measured in units of joules per mole per kelvin (J/mol·K).
• In chemical reactions, the standard molar entropy \(S_{\text{molar}}^\circ\) is often used, indicating entropy under standard conditions, typically 1 atm and 298 K.

Understanding entropy helps in determining why certain reactions happen and others do not. It shifts perspectives from just looking at energy (enthalpy) to also considering randomness and dispersal of particles.

Reaction Entropy

Reaction entropy, represented as \(\Delta S^\circ\), is the change in entropy that occurs in a chemical reaction when the reactants form the products. This value indicates whether a reaction leads to an increase or decrease in the disorder of the system.
Here's how to calculate reaction entropy:

• The formula for calculating \(\Delta S^\circ\) in reactions is: \[\Delta S^\circ = \sum (\text{S}_{\text{molar}}^\circ \times \text{moles of products}) - \sum (\text{S}_{\text{molar}}^\circ \times \text{moles of reactants})\]
• For each substance, multiply its standard molar entropy by the number of moles in the balanced equation.
• Subtract the total entropy of reactants from the total entropy of products.

In this exercise, we calculated \(\Delta S^\circ\) for the reaction \(3\ \text{H}_{2}(g)+\text{N}_{2}(g) \rightarrow 2\ \text{NH}_{3}(g)\) and found that it results in a negative value of -198.40 J/mol·K. This implies a decrease in disorder, as the reaction favors the formation of a more ordered product, ammonia (NH₃), from gaseous hydrogen and nitrogen. This negative change reflects a shift towards reduced randomness as the smaller, lighter gases are combined into a more complex molecule.

Temperature-dependent Heat Capacities

Temperature-dependent heat capacities refer to the ability of a substance to absorb heat as the temperature changes, impacting the entropy of systems. In thermodynamics, adjusting for temperature effects is critical because heat capacity can vary with temperature.
For reactions taking place under various temperatures, this characteristic dictates how the entropy and enthalpy values might alter due to the additional thermal energy involved.

• When calculating changes in entropy over different temperatures, ignoring higher order terms like \(T^2/\text{K}^2\) simplifies calculations while providing reasonable accuracy under certain conditions.
• While keeping calculations simpler, this assumption usually works when temperature changes are mild or within a limited range.

In the provided exercise, such simplifications were applied by omitting terms in temperature-dependent heat capacities higher than \(T^2/\text{K}^2\). This allows students to focus on core concepts of reaction entropy change without getting lost in deeper complexities that higher-order terms might introduce. Remember, though simplifications can make calculations easier, they also require careful judgment about when and how they are used.