Chapter 18, Problem 15QP

Using the data in Appendix 2, calculate the standard entropy changes for the following reactions at $\text{25°C:}$

$\begin{array}{l}\text{(a)}\\ {\text{H}}_{\text{2}}\text{(}g\text{) + CuO(}s\text{) }\underset{}{\to }\text{ Cu(}s{\text{) + H}}_{\text{2}}\text{O(}g\text{)}\\ \text{(b)}\\ \text{2Al(}s\text{) + 3ZnO(}s\text{) }\underset{}{\to }{\text{ Al}}_{\text{2}}{\text{O}}_{\text{3}}\text{(}s\text{) + 3Zn(}s\text{)}\\ \text{(c)}\\ {\text{CH}}_{\text{4}}\text{(}g{\text{) + 2O}}_{\text{2}}\text{(}g\text{) }\underset{}{\to }{\text{ CO}}_{\text{2}}\text{(}g{\text{) + 2H}}_{\text{2}}\text{O(}l\text{)}\end{array}$

Interpretation Introduction

Interpretation:

The standard entropy change for the given reactionis to be calculated.

Concept introduction:

Entropy is the direct measurement of randomness or disorderness. Entropy is an extensive property. It is a state function.

Entropy of a system is a measure of how spread out or how dispersed the system’s energy is. Entropy change of a system is the difference between the entropy of the final state and the entropy of theinitial state.

The entropy of the system and the entropy of the surroundings make up the entropy of the universe.

Entropy change of the reaction is the difference between the entropies of the reactants and products.

Mathematically, the entropy change can be represented as follows: $\Delta S_{rxn}^o={\displaystyle \sum n}{S}^o\left(\text{product}\right)-{\displaystyle \sum m{S}^o}\left(\text{reactant}\right)$

Here, $\Delta S_{rxn}^o$ is the standard entropy change for the reaction, $\Delta S^o$ is the standard entropy change of the substance, ${\displaystyle \sum }$ is for summation, n is the stoichiometric coefficients of the products and m is the stoichiometric coefficients of the reactants.

Answer to Problem 15QP

Solution:

a)

$47.5\text{J}/\text{K mol}$

b)

$-12.5\text{J}/\text{K mol}$

c)

$-242.8\text{J}/\text{K mol}$

Explanation of Solution

a) ${\text{H}}_2\left(g\right)+\text{CuO}\left(s\right)\to \text{Cu}\left(s\right){\text{+H}}_2\text{O}\left(g\right)$

The standard entropy change for this reaction is calculated using the expression as follows:

$\Delta S_{rxn}^o={\displaystyle \sum n}{S}^o\left(\text{product}\right)-{\displaystyle \sum m{S}^o}\left(\text{reactant}\right)$

Here, $\Delta S_{rxn}^o$ is the standard entropy change for the reaction, $\Delta S^o$ is the standard entropy change of the substance, $n$ is the stoichiometric coefficient of product, and $m$ is the stoichiometric coefficient of reactant.

$\Delta S_{rxn}^o=\left(S^0\left[\text{Cu}\right]+S^o\left[{\text{H}}_2\text{O}\right]\right)-\left(S^o\left[{\text{H}}_2\right]+S^o\left[\text{CuO}\right]\right)$

From Appendix 2, the standard entropy value of the substance is as follows:

$S^o\left[\text{Cu}\left(s\right)\right]=33.3\text{J}/\text{K}\cdot \text{mol}$

$S^o\left[{\text{H}}_2\text{O}\left(g\right)\right]=188.7\text{J}/\text{K}\cdot \text{mol}$

$S^o\left[{\text{H}}_2\left(g\right)\right]=131.0\text{J}/\text{K}\cdot \text{mol}$

$S^o\left[\text{CuO}\left(s\right)\right]=43.5\text{J}/\text{K}\cdot \text{mol}$

Substitute, $33.3\text{J}/\text{K}\cdot \text{mol}$ for standard entropy of $\text{Cu}$, $188.7\text{J}/\text{K}\cdot \text{mol}$ for standard entropy of ${\text{H}}_2\text{O}$, $131.0\text{J}/\text{K}\cdot \text{mol}$ for standard entropy of ${\text{H}}_2$, and $43.5\text{J}/\text{K}\cdot \text{mol}$ for standard entropy of $\text{CuO}$ in the above expression as follows:

$\begin{array}{c}\Delta S_{rxn}^o=\left[\left(1\right)\times \left(\text{33}\text{.3}\text{J}/\text{K mol}\right)+\left(1\right)\times \left(\text{188}\text{.7}\text{J}/\text{K mol}\right)\right]-\\ \left[\left(1\right)\times \left(\text{131}\text{.0}\text{J}/\text{K mol}\right)+\left(1\right)\times \left(43.5\text{J}/\text{K mol}\right)\right]\\ =\left(222\text{ J/K mol}\right)-\left(\text{174}\text{.5 J/K mol}\right)\\ =47.5\text{J}/\text{K mol}\end{array}$

Therefore, the standard entropy change for this reaction is $47.5\text{J}/\text{K}\cdot \text{mol}$.

b) $\text{2Al}\left(s\right)+3\text{ZnO}\left(s\right)\to {\text{Al}}_2{\text{O}}_3\left(s\right)+\text{Zn}\left(g\right)$

The standard entropy change for the reaction is calculated using the expression as follows:

$\Delta S_{rxn}^o={\displaystyle \sum n}{S}^o\left(\text{product}\right)-{\displaystyle \sum m{S}^o}\left(\text{reactant}\right)$

Here, $\Delta S_{rxn}^o$ is the standard entropy change for the reaction, $\Delta S^o$ is the standard entropy change of the substance, $n$ is the stoichiometric coefficient of product, and $m$ is the stoichiometric coefficient of reactant.

The entropy change for the reaction is calculated by the expression as follows:

$\Delta S_{rxn}^o=\left(S^0\left[{\text{Al}}_2{\text{O}}_3\right]+3\times S^o\left[\text{Zn}\right]\right)-\left(2\times S^o\left[\text{Al}\right]+3\times S^o\left[\text{ZnO}\right]\right)$

From Appendix $2$, the standard entropy value of the substance is as follows:

$S^o\left[{\text{Al}}_2{\text{O}}_3\left(s\right)\right]=50.99\text{J}/\text{K}\cdot \text{mol}$

$S^o\left[\text{Zn}\left(s\right)\right]=41.6\text{J}/\text{K}\cdot \text{mol}$

$S^o\left[\text{Al}\left(s\right)\right]=28.3\text{J}/\text{K}\cdot \text{mol}$

$S^o\left[\text{ZnO}\left(s\right)\right]=43.9\text{J}/\text{K}\cdot \text{mol}$

Substitute, $50.99\text{J}/\text{K}\cdot \text{mol}$ for standard entropy of ${\text{Al}}_2{\text{O}}_3$, $41.6\text{J}/\text{K}\cdot \text{mol}$ for standard entropy of $\text{Zn}$, $28.3\text{J}/\text{K}\cdot \text{mol}$ for standard entropy of $\text{Al}$, and $43.9\text{J}/\text{K}\cdot \text{mol}$ for standard entropy of $\text{ZnO}$ in the above expression as follows:

$\begin{array}{c}\Delta S_{rxn}^o=\left[\left(1\right)\times \left(50.99\text{J}/\text{K}\cdot \text{mol}\right)+\left(3\right)\times \left(41.6\text{J}/\text{K}\cdot \text{mol}\right)\right]-\\ \left[\left(2\right)\times \left(\text{28}\text{.3}\text{J}/\text{K}\cdot \text{mol}\right)+\left(3\right)\times \left(\text{43}\text{.9}\text{J}/\text{K}\cdot \text{mol}\right)\right]\\ =\left(151.58\text{ J/K mol}\right)-\left(188.3\text{ J/K mol}\right)\\ =-12.5\text{J}/\text{K}\cdot \text{mol}\end{array}$

Therefore, the standard entropy change for this reaction is $-12.5\text{J}/\text{K}\cdot \text{mol}$.

c) ${\text{CH}}_4\left(g\right)+2{\text{O}}_2\left(g\right)\to {\text{CO}}_2\left(g\right)+2{\text{H}}_2\text{O}\left(l\right)$

The standard entropy change for this reaction is calculated using the expression as follows:

$\Delta S_{rxn}^o={\displaystyle \sum n}{S}^o\left(\text{product}\right)-{\displaystyle \sum m{S}^o}\left(\text{reactant}\right)$

Here, $\Delta S_{rxn}^o$ is the standard entropy change for the reaction, $S^o$ is the standard entropy change of the substance, $n$ is the stoichiometric coefficient of product, and $m$ is the stoichiometric coefficient of reactant.

$\Delta S_{rxn}^o=\left(S^0\left[{\text{CO}}_2\right]+\left(2\right)\times S^o\left[{\text{H}}_2\text{O}\right]\right)-\left(S^o\left[{\text{CH}}_4\right]+\left(2\right)\times S^o\left[{\text{O}}_2\right]\right)$

From Appendix 2, the standard entropy value of the substance is as follows:

$S^o\left[{\text{CO}}_2\left(g\right)\right]=213.6\text{J}/\text{K}\cdot \text{mol}$

$S^o\left[{\text{H}}_2\text{O}\left(l\right)\right]=69.9\text{J}/\text{K}\cdot \text{mol}$

$S^o\left[{\text{O}}_2\left(g\right)\right]=205.0\text{J}/\text{K}\cdot \text{mol}$

$S^o\left[{\text{CH}}_4\left(g\right)\right]=186.2\text{J}/\text{K}\cdot \text{mol}$

Substitute, $213.6\text{J}/\text{K}\cdot \text{mol}$ for standard entropy of ${\text{CO}}_2$, $69.9\text{J}/\text{K}\cdot \text{mol}$ for standard entropy of ${\text{H}}_2\text{O}$, $205.0\text{J}/\text{K}\cdot \text{mol}$ for standard entropy of ${\text{O}}_2$, and $186.2\text{J}/\text{K}\cdot \text{mol}$ for standard entropy of ${\text{CH}}_4$ in the above expressionas follows:

$\begin{array}{c}\Delta S_{rxn}^o=\left[\left(1\right)\times \left(\text{213}\text{.6}\text{J}/\text{K}\cdot \text{mol}\right)+\left(2\right)\times \left(\text{69}\text{.9}\text{J}/\text{K}\cdot \text{mol}\right)\right]-\\ \left[\left(1\right)\times \left(\text{186}\text{.2}\text{J}/\text{K}\cdot \text{mol}\right)+\left(2\right)\times \left(205.0\text{J}/\text{K}\cdot \text{mol}\right)\right]\\ =\left(353.4\text{ J/K mol}\right)-\left(596.2\text{ J/K mol}\right)\\ =-242.8\text{J}/\text{K}\cdot \text{mol}\end{array}$

Therefore, the standard entropy change for this reaction is $-242.8\text{J}/\text{K}\cdot \text{mol}$.