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Chapter 18: Problem 87

A student looked up the \(\Delta G_{\mathrm{f}}^{\circ}, \Delta H_{\mathrm{f}}^{\circ}\), and \(S^{\circ}\) values for \(\mathrm{CO}_{2}\) in Appendix 2. Plugging these values into Equation \((18.10),\) he found that \(\Delta G_{\mathrm{f}}^{\circ} \neq \Delta H_{\mathrm{f}}^{\circ}-T S^{\circ}\) at \(298 \mathrm{~K}\). What is wrong with his approach?

Short Answer

Expert verified
The student's approach is incorrect because he used standard state values (\(\Delta G_{\mathrm{f}}^{\circ}\), \(\Delta H_{\mathrm{f}}^{\circ}\), and \(S^{\circ}\)) at a non-standard temperature condition. This approach neglects the temperature dependence of these quantities.

Step by step solution

01

Understanding Standard State Variables

It's crucial to understand that the variables \(\Delta G_{\mathrm{f}}^{\circ}\), \(\Delta H_{\mathrm{f}}^{\circ}\) and \(S^{\circ}\) are standard state values (with the circle superscript denoting standard state conditions). They represent Gibbs energy, enthalpy, and entropy at standard conditions (1 bar pressure, specific standard state concentration and temperature - usually 298 K in many tables). The student, in using these values, has assumed that the change in these variables is linear with respect to temperature, which is a common misuse.
02

Identify the Error

The equation (the Gibbs-Helmholtz equation) \(\Delta G_{\mathrm{f}}^{\circ} = \Delta H_{\mathrm{f}}^{\circ}-T S^{\circ}\) can be used to calculate the Gibbs free energy based on enthalpy and entropy values. But, this equation only applies to the standard states (where the temperature is 298 K in many tables). The student has committed an error by using the given standard state values to calculate Gibbs free energy at a different temperature.
03

Clarifying Correct Approach

In practice, to calculate Gibbs free energy, enthalpy or entropy at any temperature, we would generally use temperature-dependent equations or specific heat capacities which are beyond the scope of this specific question. To check the correctness of calculated \(\Delta G\) at any temperature, one has to rely on temperature dependence of these parameters, which often require more advanced thermodynamic understanding and resources like heat capacity data or empirical relations for temperature dependences which might not be readily available.

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Most popular questions from this chapter

Hydrogenation reactions (for example, the process of converting \(\mathrm{C}=\mathrm{C}\) bonds to \(\mathrm{C}-\mathrm{C}\) bonds in food industry) are facilitated by the use of a transition metal catalyst, such as Ni or \(\mathrm{Pt}\). The initial step is the adsorption, or binding, of hydrogen gas onto the metal surface. Predict the signs of \(\Delta H, \Delta S,\) and \(\Delta G\) when hydrogen gas is adsorbed onto the surface of Ni metal.

Use the following data to determine the normal boiling point, in kelvins, of mercury. What assumptions must you make in order to do the calculation? $$ \begin{aligned} \mathrm{Hg}(l): & \Delta H_{\mathrm{f}}^{\circ} &=0 \text { (by definition) } \\\ & S^{\circ} &=77.4 \mathrm{~J} / \mathrm{K} \cdot \mathrm{mol} \\ \mathrm{Hg}(g): & \Delta H_{\mathrm{f}}^{\circ} &=60.78 \mathrm{~kJ} / \mathrm{mol} \\ & S^{\circ} &=174.7 \mathrm{~J} / \mathrm{K} \cdot \mathrm{mol} \end{aligned} $$

Comment on the correctness of the analogy sometimes used to relate a student's dormitory room becoming untidy to an increase in entropy.

The equilibrium constant \(\left(K_{P}\right)\) for the reaction $$ \mathrm{H}_{2}(g)+\mathrm{CO}_{2}(g) \rightleftharpoons \mathrm{H}_{2} \mathrm{O}(g)+\mathrm{CO}(g) $$ is 4.40 at \(2000 \mathrm{~K}\). (a) Calculate \(\Delta G^{\circ}\) for the reaction. (b) Calculate \(\Delta G\) for the reaction when the partial pressures are \(P_{\mathrm{H}_{2}}=0.25 \mathrm{~atm}, P_{\mathrm{CO}_{2}}=0.78 \mathrm{~atm}\) \(P_{\mathrm{H}_{2} \mathrm{O}}=0.66 \mathrm{~atm},\) and \(P_{\mathrm{CO}}=1.20 \mathrm{~atm}\)

Ammonium nitrate \(\left(\mathrm{NH}_{4} \mathrm{NO}_{3}\right)\) dissolves spontaneously and endothermically in water. What can you deduce about the sign of \(\Delta S\) for the solution process?
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