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Chapter 18: Problem 24

For the autoionization of water at \(25^{\circ} \mathrm{C}\), $$ \mathrm{H}_{2} \mathrm{O}(l) \rightleftharpoons \mathrm{H}^{+}(a q)+\mathrm{OH}^{-}(a q) $$ \(K_{\mathrm{w}}\) is \(1.0 \times 10^{-14}\). What is \(\Delta G^{\circ}\) for the process?

Short Answer

Expert verified
\(\Delta G^{\circ} = 77.99 \, \mathrm{kJ} \, \mathrm{mol}^{-1}\)

Step by step solution

01

Convert the temperature from Celsius to Kelvin

The problem specifies a temperature of \(25^{\circ} \mathrm{C}\). To convert this to Kelvin, add 273.15, therefore, \( T = 25 + 273.15 = 298.15 \, \mathrm{K}\).
02

Substitute the given values into the equation

The equation for \(\Delta G^{\circ}\) is \(\Delta G^{\circ} = -RT \ln K\). Substituting \(\ln K_{\mathrm{w}} = \ln (1.0 \times 10^{-14})\), \(R = 8.314 \, \mathrm{J} \, \mathrm{mol}^{-1} \, \mathrm{K}^{-1}\) (as is standard in such calculations) and \(T = 298.15 \, \mathrm{K}\), the equation becomes \(\Delta G^{\circ} = -8.314 \times 298.15 \times \ln (1.0 \times 10^{-14})\).
03

Solve for \(\Delta G^{\circ}\)

Perform the multiplication and the natural logarithm operation to find the numerical value: \(\Delta G^{\circ} = -8.314 \times 298.15 \times (-31.42) = 77.99 \, \mathrm{kJ} \, \mathrm{mol}^{-1}\).

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Most popular questions from this chapter

What is the role of ATP in biological reactions?

Explain the difference between \(\Delta G\) and \(\Delta G^{\circ}\).

Large quantities of hydrogen are needed for the synthesis of ammonia. One preparation of hydrogen involves the reaction between carbon monoxide and steam at \(300^{\circ} \mathrm{C}\) in the presence of a copper-zinc catalyst: $$ \mathrm{CO}(g)+\mathrm{H}_{2} \mathrm{O}(g) \rightleftharpoons \mathrm{CO}_{2}(g)+\mathrm{H}_{2}(g) $$.

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For each pair of substances listed here, choose the one having the larger standard entropy value at \(25^{\circ} \mathrm{C}\). The same molar amount is used in the comparison. Explain the basis for your choice. (a) \(\operatorname{Li}(s)\) or \(\operatorname{Li}(l)\) (b) \(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}(l)\) or \(\mathrm{CH}_{3} \mathrm{OCH}_{3}(l)\) (c) \(\operatorname{Ar}(g)\) or \(\operatorname{Xe}(g)\) (d) \(\mathrm{CO}(g)\) or \(\mathrm{CO}_{2}(g)\) (e) \(\mathrm{O}_{2}(g)\) or \(\mathrm{O}_{3}(g)\) (f) \(\mathrm{NO}_{2}(g)\) or \(\mathrm{N}_{2} \mathrm{O}_{4}(g)\)
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