Question

the $\delta h_{\text{dissolve}}$ for oxygen solubility in water at 310.0 k is -14.4 kj/mol, and the $\delta g_{\text{dissolve}}$ for this process is 19.0 kj/mol. calculate the $\delta s_{\text{dissolve}}$ for oxygen solubility in water at 310.0 k (body temperature), in j/mol·k.

Explanation:

Step1: Recall the Gibbs - Helmholtz equation

The Gibbs - Helmholtz equation is \(\Delta G=\Delta H - T\Delta S\). We need to solve for \(\Delta S\), so we can rearrange the formula to \(\Delta S=\frac{\Delta H-\Delta G}{T}\).

Step2: Convert units

First, convert \(\Delta H\) from kJ/mol to J/mol. Given \(\Delta H=- 14.4\space kJ/mol=-14.4\times1000\space J/mol = - 14400\space J/mol\), \(\Delta G = 19.0\space kJ/mol=19.0\times1000\space J/mol=19000\space J/mol\), and \(T = 310.0\space K\).

Step3: Substitute values into the formula

Substitute \(\Delta H\), \(\Delta G\) and \(T\) into the formula for \(\Delta S\):
\(\Delta S=\frac{\Delta H - \Delta G}{T}=\frac{- 14400\space J/mol-19000\space J/mol}{310.0\space K}\)

Step4: Calculate the numerator

First, calculate the numerator: \(-14400 - 19000=-33400\space J/mol\)

Step5: Calculate \(\Delta S\)

Then, divide the numerator by the temperature: \(\Delta S=\frac{-33400\space J/mol}{310.0\space K}\approx - 107.74\space J/(mol\cdot K)\)

Answer:

\(\approx - 108\space J/(mol\cdot K)\) (or more precisely \(- 107.7\space J/(mol\cdot K)\) depending on the level of precision)