Chapter 22: Problem 48

Consider the formation of a dimeric protein $$ 2 \mathrm{P} \longrightarrow \mathrm{P}_{2} $$ At $25^{\circ} \mathrm{C},$ we have $\Delta H^{\circ}=17 \mathrm{~kJ} / \mathrm{mol}$ and $\Delta S^{\circ}=$ $65 \mathrm{~J} / \mathrm{K} \cdot \mathrm{mol}$. Is the dimerization favored at this temperature? Comment on the effect of lowering the temperature. Does your result explain why some enzymes lose their activities under cold conditions?

Short Answer

Expert verified

Dimerization is not favored at 25°C as $\Delta G^{\circ}$ is positive. Lowering the temperature will make $\Delta G^{\circ}$ more positive, disfavoring dimerization. This explains why some enzymes can lose their activities under cold conditions as their dimerization would be less likely.

Step by step solution

Understand Thermodynamic Favorability

A process is thermodynamically favorable, or spontaneous, when the change in Gibbs free energy $\Delta G^{\circ}$ is negative. This is calculated using the formula $\Delta G^{\circ} = \Delta H^{\circ} - T\Delta S^{\circ}$, where $\Delta H^{\circ}$ is the change in enthalpy, $T$ is the temperature in Kelvin, and $\Delta S^{\circ}$ is the change in entropy.

Compute Gibbs Free Energy Change at 25°C

To find out if the process is thermodynamically favored at 25°C, substitute the given values into the formula. Also, convert the temperature to Kelvin using $T(K) = T(°C) + 273.15$. So the temperature is $25 + 273.15 = 298.15 K$. Substituting gives $\Delta G^{\circ} = 17 kJ/mol - 298.15K * 65 J/mol * K / 1000$

Check If $\Delta G^{\circ}$ Is Negative

Calculate the value of $\Delta G^{\circ}$ to determine if dimerization is favored at this temperature. This is done by completing the subtraction to see if the result is negative.

Discuss the Effect of Lowering the Temperature

To comment on the effect of lowering the temperature, examine the equation $\Delta G^{\circ} = \Delta H^{\circ} - T\Delta S^{\circ}$. Lowering the temperature reduces the $T\Delta S^{\circ}$ term, thereby increasing the value of $\Delta G^{\circ}$. If $\Delta G^{\circ}$ is positive, the process is not thermodynamically favorable.

Relate Results to Enzyme Activity

Connect the above findings to enzyme activity in cold conditions. If dimerization is not favored in lower temperatures, as suggested by a positive $\Delta G^{\circ}$, then enzymes which require dimerization to be functional could indeed lose their activity under these conditions.

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Short Answer

Step by step solution

Understand Thermodynamic Favorability

Compute Gibbs Free Energy Change at 25°C

Check If \(\Delta G^{\circ}\) Is Negative

Discuss the Effect of Lowering the Temperature

Relate Results to Enzyme Activity

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