Chapter 15, Problem 21P

Interpretation Introduction

Interpretation:

The ratio of the reactants to the products under the intracellular as well as in the equilibrium conditions is to be stated. The way by which reaction becomes exergonic under intracellular condition and becomes endergonic under the standard condition is to be stated.

Concept introduction:

A thermodynamic potential that is utilized in the calculation of the highest reversible function taking place in the thermodynamic system at the constant value pressure and temperature is known as Gibbs-free energy. It is denoted by $\Delta G°$.

Answer to Problem 21P

The ratio of the reactants to the products under the intracellular condition is. $3.99\times {10}^{-5}$ and under the equilibrium condition is $6.742\times {10}^{-5}$.

Explanation of Solution

The reaction that is catalysis by the enzyme aldolase in the glycolytic pathway is given as follows.

$\begin{array}{c}\text{Fructose}1,6-\text{bisphosphatem}\rightleftharpoons \text{dihydroxyacetate}\text{phosphate}+\\ \text{glceraldehyde}3-\text{phosphate}\end{array}$

The value of standard $\Delta G°\text{'}$ for this given reaction is $+23.8\text{kJ}{\text{mol}}^{-1}$ or $+5.7\text{kcal}{\text{mol}}^{-1}$.

The value of $\Delta G$ inside the cell is $-1.3\text{kJ}{\text{mol}}^{-1}$ or $-0.3\text{kcal}{\text{mol}}^{-1}$.

The value of universal gas constant is $8.315\times {10}^{-3}\text{kJ}{\text{mol}}^{-1}{\text{K}}^{-1}$.

The standard temperature is $25°\text{C}$.

The conversion of degrees Celsius into Kelvin is done as,

$0°\text{C}=\text{273}\text{K}$

Thus, the given temperature becomes,

$\begin{array}{c}25°\text{C}=25+\text{273}\text{K}\\ =\text{298}\text{K}\end{array}$

The value of standard $\Delta G^{°\text{'}}$ under equilibrium condition for the given reaction is calculated by the expression as follows.

$\begin{array}{c}\Delta G^{°\text{'}}=-RT\ln K_{\text{eq}}\\ =-RT\ln \frac{\left[\text{Products}\right]}{\left[\text{Reactants}\right]}\end{array}$(1)

Where,

• $R$ is the universal gas constant.
• $T$ is the temperature.
• $K_{\text{eq}}$ is the equilibrium constant.

Substitute the values of universal gas constant, temperature and value of standard $\Delta G°\text{'}$ for this given reaction in equation (1).

$\begin{array}{c}+23.8\text{kJ}{\text{mol}}^{-1}=-8.315\times {10}^{-3}\text{kJ}{\text{mol}}^{-1}{\text{K}}^{-1}\times 298\text{K}\times \ln \frac{\left[\text{Products}\right]}{\left[\text{Reactants}\right]}\\ +23.8\text{kJ}{\text{mol}}^{-1}=-2.478\text{kJ}{\text{mol}}^{-1}\times \ln \frac{\left[\text{Products}\right]}{\left[\text{Reactants}\right]}\\ \frac{\left[\text{Products}\right]}{\left[\text{Reactants}\right]}={\text{e}}^{-9.6045}\\ =6.742\times {10}^{-5}\end{array}$

Thus, the ratio of the reactants to the products under equilibrium condition is $6.742\times {10}^{-5}$.

The value of $\Delta G$ under intracellular condition for the given reaction is calculated by the expression as follows.

$\begin{array}{c}\Delta G=\Delta G^{°\text{'}}+RT\ln K\\ =\Delta G^{°\text{'}}+RT\ln \frac{\left[\text{Products}\right]}{\left[\text{Reactants}\right]}\end{array}$(2)

Where,

• $\Delta G$ is the actual free energy change.
• $\Delta G^{°\text{'}}$ is the standard free-energy change.
• $R$ is the universal gas constant.
• $T$ is the temperature.

Substitute the values of universal gas constant, temperature and value of $\Delta G$ for this given reaction in equation (2).

$\begin{array}{c}-1.3\text{kJ}{\text{mol}}^{-1}=23.8\text{kJ}{\text{mol}}^{-1}+8.315\times {10}^{-3}\text{kJ}{\text{mol}}^{-1}{\text{K}}^{-1}\times 298\text{K}\times \ln \frac{\left[\text{Products}\right]}{\left[\text{Reactants}\right]}\\ \frac{-1.3\text{kJ}{\text{mol}}^{-1}-23.8\text{kJ}{\text{mol}}^{-1}}{8.315\times {10}^{-3}\text{kJ}{\text{mol}}^{-1}{\text{K}}^{-1}\times 298\text{K}}=\ln \frac{\left[\text{Products}\right]}{\left[\text{Reactants}\right]}\\ \frac{\left[\text{Products}\right]}{\left[\text{Reactants}\right]}={\text{e}}^{-10.13}\\ =3.99\times {10}^{-5}\end{array}$

Thus, the ratio of the reactants to the products under equilibrium condition is $6.742\times {10}^{-5}$ and under the intracellular condition is. $3.99\times {10}^{-5}$.

Thus, according to the ratio of products to the reactants, if this ratio is kept lower than the equilibrium value then this given reaction, which is endergonic under the standard conditions, is able to convert itself into exergonic.

This type of transformation is obtained only if the product of the given reaction is used in some other reaction after its immediate formation.

Conclusion

The ratio of the reactants to the products under the intracellular condition is. $3.99\times {10}^{-5}$ and under the equilibrium condition is $6.742\times {10}^{-5}$.