Problem 45 - Advanced Problems in Physical Chemistry for Competitive Examinations

Question

When \(\alpha\) -D-glucose is dissolved in water, it undergoes a partial conversion to \beta-D-glucose. This conversion, called mutarotation, stops when \(64.0 \%\) of the glucose is in the \(\beta\) -form. Assuming that equilibrium has been attained, what is \(\Delta G^{\circ}\) for the reaction: \(\alpha\) -D-glucose \(\rightleftharpoons \beta-\mathrm{D}\) glucose, at this experimental temperature? (a) \(-R T \log _{10}(1.6)\) (b) \(-R T \log _{10}(1.78)\) (c) \(-R T \log _{e}(1.78)\) (d) \(-R T \log _{e}(1.6)\)

Step-by-Step Solution

Answer

(c) \(\Delta G^{\circ} = -RT \log_e(1.78)\)

Step 1: Understand the Equilibrium and Find the Ratio

At equilibrium, the ratio of \beta-D-glucose to \(\alpha\)-D-glucose is given by the percentage that has converted: 64.0\% is \(\beta\)-form and 36.0\% (\(100\%-64.0\%\)) is \(\alpha\)-form. Thus, the ratio of \(\beta\) to \(\alpha\) is \(\frac{64.0}{36.0} = 1.78\).

Step 2: Calculate the Equilibrium Constant \(K\)

The equilibrium constant \(K\) is the ratio of the concentrations of the products to the reactants at equilibrium. In this case, \(K = \frac{[\beta-D-glucose]}{[\alpha-D-glucose]} = 1.78\).

Step 3: Use the Relationship Between \(\Delta G^{\circ}\) and \(K\)

The standard free energy change \(\Delta G^{\circ}\) can be calculated using the formula \(\Delta G^{\circ} = -RT\ln(K)\), where \(R\) is the universal gas constant, \(T\) is the temperature in Kelvin, and \(\ln\) denotes the natural logarithm.

Step 4: Identify the Correct Expression for \(\Delta G^{\circ}\)

Since \(K = 1.78\), we take the natural logarithm of \(K\) to find \(\Delta G^{\circ}\). Therefore, the correct expression for \(\Delta G^{\circ}\) is based on the value of \(K\):

Step 5: Match the Expression to the Given Options

The correct expression involves taking the natural logarithm, not the logarithm base 10. Hence, the options with \(\log_{10}\) can be immediately ruled out, leaving only the options with \(\log_e\), which is the same as \(\ln\). Therefore, the correct answer is option (c) \(\Delta G^{\circ} = -RT \ln(1.78)\).

Key Concepts

Chemical Equilibrium

In chemistry, chemical equilibrium occurs when a reversible reaction has reached a state where the concentrations of the reactants and products no longer change over time. At this point, the rate of the forward reaction equals the rate of the backward reaction. Although it may seem like the reaction has stopped, it is actually ongoing; molecules continue to react, but at a rate that sustains a balance.

Understanding chemical equilibrium is crucial because it helps predict the outcome of a chemical reaction under specific conditions. It also explains why, in the mutarotation of \(\alpha\)-D-glucose to \(\beta\)-D-glucose, the reaction stops with a specific proportion of each isomer in the mixture. By finding the ratio of \(\beta\)-D-glucose to \(\alpha\)-D-glucose at equilibrium, which is 1.78 in this case, we can deduce the equilibrium constant and further analyze the reaction dynamics.

Standard Free Energy Change

Standard free energy change (\(\Delta G^\circ\)) is a thermodynamic quantity that indicates the spontaneity of a chemical reaction at standard conditions, typically defined as 1 bar of pressure for each gas, 1 M concentration for each solution, and a specified temperature, usually 25°C (298 K). A negative \(\Delta G^\circ\) value implies that the reaction can occur spontaneously under these standard conditions.

The free energy change is directly associated with the equilibrium constant (K) via the relationship \(\Delta G^\circ = -RT\ln(K)\), where \(R\) is the universal gas constant and \(T\) is the temperature in Kelvin. The natural logarithm (\(\ln\)) of the equilibrium constant indicates whether the products or reactants are favored at equilibrium. In the mutarotation exercise, calculating \(\Delta G^\circ\) requires knowing the equilibrium constant and applying the formula appropriately.

Equilibrium Constant

The equilibrium constant (K) for a chemical reaction is the ratio of the product of the concentrations of the products to the product of the concentrations of the reactants, each raised to the power of their respective coefficients in the balanced chemical equation. It is a unitless number that provides a quantitative measure of the position of equilibrium. In the reaction involving \(\alpha\)-D-glucose and \(\beta\)-D-glucose, since there are no coefficients other than one, the equilibrium constant \(K\) simply equals the ratio of concentrations of \(\beta\) to \(\alpha\) forms, which is 1.78.

An essential characteristic of K is that it is constant for a given reaction at a constant temperature. Thus, it can be used to predict the direction of the reaction or calculate the concentrations of reactants and products at equilibrium when the reaction has reached a steady state. The equilibrium constant is also a key factor in determining the standard free energy change for the reaction.

Natural Logarithm

The natural logarithm is a mathematical function represented by \(\ln\), which is the inverse of the exponential function with base \(e\), where \(e\) is Euler's number (approximately equal to 2.71828). This function is vital in various scientific and mathematical applications because it helps describe growth processes, decay, time constants, and many phenomena in physics and chemistry.

In thermodynamics and kinetics, the natural logarithm of the equilibrium constant \(K\) is utilized to relate this variable to the standard free energy change (\(\Delta G^\circ\)) of a reaction. This relationship is expressed as \(\Delta G^\circ = -RT\ln(K)\), as shown in the solution to the mutarotation problem. Choosing the natural logarithm instead of the logarithm to the base ten (\(\log_{10}\)) is important because the formula requires it, and using \(\ln\) ensures that the units of R (energy per mole per Kelvin) are compatible with the units of \(\Delta G^\circ\) (energy per mole).