The table shows the variation of lnK with T for the...

The table shows the variation of $\ln K$ with $T$ for the equilibrium: $$\begin{aligned} &\mathrm{S}+\mathrm{O}_{2} \rightleftharpoons \mathrm{SO}_{2}\\ &\begin{array}{llllll} \hline T / \mathrm{K} & 298 & 400 & 500 & 700 & 900 \\ \ln K & 121.1 & 90.49 & 72.36 & 51.44 & 39.55 \\ \hline \end{array} \end{aligned}$$ (a) Write down an expression for $K$ indicating any assumptions made. (b) Use the data to estimate values of $K$ at 300 and $600 \mathrm{K}$ (c) Determine $\Delta_{\mathrm{f}} G^{\mathrm{O}}\left(\mathrm{SO}_{2}, \mathrm{g}\right)$ at $300 \mathrm{K}$ (d) Values of $S^{\circ}$ for $\mathrm{S}(\mathrm{s}), \mathrm{O}_{2}(\mathrm{g})$ and $\mathrm{SO}_{2}(\mathrm{g})$ at $300 \mathrm{K}$ are 32.2,205.3 and $248.5 \mathrm{JK}^{-1} \mathrm{mol}^{-1}$. Determine $\Delta_{1} H\left(\mathrm{SO}_{2}, \mathrm{g}\right)$ at $300 \mathrm{K}$ (e) Given that $S^{0}\left(\mathrm{SO}_{2}, \mathrm{g}, 400 \mathrm{K}\right)=$ $260.4 \mathrm{JK}^{-1} \mathrm{mol}^{-1},$ determine $C_{P}$ for $\mathrm{SO}_{2}(\mathrm{g})$ Comment on the validity of your answer.

The table shows the variation of $\ln K$ with $T$ for the equilibrium:
$$\begin{aligned}
&\mathrm{S}+\mathrm{O}_{2} \rightleftharpoons \mathrm{SO}_{2}\\
&\begin{array}{llllll}
\hline T / \mathrm{K} & 298 & 400 & 500 & 700 & 900 \\
\ln K & 121.1 & 90.49 & 72.36 & 51.44 & 39.55 \\
\hline
\end{array}
\end{aligned}$$
(a) Write down an expression for $K$ indicating any assumptions made.
(b) Use the data to estimate values of $K$ at 300 and $600 \mathrm{K}$
(c) Determine $\Delta_{\mathrm{f}} G^{\mathrm{O}}\left(\mathrm{SO}_{2}, \mathrm{g}\right)$ at $300 \mathrm{K}$
(d) Values of $S^{\circ}$ for $\mathrm{S}(\mathrm{s}), \mathrm{O}_{2}(\mathrm{g})$ and $\mathrm{SO}_{2}(\mathrm{g})$ at
$300 \mathrm{K}$ are 32.2,205.3 and $248.5 \mathrm{JK}^{-1} \mathrm{mol}^{-1}$.
Determine $\Delta_{1} H\left(\mathrm{SO}_{2}, \mathrm{g}\right)$ at $300 \mathrm{K}$
(e) Given that $S^{0}\left(\mathrm{SO}_{2}, \mathrm{g}, 400 \mathrm{K}\right)=$
$260.4 \mathrm{JK}^{-1} \mathrm{mol}^{-1},$ determine $C_{P}$ for $\mathrm{SO}_{2}(\mathrm{g})$
Comment on the validity of your answer.

Key Concepts

Equilibrium Constant Expression

This is the quantitative measure of the amount of products versus reactants present at equilibrium for a reversible chemical reaction. In defining it, one typically raises the activity (or concentration or partial pressure) of each species to the power of its stoichiometric coefficient. An assumption often made is that the pure phases (like solids or liquids) have an activity of one and that ideal behavior applies, simplifying the expression.

van't Hoff Equation

The van't Hoff equation relates the change in the equilibrium constant with temperature to the standard enthalpy change of the reaction. It is expressed as d(ln K)/dT = ?H°/(RT^2), or in integrated form, ln K vs. 1/T yields a straight line whose slope is proportional to -?H°/R. This relationship is useful when experimental data at different temperatures are available, as it allows for the estimation of thermodynamic quantities and interpolation of K at unmeasured temperatures.

Gibbs Free Energy and Equilibrium

The standard Gibbs free energy change of a reaction is directly related to its equilibrium constant by the equation ?G° = -RT ln K. This equation is fundamental to chemical thermodynamics as it connects the macroscopic equilibrium behavior of a chemical system with its molecular energetics. A negative ?G° indicates that the reaction proceeds spontaneously under standard conditions.

Relationship Between Enthalpy, Entropy, and Gibbs Free Energy

The interdependence of these thermodynamic state functions is captured by the relation ?G° = ?H° - T?S°. This equation allows one to determine any one of these thermodynamic quantities if the other two are known. It is critical in evaluating the feasibility of a reaction, understanding temperature effects on reaction spontaneity, and linking calorimetric data with equilibrium data.

Temperature Dependence and Data Interpolation

When experimental data is available at different temperatures, interpolation (or extrapolation) techniques are used to estimate values at temperatures where direct measurements are unavailable. Particularly in thermodynamics, linear relationships (as suggested by the van't Hoff plot) can be used for such predictions, assuming that properties like ?H° do not change significantly over the temperature range considered.

Heat Capacity and Entropy Relationship

The heat capacity at constant pressure (C_P) is related to the temperature dependence of entropy. Through the relation dS/dT = C_P/T, one can estimate how the entropy of a substance changes with temperature. This concept is essential when dealing with variations in state functions with temperature, and integrating C_P over a temperature range allows one to calculate changes in entropy accurately.

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