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Chapter 15: Problem 98

(a) Use the van't Hoff equation in Problem 15.97 to derive the following expression, which relates the equilibrium constants at two different temperatures $$ \ln \frac{K_{1}}{K_{2}}=\frac{\Delta H^{\circ}}{R}\left(\frac{1}{T_{2}}-\frac{1}{T_{1}}\right) $$ How does this equation support the prediction based on Le Châtelier's principle about the shift in equilibrium with temperature? (b) The vapor pressures of water are \(31.82 \mathrm{mmHg}\) at \(30^{\circ} \mathrm{C}\) and \(92.51 \mathrm{mmHg}\) at \(50^{\circ} \mathrm{C} .\) Calculate the molar heat of vaporization of water.

Short Answer

Expert verified
Deriving from the van't Hoff equation, we get \(\ln \frac{K_{1}}{K_{2}}=\frac{\Delta H^{\circ}}{R}\left(\frac{1}{T_{2}}-\frac{1}{T_{1}}\right)\), which effectively shows how equilibrium constant changes with temperature. This supports Le Châtelier's principle, predicting shifts in equilibrium with temperature changes. The molar heat of vaporization can be calculated using the Clausius–Clapeyron equation, given the vapor pressures at two temperature points.

Step by step solution

01

Understanding Van't Hoff Equation and Le Châtelier's Principle

The Van't Hoff equation demonstrates the temperature dependence of equilibrium constants. Le Châtelier's principle states that if a system in equilibrium suffers a disturbance (such as temperature and pressure changes), the system will adjust to partially counter that disturbance. In this context, if \(\Delta H^{\circ}\) is positive (meaning an endothermic reaction), the value of K (equilibrium constant) will increase as the temperature(T) rises to absorb excess heat, according to Le Châtelier's principle. Thus, the van’t Hoff equation gives a quantitative means of expressing this dependence.
02

Deriving the given expression

The Van 't Hoff equation is generally written as \(\frac {d(lnK)}{dT}=\frac {\Delta H^{\circ}}{RT^2}\). Integrating this between two temperatures \(T_{1}\) and \(T_{2}\) with corresponding equilibrium constants \(K_{1}\) and \(K_{2}\) would lead to the required expression.
03

Calculate the molar heat of vaporization

The Clausius–Clapeyron equation can be used to find the molar heat of vaporization (\(\Delta H^{\circ}\)). This is given as \(\ln \frac{P_{2}}{P_{1}}=-\frac{\Delta H^{\circ}}{R}\left(\frac{1}{T_{2}}-\frac{1}{T_{1}}\right)\), where \(P_{1}\) and \(P_{2}\) are the vapor pressures at temperatures \(T_{1}\) and \(T_{2}\). Plug in the given pressure and temperature values to find \(\Delta H^{\circ}\). Remember to convert the temperatures to Kelvin before substituting the values in to the formula.

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Most popular questions from this chapter

Outline the steps for calculating the concentrations of reacting species in an equilibrium reaction.

The decomposition of ammonium hydrogen sulfide $$ \mathrm{NH}_{4} \mathrm{HS}(s) \rightleftharpoons \mathrm{NH}_{3}(g)+\mathrm{H}_{2} \mathrm{~S}(g) $$ is an endothermic process. A \(6.1589-\mathrm{g}\) sample of the solid is placed in an evacuated 4.000 - \(L\) vessel at exactly \(24^{\circ} \mathrm{C}\). After equilibrium has been established, the total pressure inside is \(0.709 \mathrm{~atm}\). Some solid \(\mathrm{NH}_{4} \mathrm{HS}\) remains in the vessel. (a) What is the \(K_{P}\) for the reaction? (b) What percentage of the solid has decomposed? (c) If the volume of the vessel were doubled at constant temperature, what would happen to the amount of solid in the vessel?

Consider this reaction at equilibrium in a closed container: $$ \mathrm{CaCO}_{3}(s) \rightleftharpoons \mathrm{CaO}(s)+\mathrm{CO}_{2}(g) $$ What would happen if (a) the volume is increased, (b) some \(\mathrm{CaO}\) is added to the mixture, (c) some \(\mathrm{CaCO}_{3}\) is removed, (d) some \(\mathrm{CO}_{2}\) is added to the mixture, (e) a few drops of an \(\mathrm{NaOH}\) solution are added to the mixture, (f) a few drops of an \(\mathrm{HCl}\) solution are added to the mixture (ignore the reaction between \(\mathrm{CO}_{2}\) and water \(),(\mathrm{g})\) the temperature is increased?

What do the symbols \(K_{c}\) and \(K_{P}\) represent?

The equilibrium constant \(K_{\mathrm{c}}\) for the following reaction is 0.65 at \(395^{\circ} \mathrm{C}\). $$ \mathrm{N}_{2}(g)+3 \mathrm{H}_{2}(g) \rightleftharpoons 2 \mathrm{NH}_{3}(g) $$ (a) What is the value of \(K_{P}\) for this reaction? (b) What is the value of the equilibrium constant \(K_{\mathrm{c}}\) for \(2 \mathrm{NH}_{3}(g) \rightleftharpoons \mathrm{N}_{2}(g)+3 \mathrm{H}_{2}(g) ?\) (c) What is \(K_{\mathrm{c}}\) for \(\frac{1}{2} \mathrm{~N}_{2}(g)+\frac{3}{2} \mathrm{H}_{2}(g) \rightleftharpoons \mathrm{NH}_{3}(g) ?\) (d) What are the values of \(K_{P}\) for the reactions described in (b) and (c)?
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