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Chapter 14: Problem 33

Based on rate constant considerations, explain why the equilibrium constant depends on temperature.

Short Answer

Expert verified
The equilibrium constant depends on temperature because the rates of the forward and reverse reactions, which define the equilibrium, depend on temperature. This is seen through the Arrhenius equation, which relates the rate constant to temperature, and the Van't Hoff equation, which shows how the equilibrium constant changes with temperature. In an endothermic reaction, higher temperature favours formation of products and \(K\) increases, whereas in an exothermic reaction, higher temperature favours formation of reactants and \(K\) decreases.

Step by step solution

01

Recognise the Arrhenius Equation

The Arrhenius equation is given by \(k = Ae^{-E_{a}/RT}\), where \(k\) is the rate constant, \(A\) is the pre-exponential factor, \(E_{a}\) is the activation energy, \(R\) is the gas constant, and \(T\) is the temperature in Kelvin. This equation shows that the rate constant \(k\) is dependent on the temperature \(T\). As temperature increases, the rate constant also increases.
02

Understand Equilibrium and the Van't Hoff Equation

In a chemical reaction at equilibrium, the rates of the forward and reverse reactions are equal. The equilibrium constant \(K\) is given by the ratio of the concentrations of products to reactants, all raised to their stoichiometric coefficients. The Van't Hoff equation, derived from the change of Gibbs free energy with temperature, states that \(\ln K = -\Delta H / RT + \Delta S / R\), where \(\Delta H\) is the change in enthalpy, \(\Delta S\) is the change in entropy, and \(R\) and \(T\) are as defined before. This equation shows that the equilibrium constant \(K\) is also dependent on temperature \(T\).
03

Connect Arrhenius Equation and Van't Hoff Equation

Because what characterises equilibrium is the equality of rate constants for the forward and reverse reactions, and because the rate constants themselves depend on temperature according to the Arrhenius equation, it follows that the equilibrium constant \(K\), which expresses this equality of rates, also depends on temperature. In terms of the Van't Hoff equation, an increase in temperature can shift the equilibrium \(K\) depending on whether \(\Delta H\) is positive or negative. If \(\Delta H > 0\) (endothermic reaction), increasing temperature increases the equilibrium constant. If \(\Delta H < 0\) (exothermic reaction), increasing temperature decreases the equilibrium constant.

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Most popular questions from this chapter

Consider this equilibrium reaction in a closed container: $$\mathrm{CaCO}_{3}(s) \rightleftharpoons \mathrm{CaO}(s)+\mathrm{CO}_{2}(g)$$ What will happen if the following occurs? (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 a \(\mathrm{NaOH}\) solution are added to the mixture. (f) A few drops of a \(\mathrm{HCl}\) solution are added to the mixture (ignore the reaction between \(\mathrm{CO}_{2}\) and water). (g) Temperature is increased.

The vapor pressure of mercury is \(0.0020 \mathrm{mmHg}\) at \(26^{\circ} \mathrm{C}\). (a) Calculate \(K_{\mathrm{c}}\) and \(K_{P}\) for the process \(\mathrm{Hg}(l) \rightleftharpoons \mathrm{Hg}(g) .\) (b) A chemist breaks a thermometer and spills mercury onto the floor of a laboratory measuring \(6.1 \mathrm{~m}\) long, \(5.3 \mathrm{~m}\) wide, and \(3.1 \mathrm{~m}\) high. Calculate the mass of mercury (in grams) vaporized at equilibrium and the concentration of mercury vapor in \(\mathrm{mg} / \mathrm{m}^{3}\). Does this concentration exceed the safety limit of \(0.05 \mathrm{mg} / \mathrm{m}^{3} ?\) (Ignore the volume of furniture and other objects in the laboratory.)

A reaction vessel contains \(\mathrm{NH}_{3}, \mathrm{~N}_{2},\) and \(\mathrm{H}_{2}\) at equilibrium at a certain temperature. The equilibrium concentrations are \(\left[\mathrm{NH}_{3}\right]=0.25 M,\left[\mathrm{~N}_{2}\right]=0.11 M\) and \(\left[\mathrm{H}_{2}\right]=1.91 \mathrm{M} .\) Calculate the equilibrium constant \(K_{\mathrm{c}}\) for the synthesis of ammonia if the reaction is represented as (a) \(\mathrm{N}_{2}(g)+3 \mathrm{H}_{2}(g) \rightleftharpoons 2 \mathrm{NH}_{3}(g)\) (b) \(\frac{1}{2} \mathrm{~N}_{2}(g)+\frac{3}{2} \mathrm{H}_{2}(g) \rightleftharpoons \mathrm{NH}_{3}(g)\)

Determine the initial and equilibrium concentrations of HI if the initial concentrations of \(\mathrm{H}_{2}\) and \(\mathrm{I}_{2}\) are both \(0.16 M\) and their equilibrium concentrations are both \(0.072 M\) at \(430^{\circ} \mathrm{C}\). The equilibrium constant \(\left(K_{\mathrm{c}}\right)\) for the reaction \(\mathrm{H}_{2}(g)+\mathrm{I}_{2}(g) \rightleftharpoons 2 \mathrm{HI}(g)\) is 54.2 at \(430^{\circ} \mathrm{C}\)

A mixture of 0.47 mole of \(\mathrm{H}_{2}\) and 3.59 moles of \(\mathrm{HCl}\) is heated to \(2800^{\circ} \mathrm{C}\). Calculate the equilibrium partial pressures of \(\mathrm{H}_{2}, \mathrm{Cl}_{2},\) and \(\mathrm{HCl}\) if the total pressure is 2.00 atm. For the reaction $$\mathrm{H}_{2}(g)+\mathrm{Cl}_{2}(g) \rightleftharpoons 2 \mathrm{HCl}(g)$$ \(K_{P}\) is 193 at \(2800^{\circ} \mathrm{C}\)
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