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Chapter 15: Problem 97

Write the equilibrium-constant expression for the equilibrium $$\mathrm{C}(s)+\mathrm{CO}_{2}(g) \rightleftharpoons 2 \mathrm{CO}(g)$$ The table that follows shows the relative mole percentages of \(\mathrm{CO}_{2}(g)\) and \(\mathrm{CO}(g)\) at a total pressure of 1 atm for several temperatures. Calculate the value of \(K_{p}\) at each temperature. Is the reaction exothermic or endothermic? Explain. $$\begin{array}{rll}\hline{\text { Temperature }\left({ }^{\circ} \mathrm{C}\right)} & \mathrm{CO}_{2}(\mathrm{~mol} \%) & \mathrm{CO}(\mathrm{mol} \%) \\ \hline 850 & 6.23 & 93.77 \\ 950 & 1.32 & 98.68 \\\1050 & 0.37 & 99.63 \\ 1200 & 0.06 & 99.94 \\\\\hline\end{array}$$

Short Answer

Expert verified
The equilibrium constant expression \(K_{p}\) for the given reaction is \(K_{p} = \frac{P_{\mathrm{CO}}^2}{P_{\mathrm{CO}_2}}\). Upon calculating the values of \(K_{p}\) at different temperatures, we find that it increases as the temperature rises, indicating that the reaction is endothermic.

Step by step solution

01

Write the equilibrium constant expression

For the given reaction: $$\mathrm{C}(s)+\mathrm{CO}_{2}(g) \rightleftharpoons 2 \mathrm{CO}(g)$$ We can write the equilibrium constant expression (\(K_{p}\)) using partial pressures as follows: $$K_{p} = \frac{P_{\mathrm{CO}}^2}{P_{\mathrm{CO}_2}}$$
02

Convert mole percentages to partial pressures

We are given the mole percentages of \(\mathrm{CO}_{2}(g)\) and \(\mathrm{CO}(g)\), and the total pressure is 1 atm. We can convert these mole percentages to partial pressures using the fractions of the total pressure: $$P_{\mathrm{CO}_2} = (\text{mol % of CO}_2) \times 1~\text{atm}$$ $$P_{\mathrm{CO}} = (\text{mol % of CO}) \times 1~\text{atm}$$
03

Calculate the value of \(K_{p}\) at each temperature

Using the partial pressures calculated in Step 2, we can substitute them into the \(K_{p}\) equation for each temperature: $$850^\circ C: \, K_p = \frac{P_{\mathrm{CO}}^2}{P_{\mathrm{CO}_2}} = \frac{(0.9377)^2}{0.0623} \approx 14.17$$ $$950^\circ C: \, K_p = \frac{P_{\mathrm{CO}}^2}{P_{\mathrm{CO}_2}} = \frac{(0.9868)^2}{0.0132} \approx 73.31$$ $$1050^\circ C: \, K_p = \frac{P_{\mathrm{CO}}^2}{P_{\mathrm{CO}_2}} = \frac{(0.9963)^2}{0.0037} \approx 267.80$$ $$1200^\circ C: \, K_p = \frac{P_{\mathrm{CO}}^2}{P_{\mathrm{CO}_2}} = \frac{(0.9994)^2}{0.0006} \approx 1666.02$$
04

Determine if the reaction is exothermic or endothermic

To determine if the reaction is exothermic or endothermic, we need to consider the effect of temperature on the equilibrium constant \(K_p\). If an increase in temperature causes the value of \(K_p\) to increase, the reaction is endothermic. If an increase in temperature causes the value of \(K_p\) to decrease, the reaction is exothermic. From our calculations in Step 3, we can see that the value of \(K_p\) is increasing with rising temperature. Therefore, the reaction is endothermic.

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Most popular questions from this chapter

Water molecules in the atmosphere can form hydrogenbonded dimers, \(\left(\mathrm{H}_{2} \mathrm{O}\right)_{2} .\) The presence of these dimers is thought to be important in the nucleation of ice crystals in the atmosphere and in the formation of acid rain. (a) Using VSEPR theory, draw the structure of a water dimer, using dashed lines to indicate intermolecular interactions. (b) What kind of intermolecular forces are involved in water dimer formation? (c) The \(K_{p}\) for water dimer formation in the gas phase is 0.050 at \(300 \mathrm{~K}\) and 0.020 at \(350 \mathrm{~K}\). Is water dimer formation endothermic or exothermic?

At \(2000^{\circ} \mathrm{C}\) the equilibrium constant for the reaction $$2 \mathrm{NO}(g) \rightleftharpoons \mathrm{N}_{2}(g)+\mathrm{O}_{2}(g)$$ is \(K_{c}=2.4 \times 10^{3}\). If the initial concentration of \(\mathrm{NO}\) is \(0.175 \mathrm{M}\) what are the equilibrium concentrations of \(\mathrm{NO}, \mathrm{N}_{2},\) and \(\mathrm{O}_{2} ?\)

If \(K_{c}=0.042\) for \(\mathrm{PCl}_{3}(g)+\mathrm{Cl}_{2}(g) \rightleftharpoons \mathrm{PCl}_{5}(g)\) at \(500 \mathrm{~K}\) what is the value of \(K_{p}\) for this reaction at this temperature?

Write the expression for \(K_{c}\) for the following reactions. In each case indicate whether the reaction is homogeneous or heterogeneous. (a) \(3 \mathrm{NO}(g) \rightleftharpoons \mathrm{N}_{2} \mathrm{O}(g)+\mathrm{NO}_{2}(g)\) (b) \(\mathrm{CH}_{4}(g)+2 \mathrm{H}_{2} \mathrm{~S}(g) \rightleftharpoons \mathrm{CS}_{2}(g)+4 \mathrm{H}_{2}(g)\) (c) \(\mathrm{Ni}(\mathrm{CO})_{4}(g) \rightleftharpoons \mathrm{Ni}(s)+4 \mathrm{CO}(g)\) (d) \(\mathrm{HF}(a q) \rightleftharpoons \mathrm{H}^{+}(a q)+\mathrm{F}^{-}(a q)\) (e) \(2 \mathrm{Ag}(s)+\mathrm{Zn}^{2+}(a q) \rightleftharpoons 2 \mathrm{Ag}^{+}(a q)+\mathrm{Zn}(s)\) (f) \(\mathrm{H}_{2} \mathrm{O}(l) \rightleftharpoons \mathrm{H}^{+}(a q)+\mathrm{OH}^{-}(a q)\) (g) \(2 \mathrm{H}_{2} \mathrm{O}(l) \rightleftharpoons 2 \mathrm{H}^{+}(a q)+2 \mathrm{OH}^{-}(a q)\)

For the equilibrium $$2 \mathrm{IBr}(g) \rightleftharpoons \mathrm{I}_{2}(g)+\mathrm{Br}_{2}(g)$$ \(K_{p}=8.5 \times 10^{-3}\) at \(150^{\circ} \mathrm{C}\). If 0.025 atm of IBr is placed in a 2.0-L container, what is the partial pressure of all substances after equilibrium is reached?
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