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Chapter 15: Problem 83

When \(1.50 \mathrm{~mol} \mathrm{CO}_{2}\) and \(1.50 \mathrm{~mol} \mathrm{H}_{2}\) are placed in a 3.00-L container at \(395^{\circ} \mathrm{C},\) the following reaction occurs: \(\mathrm{CO}_{2}(g)+\mathrm{H}_{2}(g) \rightleftharpoons \mathrm{CO}(g)+\mathrm{H}_{2} \mathrm{O}(g) . \quad\) If \(\quad K_{c}=0.802\) what are the concentrations of each substance in the equilibrium mixture?

Short Answer

Expert verified
At equilibrium, the concentrations of the species are: \([\mathrm{CO}_{2}] = 0.277\,\mathrm{M}\), \([\mathrm{H}_{2}] = 0.277\,\mathrm{M}\), \([\mathrm{CO}] = 0.223\,\mathrm{M}\), and \([\mathrm{H}_{2}\mathrm{O}] = 0.223\,\mathrm{M}\).

Step by step solution

01

Write the balanced chemical equation

The balanced chemical equation is given as: \(\mathrm{CO}_{2}(g) + \mathrm{H}_{2}(g) \rightleftharpoons \mathrm{CO}(g) + \mathrm{H}_{2}\mathrm{O}(g)\)
02

Set up an ICE table

An ICE (Initial, Change, Equilibrium) table helps track the changes in species concentrations throughout the reaction process. Initial concentrations: \(\frac{1.50\,\mathrm{mol}}{3.00\,\mathrm{L}} = 0.500\,\mathrm{M}\) for \(\mathrm{CO}_{2}\) and \(\mathrm{H}_{2}\) Initial concentrations of \(\mathrm{CO}\) and \(\mathrm{H}_{2}\mathrm{O}\) are 0 M since they are not given in the problem statement. ``` CO2 H2 CO H2O Initial 0.500 0.500 0 0 Change -x -x +x +x Final 0.500-x 0.500-x x x ```
03

Write the equilibrium constant expression

\(K_c = \frac{[\mathrm{CO}] [\mathrm{H}_{2}\mathrm{O}]}{[\mathrm{CO}_{2}] [\mathrm{H}_{2}]}\)
04

Substitute ICE table values into the equilibrium expression

We substitute the equilibrium concentrations from the ICE table into the \(K_c\) expression: \(0.802 = \frac{x^2}{(0.500-x)^2}\)
05

Solve for x

To solve for x, we first notice that the denominator and numerator have the same exponent. Thus, we can simplify the equation: \(0.802 = \frac{x}{0.500-x}\) Now, multiply both sides by \((0.500-x)\): \(0.802(0.500-x) = x\) Distribute the 0.802 on the left-hand side: \(0.401 - 0.802x = x\) Combine the x terms and solve for x: \(x = \frac{0.401}{1.802} \approx 0.223\,\mathrm{M}\)
06

Calculate the equilibrium concentrations

Now that we have the value of x, we can find the equilibrium concentrations of all species: \([\mathrm{CO}]_{eq} = [\mathrm{H}_{2}\mathrm{O}]_{eq} = x = 0.223\,\mathrm{M}\) \([\mathrm{CO}_{2}]_{eq} = [\mathrm{H}_{2}]_{eq} = 0.500 - x = 0.500 - 0.223 = 0.277\,\mathrm{M}\) So, at equilibrium, the concentrations of the species are: \([\mathrm{CO}_{2}] = 0.277\,\mathrm{M}\) \([\mathrm{H}_{2}] = 0.277\,\mathrm{M}\) \([\mathrm{CO}] = 0.223\,\mathrm{M}\) \([\mathrm{H}_{2}{\mathrm{O}}] = 0.223\,\mathrm{M}\)

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Most popular questions from this chapter

The equilibrium \(2 \mathrm{NO}(g)+\mathrm{Cl}_{2}(g) \rightleftharpoons 2 \mathrm{NOCl}(g)\) is estab- lished at \(500 \mathrm{~K}\). An equilibrium mixture of the three gases has partial pressures of 0.095 atm, 0.171 atm, and 0.28 atm for \(\mathrm{NO}, \mathrm{Cl}_{2},\) and \(\mathrm{NOCl}\), respectively. (a) Calculate \(K_{p}\) for this reaction at \(500.0 \mathrm{~K}\). (b) If the vessel has a volume of \(5.00 \mathrm{~L}\) calculate \(K_{c}\) at this temperature.

Suppose that the gas-phase reactions \(\mathrm{A} \longrightarrow \mathrm{B}\) and \(\mathrm{B} \longrightarrow \mathrm{A}\) are both elementary processes with rate constants of \(4.7 \times 10^{-3} \mathrm{~s}^{-1}\) and \(5.8 \times 10^{-1} \mathrm{~s}^{-1}\), respectively. (a) What is the value of the equilibrium constant for the equilibrium \(\mathrm{A}(g) \rightleftharpoons \mathrm{B}(g) ?\) (b) Which is greater at equilibrium, the partial pressure of \(\mathrm{A}\) or the partial pressure of \(\mathrm{B}\) ? Explain.

For the equilibrium $$\mathrm{Br}_{2}(g)+\mathrm{Cl}_{2}(g) \rightleftharpoons 2 \operatorname{BrCl}(g)$$ at \(400 \mathrm{~K}, K_{c}=7.0 .\) If \(0.25 \mathrm{~mol}\) of \(\mathrm{Br}_{2}\) and \(0.55 \mathrm{~mol}\) of \(\mathrm{Cl}_{2}\) are introduced into a \(3.0-\mathrm{L}\) container at \(400 \mathrm{~K},\) what will be the equilibrium concentrations of \(\mathrm{Br}_{2}, \mathrm{Cl}_{2},\) and \(\mathrm{BrCl} ?\)

(a) How is a reaction quotient used to determine whether a system is at equilibrium? (b) If \(Q_{c}>K_{c}\), how must the reaction proceed to reach equilibrium? (c) At the start of a certain reaction, only reactants are present; no products have been formed. What is the value of \(Q_{c}\) at this point in the reaction?

A mixture of \(0.2000 \mathrm{~mol}\) of \(\mathrm{CO}_{2}, 0.1000 \mathrm{~mol}\) of \(\mathrm{H}_{2},\) and \(0.1600 \mathrm{~mol}\) of \(\mathrm{H}_{2} \mathrm{O}\) is placed in a \(2.000-\mathrm{L}\) vessel. The following equilibrium is established at \(500 \mathrm{~K}\) : $$\mathrm{CO}_{2}(g)+\mathrm{H}_{2}(g) \rightleftharpoons \mathrm{CO}(g)+\mathrm{H}_{2} \mathrm{O}(g)$$ (a) Calculate the initial partial pressures of \(\mathrm{CO}_{2}, \mathrm{H}_{2},\) and \(\mathrm{H}_{2} \mathrm{O} .\) (b) At equilibrium \(P_{\mathrm{H}_{2} \mathrm{O}}=3.51 \mathrm{~atm} .\) Calculate the equilibrium partial pressures of \(\mathrm{CO}_{2}, \mathrm{H}_{2},\) and \(\mathrm{CO} .\) (c) Calculate \(K_{p}\) for the reaction. (d) Calculate \(K_{c}\) for the reaction.
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