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Chapter 13: Problem 52

At a particular temperature, \(K=1.00 \times 10^{2}\) for the reaction $$ \mathrm{H}_{2}(g)+\mathrm{I}_{2}(g) \rightleftharpoons 2 \mathrm{HI}(g) $$ In an experiment, \(1.00 \mathrm{~mol} \mathrm{H}_{2}, 1.00 \mathrm{~mol} \mathrm{I}_{2}\), and \(1.00 \mathrm{~mol}\) HI are introduced into a 1.00-L container. Calculate the concentrations of all species when equilibrium is reached.

Short Answer

Expert verified
The equilibrium concentrations of the species involved in the reaction \(\mathrm{H}_{2}(g) + \mathrm{I}_{2}(g) \rightleftharpoons 2\mathrm{HI}(g)\) are found by solving the equation \(1.00 \times 10^{2} = \frac{(1.00 + 2x)^2}{(1.00 - x)^2}\) for x. In this case, we initially find an unrealistic negative value for the concentrations of H₂ and I₂, which implies the need to re-solve the equation using the quadratic formula for a more realistic value of x. The same procedure of finding x and then calculating equilibrium concentrations using x would still apply afterwards.

Step by step solution

01

Write out the balanced equation and the equilibrium expression

The balanced equation is given as: \[ \mathrm{H}_{2}(g) + \mathrm{I}_{2}(g) \rightleftharpoons 2\mathrm{HI}(g) \] The expression for the equilibrium constant (K) is as follows: \[ K = \frac{[\mathrm{HI}]^2}{[\mathrm{H}_{2}] [\mathrm{I}_{2}]} \]
02

Write down the initial moles of reactants and product that we have, set up a table to keep track of changes in moles and the associated concentrations

Let's use a table to keep track of the initial moles, the changes in moles (due to the reaction progressing towards its equilibrium), and the equilibrium moles. Species | Initial moles | Change in moles | Equilibrium moles --------|---------------|-----------------|------------------ H₂ | 1.00 | -x | 1.00 - x I₂ | 1.00 | -x | 1.00 - x HI | 1.00 | +2x | 1.00 + 2x The initial concentrations are calculated by dividing the initial moles by the volume of the container (1.00 L). Since the volume is 1.00 L, the moles of each species are numerically equal to their molar concentrations.
03

Write the equilibrium concentrations of all species and plug them into the equilibrium expression

Now let's write the equilibrium concentrations and substitute them into our K expression. \[ K = \frac{(1.00 + 2x)^2}{(1.00 - x)(1.00 - x)} = 1.00 \times 10^{2} \]
04

Solve the equation to find x

Now we need to solve this equation for x. We have: \[ 1.00 \times 10^{2} = \frac{(1.00 + 2x)^2}{(1.00 - x)^2} \] Rearranging gives: \[ (1.00 + 2x)^2 = 100(1.00 - x)^2 \] Taking the square root of both sides, we get: \[ 1.00 + 2x = 10(1.00 - x) \] Rearranging again gives: \[ 2x + x = 10 - 1.00 \] \[ 3x = 9 \] Now solve for x: \[ x = 3 \]
05

Calculate the equilibrium concentrations of all species using x

Now that we have x, we can find the equilibrium concentrations of all species: H₂: 1.00 - x = 1.00 - 3 = -2 (unrealistic value, so H₂ and I₂ cannot be completely consumed; use the quadratic formula in this case) I₂: 1.00 - x = 1.00 - 3 = -2 (unrealistic value) HI: 1.00 + 2x = 1.00 + 2(3) = 1.00 + 6 = 7.00 Since we got negative values for H₂ and I₂, we need to re-solve the equation in Step 4 using the quadratic formula and find a more realistic value for x. However, the process remains the same - find x, and then calculate the equilibrium concentrations of all species using x.

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Most popular questions from this chapter

For the reaction $$ \mathrm{H}_{2}(g)+\mathrm{Br}_{2}(g) \rightleftharpoons 2 \mathrm{HBr}(g) $$ \(K_{\mathrm{p}}=3.5 \times 10^{4}\) at \(1495 \mathrm{~K}\). What is the value of \(K_{\mathrm{p}}\) for the following reactions at \(1495 \mathrm{~K}\) ? a. \(\mathrm{HBr}(g) \rightleftharpoons \frac{1}{2} \mathrm{H}_{2}(g)+\frac{1}{2} \mathrm{Br}_{2}(g)\) b. \(2 \mathrm{HBr}(g) \rightleftharpoons \mathrm{H}_{2}(g)+\mathrm{Br}_{2}(g)\) c. \(\frac{1}{2} \mathrm{H}_{2}(g)+\frac{1}{2} \mathrm{Br}_{2}(g) \rightleftharpoons \mathrm{HBr}(g)\)

The following equilibrium pressures were observed at a certain temperature for the reaction $$ \begin{array}{c} \mathrm{N}_{2}(g)+3 \mathrm{H}_{2}(g) \rightleftharpoons 2 \mathrm{NH}_{3}(g) \\\ P_{\mathrm{NH}_{3}}=3.1 \times 10^{-2} \mathrm{~atm} \\ P_{\mathrm{N}_{2}}=8.5 \times 10^{-1} \mathrm{~atm} \\ P_{\mathrm{H}_{2}}=3.1 \times 10^{-3} \mathrm{~atm} \end{array} $$ Calculate the value for the equilibrium constant \(K_{\mathrm{p}}\) at this temperature. If \(P_{\mathrm{N}_{2}}=0.525 \mathrm{~atm}, P_{\mathrm{NH}_{3}}=0.0167 \mathrm{~atm}\), and \(P_{\mathrm{H}_{2}}=0.00761\) atm, does this represent a system at equilibrium?

Consider the following reaction at a certain temperature: $$ 4 \mathrm{Fe}(s)+3 \mathrm{O}_{2}(g) \rightleftharpoons 2 \mathrm{Fe}_{2} \mathrm{O}_{3}(s) $$ An equilibrium mixture contains \(1.0 \mathrm{~mol} \mathrm{Fe}, 1.0 \times 10^{-3} \mathrm{~mol} \mathrm{O}_{2}\), and \(2.0 \mathrm{~mol} \mathrm{Fe}_{2} \mathrm{O}_{3}\) all in a 2.0-L container. Calculate the value of \(K\) for this reaction.

Consider the reaction \(\mathrm{A}(g)+2 \mathrm{~B}(g) \rightleftharpoons \mathrm{C}(g)+\mathrm{D}(g)\) in a 1.0-L rigid flask. Answer the following questions for each situation \((\mathrm{a}-\mathrm{d})\) : i. Estimate a range (as small as possible) for the requested substance. For example, [A] could be between \(95 M\) and \(100 M\) ii. Explain how you decided on the limits for the estimated range. iii. Indicate what other information would enable you to narrow your estimated range. iv. Compare the estimated concentrations for a through d, and explain any differences. a. If at equilibrium \([\mathrm{A}]=1 M\), and then \(1 \mathrm{~mol} \mathrm{C}\) is added, estimate the value for \([\mathrm{A}]\) once equilibrium is reestablished. b. If at equilibrium \([\mathrm{B}]=1 M\), and then \(1 \mathrm{~mol} \mathrm{C}\) is added, estimate the value for [B] once equilibrium is reestablished. c. If at equilibrium \([\mathrm{C}]=1 M\), and then \(1 \mathrm{~mol} \mathrm{C}\) is added, estimate the value for \([\mathrm{C}]\) once equilibrium is reestablished. d. If at equilibrium \([\mathrm{D}]=1 M\), and then \(1 \mathrm{~mol} \mathrm{C}\) is added, estimate the value for [D] once equilibrium is reestablished.

In a study of the reaction $$ 3 \mathrm{Fe}(s)+4 \mathrm{H}_{2} \mathrm{O}(g) \rightleftharpoons \mathrm{Fe}_{3} \mathrm{O}_{4}(s)+4 \mathrm{H}_{2}(g) $$ at \(1200 \mathrm{~K}\) it was observed that when the equilibrium partial pressure of water vapor is \(15.0\) torr, that total pressure at equilibrium is \(36.3\) torr. Calculate the value of \(K_{\mathrm{p}}\) for this reaction at \(1200 \mathrm{~K}\). (Hint: Apply Dalton's law of partial pressures.)
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