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Chapter 13: Problem 66

The creation of shells by mollusk species is a fascinating process. By utilizing the \(\mathrm{Ca}^{2+}\) in their food and aqueous environment, as well as some complex equilibrium processes, a hard calcium carbonate shell can be produced. One important equilibrium reaction in this complex process is $$\mathrm{HCO}_{3}^{-}(a q) \leftrightharpoons \mathrm{H}^{+}(a q)+\mathrm{CO}_{3}^{2-}(a q) K=5.6 \times 10^{-11}$$ If 0.16 mole of \(\mathrm{HCO}_{3}^{-}\) is placed into 1.00 \(\mathrm{L}\) of solution, what will be the equilibrium concentration of \(\mathrm{CO}_{3}^{2-2}\) ?

Short Answer

Expert verified
The equilibrium concentration of \(CO_{3}^{2-}\) is approximately \(9.48 \times 10^{-7}\, M\).

Step by step solution

01

Write the given reaction and equilibrium constant

The reaction and equilibrium constant given are: \(HCO_{3}^{-}(aq) \leftrightharpoons H^{+}(aq) + CO_{3}^{2-}(aq)\;\;\;\;K = 5.6 \times 10^{-11}\)
02

Calculate initial concentrations

The initial concentration of bicarbonate ion is: \([HCO_{3}^{-}]_{initial} = \frac{0.16\, mol}{1.00\, L} = 0.16\, M\) Initially, there is no \(H^{+}\) or \(CO_{3}^{2-}\) from the reaction, so their concentrations are 0.
03

Create an ICE table

We can set up an ICE table (Initial, Change, Equilibrium) to find the equilibrium concentrations. The table looks like this: | | \(HCO_{3}^{-}\) | \(H^{+}\) | \(CO_{3}^{2-}\) | |-------|--------------|-------|--------------| |Initial| 0.16 | 0 | 0 | |Change | -x | +x | +x | |Equilibrium| 0.16-x | x | x | Where "x" represents the change in concentrations at equilibrium.
04

Write the equilibrium expression and substitute values

The equilibrium expression for the given reaction is: \(K = \frac{[H^{+}][CO_{3}^{2-}]}{[HCO_{3}^{-}]}\) Substitute the values from the ICE table: \(5.6 \times 10^{-11} = \frac{x\cdot x}{0.16-x}\)
05

Solve the equation for x

Assuming that x is much smaller than 0.16, we can simplify the equation as follows: \(5.6 \times 10^{-11} = \frac{x^2}{0.16}\) Now, solve for x: \(x^2 = 0.16\times (5.6\times 10^{-11}) \) \(x^2 = 8.96 \times 10^{-12}\) \(x = \sqrt{8.96 \times 10^{-12}}\) \(x = \approx 9.48 \times 10^{-7}\)
06

Determine the equilibrium concentration of \(CO_{3}^{2-}\)

The equilibrium concentration of \(CO_{3}^{2-}\) is equal to x. Therefore, \([CO_{3}^{2-}]_{equilibrium} \approx 9.48 \times 10^{-7}\, M\)

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Most popular questions from this chapter

For the reaction $$2 \mathrm{NO}(g)+2 \mathrm{H}_{2}(g) \rightleftharpoons \mathrm{N}_{2}(g)+2 \mathrm{H}_{2} \mathrm{O}(g)$$ it is determined that, at equilibrium at a particular temperature, the concentrations are as follows: $[\mathrm{NO}(g)]=8.1 \times 10^{-3} \mathrm{M}\( \)\left[\mathrm{H}_{2}(g)\right]=4.1 \times 10^{-5} M,\left[\mathrm{N}_{2}(g)\right]=5.3 \times 10^{-2} M,$ and \(\left[\mathrm{H}_{2} \mathrm{O}(g)\right]-2.9 \times 10^{-3} \mathrm{M} .\) Calculate the value of \(K\) for the reaction at this temperature.

For the reaction below, \(K_{\mathrm{p}}=1.16\) at \(800 .^{\circ} \mathrm{C}\) $$\mathrm{CaCO}_{3}(s) \rightleftharpoons \mathrm{CaO}(s)+\mathrm{CO}_{2}(g)$$ If a 20.0 -g sample of \(\mathrm{CaCO}_{3}\) is put into a 10.0 -L container and heated to \(800 .^{\circ} \mathrm{C},\) what percentage by mass of the \(\mathrm{CaCO}_{3}\) will react to reach equilibrium?

An initial mixture of nitrogen gas and hydrogen gas is reacted in a rigid container at a certain temperature by the reaction $$3 \mathrm{H}_{2}(g)+\mathrm{N}_{2}(g) \rightleftharpoons 2 \mathrm{NH}_{3}(g)$$ At equilibrium, the concentrations are $\left[\mathrm{H}_{2}\right]=5.0 M,\left[\mathrm{N}_{2}\right]=$ \(8.0 M,\) and \(\left[\mathrm{NH}_{3}\right]=4.0 \mathrm{M} .\) What were the concentrations of nitrogen gas and hydrogen gas that were reacted initially?

Ethyl acetate is synthesized in a nonreacting solvent (not water) according to the following reaction: $$\mathrm{CH}_{3} \mathrm{CO}_{2} \mathrm{H}+\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH} \rightleftharpoons \mathrm{CH}_{3} \mathrm{CO}_{2} \mathrm{C}_{2} \mathrm{H}_{5}+\mathrm{H}_{2} \mathrm{O} \quad K=2.2$$ For the following mixtures (a-d), will the concentration of $\mathrm{H}_{2} \mathrm{O}$ increase, decrease, or remain the same as equilibrium is established? a. $\left[\mathrm{CH}_{3} \mathrm{CO}_{2} \mathrm{C}_{2} \mathrm{H}_{5}\right]=0.22 M,\left[\mathrm{H}_{2} \mathrm{O}\right]=0.10 M$ $\quad\left[\mathrm{CH}_{3} \mathrm{CO}_{2} \mathrm{H}\right]=0.010 M,\left[\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}\right]=0.010 M$ b. $\left[\mathrm{CH}_{3} \mathrm{CO}_{2} \mathrm{C}_{2} \mathrm{H}_{5}\right]=0.22 M,\left[\mathrm{H}_{2} \mathrm{O}\right]=0.0020 M$ $\quad\left[\mathrm{CH}_{3} \mathrm{CO}_{2} \mathrm{H}\right]=0.0020 M,\left[\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}\right]=0.10 M$ c. $\left[\mathrm{CH}_{3} \mathrm{CO}_{2} \mathrm{C}_{2} \mathrm{H}_{5}\right]=0.88 M,\left[\mathrm{H}_{2} \mathrm{O}\right]=0.12 M$ $\quad\left[\mathrm{CH}_{3} \mathrm{CO}_{2} \mathrm{H}\right]=0.044 M,\left[\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}\right]=6.0 M$ d. $\left[\mathrm{CH}_{3} \mathrm{CO}_{2} \mathrm{C}_{2} \mathrm{H}_{5}\right]=4.4 M,\left[\mathrm{H}_{2} \mathrm{O}\right]=4.4 M$ $\quad\left[\mathrm{CH}_{3} \mathrm{CO}_{2} \mathrm{H}\right]=0.88 M,\left[\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}\right]=10.0 M$ e. What must the concentration of water be for a mixture with $\left[\mathrm{CH}_{3} \mathrm{CO}_{2} \mathrm{C}_{2} \mathrm{H}_{5}\right]=2.0 M,\left[\mathrm{CH}_{3} \mathrm{CO}_{2} \mathrm{H}\right]=0.10 M,$ and \(\left[\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}\right]=5.0 M\) to be at equilibrium? f. Why is water included in the equilibrium expression for this reaction?

Write expressions for \(K\) and \(K_{\mathrm{p}}\) for the following reactions. a. $2 \mathrm{NH}_{3}(g)+\mathrm{CO}_{2}(g) \rightleftharpoons \mathrm{N}_{2} \mathrm{CH}_{4} \mathrm{O}(s)+\mathrm{H}_{2} \mathrm{O}(g)$ b. $2 \mathrm{NBr}_{3}(s) \Longrightarrow \mathrm{N}_{2}(g)+3 \mathrm{Br}_{2}(g)$ c. $2 \mathrm{KClO}_{3}(s) \Longrightarrow 2 \mathrm{KCl}(s)+3 \mathrm{O}_{2}(g)$ d. $\mathrm{CuO}(s)+\mathrm{H}_{2}(g) \rightleftharpoons \mathrm{Cu}(l)+\mathrm{H}_{2} \mathrm{O}(g)$
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