### Learning Outcomes

- Identify the changes in concentration or pressure that occur for chemical species in equilibrium systems
- Calculate equilibrium concentrations or pressures and equilibrium constants, using various algebraic approaches

Having covered the essential concepts of chemical equilibria in the preceding sections of this chapter, this final section will demonstrate the more practical aspect of using these concepts and appropriate mathematical strategies to perform various equilibrium calculations. These types of computations are essential to many areas of science and technology—for example, in the formulation and dosing of pharmaceutical products. After a drug is ingested or injected, it is typically involved in several chemical equilibria that affect its ultimate concentration in the body system of interest. Knowledge of the quantitative aspects of these equilibria is required to compute a dosage amount that will solicit the desired therapeutic effect.

Many of the useful equilibrium calculations that will be demonstrated here require terms representing changes in reactant and product concentrations. These terms are derived from the stoichiometry of the reaction, as illustrated by decomposition of ammonia:

[latex]2{\text{NH}}_{3}\left(g\right){\rightleftharpoons}{\text{N}}_{2}\left(g\right)+3{\text{H}}_{2}\left(g\right)[/latex]

As shown earlier in this chapter, this equilibrium may be established within a sealed container that initially contains either NH_{3} only, or a mixture of any two of the three chemical species involved in the equilibrium. Regardless of its initial composition, a reaction mixture will show the same relationships between changes in the concentrations of the three species involved, as dictated by the reaction stoichiometry (see also the related content on expressing reaction rates in the chapter on kinetics). For example, if the nitrogen concentration increases by an amount *x*:

the corresponding changes in the other species concentrations are

where the negative sign indicates a decrease in concentration.

### Example 1: Determining Relative Changes in Concentration

Derive the missing terms representing concentration changes for each of the following reactions.

- [latex]\begin{array}{ccccc}{\text{C}}_{2}{\text{H}}_{2}\text{(}g\text{)}&+\hfill & 2{\text{Br}}_{2}\text{(}g\text{)}\hfill & \rightleftharpoons& {\text{C}}_{2}{\text{H}}_{2}{\text{Br}}_{4}\text{(}g\text{)}\hfill \\ x\end{array}[/latex]
- [latex]\begin{array}{ccccc}{\text{I}}_{2}\text{(}aq\text{)}&+\hfill & {\text{I}}^{-}\text{(}aq\text{)}\hfill & \rightleftharpoons\hfill & {\text{I}}_{3}{}^{-}\text{(}aq\text{)}\hfill \\ & & & & x \end{array}[/latex]
- [latex]\begin{array}{cccccc}{\text{C}}_{3}{\text{H}}_{8}\text{(}g\text{)}&+\hfill & 5{\text{O}}_{2}\text{(}g\text{)}\hfill & \rightleftharpoons\hfill & 3{\text{CO}}_{2}\text{(}g\text{)}+\hfill & 4{\text{H}}_{2}\text{O}\text{(}g\text{)}\hfill \\ x \end{array}[/latex]

#### Check Your Learning

Complete the changes in concentrations for each of the following reactions:

- [latex]\begin{array}{ccccc}2{\text{SO}}_{2}\text{(}g\text{)}&+\hfill & {\text{O}}_{2}\text{(}g\text{)}\hfill & \rightleftharpoons\hfill & 2{\text{SO}}_{3}\text{(}g\text{)}\hfill \\ & & x & \hfill \end{array}[/latex]
- [latex]\begin{array}{ccc}{\text{C}}_{4}{\text{H}}_{8}\text{(}g\text{)}\hfill & \rightleftharpoons\hfill & 2{\text{C}}_{2}{\text{H}}_{4}\text{(}g\text{)}\hfill \\ & & -2x\hfill \end{array}[/latex]
- [latex]\begin{array}{ccccccc}4{\text{NH}}_{3}\text{(}g\text{)}&+\hfill & 7{\text{H}}_{2}\text{O}\text{(}g\text{)}\hfill & \rightleftharpoons\hfill & 4{\text{NO}}_{2}\text{(}g\text{)}&+\hfill & 6{\text{H}}_{2}\text{O}\text{(}g\text{)}\end{array}[/latex]

### Try It

- A reaction is represented by this equation: [latex]\text{A}\left(aq\right)+2\text{B}\left(aq\right)\rightleftharpoons2\text{C}\left(aq\right){K}_{c}=1\times {10}^{3}[/latex]
- Write the mathematical expression for the equilibrium constant.
- Using concentrations ≤1
*M*, make up two sets of concentrations that describe a mixture of A, B, and C at equilibrium.

- A reaction is represented by this equation: [latex]2\text{W}\left(aq\right)\rightleftharpoons\text{X}\left(aq\right)+2\text{Y}\left(aq\right){K}_{c}=5\times {10}^{-4}[/latex]
- Write the mathematical expression for the equilibrium constant.
- Using concentrations of ≤1
*M*, make up two sets of concentrations that describe a mixture of W, X, and Y at equilibrium.

## Calculations of an Equilibrium Constant

The equilibrium constant for a reaction is calculated from the equilibrium concentrations (or pressures) of its reactants and products. If these concentrations are known, the calculation simply involves their substitution into the K expression, as was illustrated by Example 2. A slightly more challenging example is provided next, in which the reaction stoichiometry is used to derive equilibrium concentrations from the information provided. The basic strategy of this computation is helpful for many types of equilibrium computations and relies on the use of terms for the reactant and product concentrations *initially* present, for how they *change* as the reaction proceeds, and for what they are when the system reaches *equilibrium*. The acronym ICE is commonly used to refer to this mathematical approach, and the concentrations terms are usually gathered in a tabular format called an ICE table.

### Example 2: Calculation of an Equilibrium Constant

Iodine molecules react reversibly with iodide ions to produce triiodide ions.

[latex]{\text{I}}_{2}\left(aq\right)+{\text{I}}^{-}\left(aq\right)\rightleftharpoons{\text{I}}_{3}{}^{-}\left(aq\right)[/latex]

If a solution with the concentrations of I_{2} and I^{−} both equal to 1.000 × 10^{−3}*M* before reaction gives an equilibrium concentration of I_{2} of 6.61 × 10^{−4}*M*, what is the equilibrium constant for the reaction?

### Try It

- What is the value of the equilibrium constant at 500 °C for the formation of NH
_{3}according to the following equation? [latex]{\text{N}}_{2}\left(g\right)+3{\text{H}}_{2}\left(g\right)\rightleftharpoons2{\text{NH}}_{3}\left(g\right)[/latex]

An equilibrium mixture of NH_{3}(*g*), H_{2}(*g*), and N_{2}(*g*) at 500 °C was found to contain 1.35*M*H_{2}, 1.15*M*N_{2}, and 4.12 × 10^{-1}*M*NH_{3}. - Hydrogen is prepared commercially by the reaction of methane and water vapor at elevated temperatures.

[latex]{\text{CH}}_{4}\left(g\right)+{\text{H}}_{2}\text{O}\left(g\right)\rightleftharpoons3{\text{H}}_{2}\left(g\right)+\text{CO}\left(g\right)[/latex]

What is the equilibrium constant for the reaction if a mixture at equilibrium contains gases with the following concentrations: CH_{4}, 0.126*M*; H_{2}O, 0.242*M*; CO, 0.126*M*; H_{2}1.15*M*, at a temperature of 760 °C? - A 0.72-mol sample of PCl
_{5}is put into a 1.00-L vessel and heated. At equilibrium, the vessel contains 0.40 mol of PCl_{3}(*g*) and 0.40 mol of Cl_{2}(*g*). Calculate the value of the equilibrium constant for the decomposition of PCl_{5}to PCl_{3}and Cl_{2}at this temperature. - At 1 atm and 25 °C, NO
_{2}with an initial concentration of 1.00*M*is 3.3 × 10^{-3}% decomposed into NO and O_{2}. Calculate the value of the equilibrium constant for the reaction

[latex]2{\text{NO}}_{2}\left(g\right)\rightleftharpoons2\text{NO}\left(g\right)+{\text{O}}_{2}\left(g\right)[/latex] - Calculate the value of the equilibrium constant
*K*for the reaction [latex]2\text{NO}\left(g\right)+{\text{Cl}}_{2}\left(g\right)\rightleftharpoons2\text{NOCl}\left(g\right)[/latex] from these equilibrium pressures: NO, 0.050 atm; Cl_{P}_{2}, 0.30 atm; NOCl, 1.2 atm. - When heated, iodine vapor dissociates according to this equation: [latex]{\text{I}}_{2}\left(g\right)\rightleftharpoons2\text{I}\left(g\right)[/latex]

At 1274 K, a sample exhibits a partial pressure of I_{2}of 0.1122 atm and a partial pressure due to I atoms of 0.1378 atm. Determine the value of the equilibrium constant,*K*, for the decomposition at 1274 K._{P} - A sample of ammonium chloride was heated in a closed container: [latex]{\text{NH}}_{4}\text{Cl}\left(s\right)\rightleftharpoons{\text{NH}}_{3}\left(g\right)+\text{HCl}\left(g\right)[/latex]

At equilibrium, the pressure of NH_{3}(*g*) was found to be 1.75 atm. What is the value of the equilibrium constant*K*for the decomposition at this temperature?_{P} - At a temperature of 60 °C, the vapor pressure of water is 0.196 atm. What is the value of the equilibrium constant
*K*for the transformation at 60 °C?_{P}

[latex]{\text{H}}_{2}\text{O}\left(l\right)\rightleftharpoons{\text{H}}_{2}\text{O}\left(g\right)[/latex]

### Calculation of a Missing Equilibrium Concentration

When the equilibrium constant and all but one equilibrium concentration are provided, the other equilibrium concentration(s) may be calculated. A computation of this sort is illustrated in the next example exercise.

### Example 3: Calculation of a Missing Equilibrium Concentration

Nitrogen oxides are air pollutants produced by the reaction of nitrogen and oxygen at high temperatures. At 2000 °C, the value of the equilibrium constant for the reaction, [latex]{\text{N}}_{2}\left(g\right)+{\text{O}}_{2}\left(g\right)\rightleftharpoons2\text{NO}\left(g\right)[/latex], is 4.1 × 10^{−4}. Find the concentration of NO(*g*) in an equilibrium mixture with air at 1 atm pressure at this temperature. In air, [N_{2}] = 0.036 mol/L and [O_{2}] 0.0089 mol/L.

#### Check Your Learning

The equilibrium constant for the reaction of nitrogen and hydrogen to produce ammonia at a certain temperature is 6.00 × 10^{−2}. Calculate the equilibrium concentration of ammonia if the equilibrium concentrations of nitrogen and hydrogen are 4.26 *M* and 2.09 *M*, respectively.

### Try It

- Analysis of the gases in a sealed reaction vessel containing NH
_{3}, N_{2}, and H_{2}at equilibrium at 400 °C established the concentration of N_{2}to be 1.2*M*and the concentration of H_{2}to be 0.24*M*.

[latex]{\text{N}}_{2}\left(g\right)+3{\text{H}}_{2}\left(g\right)\rightleftharpoons2{\text{NH}}_{3}\left(g\right){K}_{c}=0.50\text{ at }400^{\circ}\text{C}[/latex]

Calculate the equilibrium molar concentration of NH_{3}. - Carbon reacts with water vapor at elevated temperatures.

[latex]\text{C}\left(s\right)+{\text{H}}_{2}\text{O}\left(g\right)\rightleftharpoons\text{CO}\left(g\right)+{\text{H}}_{2}\left(g\right){K}_{c}=0.2\text{ at }1000^{\circ}\text{C}[/latex]

What is the concentration of CO in an equilibrium mixture with [H_{2}O] = 0.500*M*at 1000 °C? - Cobalt metal can be prepared by reducing cobalt(II) oxide with carbon monoxide.

[latex]\text{CoO}\left(s\right)+\text{CO}\left(g\right)\rightleftharpoons\text{Co}\left(s\right)+{\text{CO}}_{2}\left(g\right){K}_{c}=4.90\times {10}^{2}\text{ at }550^{\circ}\text{C}[/latex]

What concentration of CO remains in an equilibrium mixture with [CO_{2}] = 0.100*M*? - A student solved the following problem and found [N
_{2}O_{4}] = 0.16*M*at equilibrium. How could this student recognize that the answer was wrong without reworking the problem? The problem was: What is the equilibrium concentration of N_{2}O_{4}in a mixture formed from a sample of NO_{2}with a concentration of 0.10*M*?

[latex]2{\text{NO}}_{2}\left(g\right)\rightleftharpoons{\text{N}}_{2}{\text{O}}_{4}\left(g\right){K}_{c}=160[/latex] - A student solved the following problem and found the equilibrium concentrations to be [SO
_{2}] = 0.590*M*, [O_{2}] = 0.0450*M*, and [SO_{3}] = 0.260*M*. How could this student check the work without reworking the problem? The problem was: For the following reaction at 600 °C:

[latex]2{\text{SO}}_{2}\left(g\right)+{\text{O}}_{2}\left(g\right)\rightleftharpoons2{\text{SO}}_{3}\left(g\right){K}_{c}=4.32[/latex]

What are the equilibrium concentrations of all species in a mixture that was prepared with [SO_{3}] = 0.500*M*, [SO_{2}] = 0*M*, and [O_{2}] = 0.350*M*

### Calculation of Changes in Concentration

Perhaps the most challenging type of equilibrium calculation can be one in which equilibrium concentrations are derived from initial concentrations and an equilibrium constant. For these calculations, a four-step approach is typically useful:

- Identify the direction in which the reaction will proceed to reach equilibrium.
- Develop an ICE table.
- Calculate the concentration changes and, subsequently, the equilibrium concentrations.
- Confirm the calculated equilibrium concentrations.

The last two example exercises of this chapter demonstrate the application of this strategy.

### Example 4: Calculation of Concentration Changes as a Reaction Goes to Equilibrium

Under certain conditions, the equilibrium constant for the decomposition of PCl_{5}(*g*) into PCl_{3}(*g*) and Cl_{2}(*g*) is 0.0211. What are the equilibrium concentrations of PCl_{5}, PCl_{3}, and Cl_{2} if the initial concentration of PCl_{5} was 1.00 *M*?

#### Check Your Learning

Acetic acid, CH_{3}CO_{2}H, reacts with ethanol, C_{2}H_{5}OH, to form water and ethyl acetate, CH_{3}CO_{2}C_{2}H_{5}.

[latex]{\text{CH}}_{3}{\text{CO}}_{2}\text{H}+{\text{C}}_{2}{\text{H}}_{5}\text{OH}\rightleftharpoons{\text{CH}}_{3}{\text{CO}}_{2}{\text{C}}_{2}{\text{H}}_{5}+{\text{H}}_{2}\text{O}[/latex]

The equilibrium constant for this reaction with dioxane as a solvent is 4.0. What are the equilibrium concentrations when a mixture that is 0.15 *M* in CH_{3}CO_{2}H, 0.15 *M* in C_{2}H_{5}OH, 0.40 *M* in CH_{3}CO_{2}C_{2}H_{5}, and 0.40 *M* in H_{2}O are mixed in enough dioxane to make 1.0 L of solution?

#### Check Your Learning

A 1.00-L flask is filled with 1.00 moles of H_{2} and 2.00 moles of I_{2}. The value of the equilibrium constant for the reaction of hydrogen and iodine reacting to form hydrogen iodide is 50.5 under the given conditions. What are the equilibrium concentrations of H_{2}, I_{2}, and HI in moles/L?

[latex]{\text{H}}_{2}\left(g\right)+{\text{I}}_{2}\left(g\right)\rightleftharpoons2\text{HI}\left(g\right)[/latex]

### EXAMPLE 5: Calculation of Equilibrium Concentrations Using an Algebra-Simplifying Assumption

What are the concentrations at equilibrium of a 0.15 *M* solution of HCN?

[latex]\text{HCN}(aq) \rightleftharpoons \text{H}^+ (aq) + \text{CN}^- (aq)[/latex] [latex]K_c = 4.9 \text{x} 10^{-10}[/latex]

#### Check Your Learning

What are the equilibrium concentrations in a 0.25 *M* NH_{3} solution?

[latex] \text{NH}_3(aq) + \text{H}_2\text{O}(l) \rightleftharpoons \text{NH}_4+(aq) +\text{OH}-(aq)\qquad{K}_{c} = 1.8 \times 10^{-5}[/latex]

### Try It

- Complete the changes in concentrations (or pressure, if requested) for each of the following reactions.
- [latex]\begin{array}{llll}2{\text{SO}}_{3}\left(g\right)\hfill & \rightleftharpoons\hfill & 2{\text{SO}}_{2}\left(g\right)+\hfill & {\text{O}}_{2}\left(g\right)\hfill \\ \text{ }\hfill & & \text{ }\hfill & +x\hfill \\ \text{ }\hfill & & \text{ }\hfill & 0.125M\hfill \end{array}[/latex]
- [latex]\begin{array}{lllll}4{\text{NH}}_{3}\left(g\right)\hfill & +3{\text{O}}_{2}\left(g\right)\hfill & \rightleftharpoons\hfill & 2{\text{N}}_{2}\left(g\right)+\hfill & 6{\text{H}}_{2}\text{O}\left(g\right)\hfill \\ \text{ }\hfill & 3x\hfill & & \text{ }\hfill & \text{ }\hfill \\ \text{ }\hfill & 0.24M\hfill & & \text{ }\hfill & \text{ }\hfill \end{array}[/latex]
- Change in pressure:

[latex]\begin{array}{llll}2{\text{CH}}_{4}\left(g\right)\hfill & \rightleftharpoons\hfill & {\text{C}}_{2}{\text{H}}_{2}\left(g\right)+\hfill & 3{\text{H}}_{2}\left(g\right)\hfill \\ \text{ }\hfill & & x\hfill & \text{ }\hfill \\ \text{ }\hfill & & 25\text{torr}\hfill & \text{ }\hfill \end{array}[/latex] - Change in pressure:

[latex]\begin{array}{lllll}{\text{CH}}_{4}\left(g\right)+\hfill & {\text{H}}_{2}\text{O}\left(g\right)\hfill & \rightleftharpoons\hfill & \text{CO}\left(g\right)+\hfill & 3{\text{H}}_{2}\left(g\right)\hfill \\ \text{ }\hfill & x\hfill & & \text{ }\hfill & \text{ }\hfill \\ \text{ }\hfill & 5\text{atm}\hfill & & \text{ }\hfill & \text{ }\hfill \end{array}[/latex] - [latex]\begin{array}{llll}{\text{NH}}_{4}\text{Cl}\left(s\right)\hfill & \rightleftharpoons\hfill & {\text{NH}}_{3}\left(g\right)+\hfill & \text{HCl}\left(g\right)\hfill \\ & & x\hfill & \text{ }\hfill \\ & & \hfill 1.03\times {10}^{-4}M\hfill & \text{ }\hfill \end{array}[/latex]
- change in pressure:

[latex]\begin{array}{cccc}\text{Ni}\left(s\right)+\hfill & 4\text{CO}\left(g\right)\hfill & \rightleftharpoons\hfill & \text{Ni}{\left(\text{CO}\right)}_{4}\left(g\right)\hfill \\ & 4x\hfill & & \text{ }\hfill \\ & \hfill 0.40\text{atm}\hfill & & \text{ }\hfill \end{array}[/latex]

- Complete the changes in concentrations (or pressure, if requested) for each of the following reactions.
- [latex]\begin{array}{cccc}2{\text{H}}_{2}\left(g\right)+\hfill & {\text{O}}_{2}\left(g\right)\hfill & \rightleftharpoons\hfill & 2{\text{H}}_{2}\text{O}\left(g\right)\hfill \\ \text{ }\hfill & \text{ }\hfill & & +2x\hfill \\ \text{ }\hfill & \text{ }\hfill & & 1.50M\hfill \end{array}[/latex]
- [latex]\begin{array}{ccccc}{\text{CS}}_{2}\left(g\right)+\hfill & 4{\text{H}}_{2}\left(g\right)\hfill & \rightleftharpoons\hfill & {\text{CH}}_{4}\left(g\right)+\hfill & 2{\text{H}}_{2}\text{S}\left(g\right)\hfill \\ x\hfill & \text{ }\hfill & & \text{ }\hfill & \text{ }\hfill \\ 0.020M\hfill & \text{ }\hfill & & \text{ }\hfill & \text{ }\hfill \end{array}[/latex]
- Change in pressure:

[latex]\begin{array}{cccc}{\text{H}}_{2}\left(g\right)+\hfill & {\text{Cl}}_{2}\left(g\right)\hfill & \rightleftharpoons\hfill & 2\text{HCl}\left(g\right)\hfill \\ x\hfill & \text{ }\hfill & & \text{ }\hfill \\ 1.50\text{atm}\hfill & \text{ }\hfill & & \text{ }\hfill \end{array}[/latex] - Change in pressure:

[latex]\begin{array}{ccccc}2{\text{NH}}_{3}\left(g\right)\hfill & +2{\text{O}}_{2}\left(g\right)\hfill & \rightleftharpoons\hfill & {\text{N}}_{2}\text{O}\left(g\right)+\hfill & 3{\text{H}}_{2}\text{O}\left(g\right)\hfill \\ \text{ }\hfill & \text{ }\hfill & & \text{ }\hfill & x\hfill \\ \text{ }\hfill & \text{ }\hfill & & \text{ }\hfill & 60.6\text{torr}\hfill \end{array}[/latex] - [latex]\begin{array}{cccc}{\text{NH}}_{4}\text{HS}\left(s\right)\hfill & \rightleftharpoons\hfill & {\text{NH}}_{3}\left(g\right)+\hfill & {\text{H}}_{2}\text{S}\left(g\right)\hfill \\ & & x\hfill & \text{ }\hfill \\ & & 9.8\times {10}^{-6}M\hfill & \text{ }\hfill \end{array}[/latex]
- Change in pressure:

[latex]\begin{array}{cccc}\text{Fe}\left(s\right)+\hfill & 5\text{CO}\left(g\right)\hfill & \rightleftharpoons\hfill & \text{Fe}{\left(\text{CO}\right)}_{4}\left(g\right)\hfill \\ & \text{ }\hfill & & x\hfill \\ & \text{ }\hfill & & 0.012\text{atm}\hfill \end{array}[/latex]

- Why are there no changes specified for Ni in question 11 , part (f)? What property of Ni does change?
- Why are there no changes specified for NH
_{4}HS in question 12, part (e)? What property of NH_{4}HS does change?

### Try It

- Assume that the change in concentration of N
_{2}O_{4}is small enough to be neglected in the following problem.- Calculate the equilibrium concentration of both species in 1.00 L of a solution prepared from 0.129 mol of N
_{2}O_{4}with chloroform as the solvent.

[latex]{\text{N}}_{2}{\text{O}}_{4}\left(g\right)\rightleftharpoons2{\text{NO}}_{2}\left(g\right){K}_{c}=1.07\times {10}^{-5}[/latex] in chloroform - Show that the change is small enough to be neglected.

- Calculate the equilibrium concentration of both species in 1.00 L of a solution prepared from 0.129 mol of N
- Assume that the change in concentration of COCl
_{2}is small enough to be neglected in the following problem.- Calculate the equilibrium concentration of all species in an equilibrium mixture that results from the decomposition of COCl
_{2}with an initial concentration of 0.3166*M*.

[latex]{\text{COCl}}_{2}\left(g\right)\rightleftharpoons\text{CO}\left(g\right)+{\text{Cl}}_{2}\left(g\right){K}_{c}=2.2\times {10}^{-10}[/latex] - Show that the change is small enough to be neglected.

- Calculate the equilibrium concentration of all species in an equilibrium mixture that results from the decomposition of COCl
- Assume that the change in pressure of H
_{2}S is small enough to be neglected in the following problem.- Calculate the equilibrium pressures of all species in an equilibrium mixture that results from the decomposition of H
_{2}S with an initial pressure of 0.824 atm.

[latex]2{\text{H}}_{2}\text{S}\left(g\right)\rightleftharpoons2{\text{H}}_{2}\left(g\right)+{\text{S}}_{2}\left(g\right){K}_{P}=2.2\times {10}^{-6}[/latex] - Show that the change is small enough to be neglected.

- Calculate the equilibrium pressures of all species in an equilibrium mixture that results from the decomposition of H
- What are all concentrations after a mixture that contains [H
_{2}O] = 1.00*M*and [Cl_{2}O] = 1.00*M*comes to equilibrium at 25 °C?

[latex]{\text{H}}_{2}\text{O}\left(g\right)+{\text{Cl}}_{2}\text{O}\left(g\right)\rightleftharpoons2\text{HOCl}\left(g\right){K}_{c}=0.0900[/latex] - What are the concentrations of PCl
_{5}, PCl_{3}, and Cl_{2}in an equilibrium mixture produced by the decomposition of a sample of pure PCl_{5}with [PCl_{5}] = 2.00*M*?

[latex]{\text{PCl}}_{5}\left(g\right)\rightleftharpoons{\text{PCl}}_{3}\left(g\right)+{\text{Cl}}_{2}\left(g\right){K}_{c}=0.0211[/latex]

### Key Concepts and Summary

Calculating values for equilibrium constants and/or equilibrium concentrations is of practical benefit to many applications. A mathematical strategy that uses initial concentrations, changes in concentrations, and equilibrium concentrations (and goes by the acronym ICE) is useful for several types of equilibrium calculations.

### Try It

- Calculate the number of moles of HI that are at equilibrium with 1.25 mol of H
_{2}and 1.25 mol of I_{2}in a 5.00-L flask at 448 °C.

[latex]{\text{H}}_{2}+{\text{I}}_{2}\rightleftharpoons2\text{HI}{K}_{c}=50.2\text{ at }448^{\circ}\text{C}[/latex] - What is the pressure of BrCl in an equilibrium mixture of Cl
_{2}, Br_{2}, and BrCl if the pressure of Cl_{2}in the mixture is 0.115 atm and the pressure of Br_{2}in the mixture is 0.450 atm?

[latex]{\text{Cl}}_{2}\left(g\right)+{\text{Br}}_{2}\left(g\right)\rightleftharpoons2\text{BrCl}\left(g\right){K}_{P}=4.7\times {10}^{-2}[/latex] - What is the pressure of CO
_{2}in a mixture at equilibrium that contains 0.50 atm H_{2}, 2.0 atm of H_{2}O, and 1.0 atm of CO at 990 °C?

[latex]{\text{H}}_{2}\left(g\right)+{\text{CO}}_{2}\left(g\right)\rightleftharpoons{\text{H}}_{2}\text{O}\left(g\right)+\text{CO}\left(g\right){K}_{P}=1.6\text{ at }990^{\circ}\text{C}[/latex] - Sodium sulfate 10-hydrate, Na
_{2}SO_{4}[latex]\cdot [/latex] 10H_{2}O, dehydrates according to the equation

[latex]{\text{Na}}_{2}{\text{SO}}_{4}\cdot 10{\text{H}}_{2}\text{O}\left(s\right)\rightleftharpoons{\text{Na}}_{2}{\text{SO}}_{4}\left(s\right)+10{\text{H}}_{2}\text{O}\left(g\right){K}_{P}=4.08\times {10}^{-25}\text{ at }25^{\circ}\text{C}[/latex]

What is the pressure of water vapor at equilibrium with a mixture of Na_{2}SO_{4}[latex]\cdot [/latex] 10H_{2}O and NaSO_{4}? - Calcium chloride 6-hydrate, CaCl
_{2}[latex]\cdot [/latex] 6H_{2}O, dehydrates according to the equation

[latex]{\text{CaCl}}_{2}\cdot 6{\text{H}}_{2}\text{O}\left(s\right)\rightleftharpoons{\text{CaCl}}_{2}\left(s\right)+6{\text{H}}_{2}\text{O}\left(g\right){K}_{P}=5.09\times {10}^{-44}\text{ at }25^{\circ}\text{C}[/latex]

What is the pressure of water vapor at equilibrium with a mixture of CaCl_{2}[latex]\cdot [/latex] 6H_{2}O and CaCl_{2}? - Calculate the pressures of all species at equilibrium in a mixture of NOCl, NO, and Cl
_{2}produced when a sample of NOCl with a pressure of 0.500 atm comes to equilibrium according to this reaction:

[latex]2\text{NOCl}\left(g\right)\rightleftharpoons 2\text{NO}\left(g\right)+{\text{Cl}}_{2}\left(g\right){K}_{P}=4.0\times {10}^{-4}[/latex] - Calculate the equilibrium concentrations of NO, O
_{2}, and NO_{2}in a mixture at 250 °C that results from the reaction of 0.20*M*NO and 0.10*M*O_{2}. (Hint:*K*is large; assume the reaction goes to completion then comes back to equilibrium.)

[latex]2\text{NO}\left(g\right)+{\text{O}}_{2}\left(g\right)\rightleftharpoons2{\text{NO}}_{2}\left(g\right){K}_{c}=2.3\times {10}^{5}\text{ at }250^{\circ}\text{C}[/latex] - Calculate the equilibrium concentrations that result when 0.25
*M*O_{2}and 1.0*M*HCl react and come to equilibrium. (Hint:*K*is large; assume the reaction goes to completion, then comes back to equilibrium.)_{c}

[latex]4\text{HCl}\left(g\right)+{\text{O}}_{2}\left(g\right)\rightleftharpoons2{\text{Cl}}_{2}\left(g\right)+2{\text{H}}_{2}\text{O}\left(g\right){K}_{c}=3.1\times {10}^{13}[/latex] - One of the important reactions in the formation of smog is represented by the equation

[latex]{\text{O}}_{3}\left(g\right)+\text{NO}\left(g\right)\rightleftharpoons{\text{NO}}_{2}\left(g\right)+{\text{O}}_{2}\left(g\right){K}_{P}=6.0\times {10}^{34}[/latex]

What is the pressure of O_{3}remaining after a mixture of O_{3}with a pressure of 1.2 × 10^{-8}atm and NO with a pressure of 1.2 × 10^{-8}atm comes to equilibrium? (Hint:*K*is large; assume the reaction goes to completion then comes back to equilibrium.)_{P} - Calculate the pressures of NO, Cl
_{2}, and NOCl in an equilibrium mixture produced by the reaction of a starting mixture with 4.0 atm NO and 2.0 atm Cl_{2}. (Hint:*K*is small; assume the reverse reaction goes to completion then comes back to equilibrium.)_{P} - Calculate the number of grams of HI that are at equilibrium with 1.25 mol of H
_{2}and 63.5 g of iodine at 448 °C.

[latex]{\text{H}}_{2}+{\text{I}}_{2}\rightleftharpoons2\text{HI}{K}_{c}=50.2\text{ at }448^{\circ}\text{C}[/latex] - Butane exists as two isomers,
*n*-butane and isobutane.

*K*= 2.5 at 25 °CWhat is the pressure of isobutane in a container of the two isomers at equilibrium with a total pressure of 1.22 atm?_{P} - What is the minimum mass of CaCO
_{3}required to establish equilibrium at a certain temperature in a 6.50-L container if the equilibrium constant (*K*) is 0.050 for the decomposition reaction of CaCO_{c}_{3}at that temperature?

[latex]{\text{CaCO}}_{3}\left(s\right)\rightleftharpoons\text{CaO}\left(s\right)+{\text{CO}}_{2}\left(g\right)[/latex] - The equilibrium constant (
*K*) for this reaction is 1.60 at 990 °C: [latex]{\text{H}}_{2}\left(g\right)+{\text{CO}}_{2}\left(g\right)\rightleftharpoons{\text{H}}_{2}\text{O}\left(g\right)+\text{CO}\left(g\right)[/latex]_{c}

Calculate the number of moles of each component in the final equilibrium mixture obtained from adding 1.00 mol of H_{2}, 2.00 mol of CO_{2}, 0.750 mol of H_{2}O, and 1.00 mol of CO to a 5.00-L container at 990 °C.

### Try It

- At 25 °C and at 1 atm, the partial pressures in an equilibrium mixture of N
_{2}O_{4}and NO_{2}are [latex]{\text{P}}_{{\text{N}}_{2}{\text{O}}_{4}}=0.70\text{atm}[/latex] and [latex]{\text{P}}_{{\text{NO}}_{2}}=0.30\text{atm.}[/latex]- Predict how the pressures of NO
_{2}and N_{2}O_{4}will change if the total pressure increases to 9.0 atm. Will they increase, decrease, or remain the same? - Calculate the partial pressures of NO
_{2}and N_{2}O_{4}when they are at equilibrium at 9.0 atm and 25 °C.

- Predict how the pressures of NO
- In a 3.0-L vessel, the following equilibrium partial pressures are measured: N
_{2}, 190 torr; H_{2}, 317 torr; NH_{3}, 1.00 × 10^{3}torr: [latex]{\text{N}}_{2}\left(g\right)+3{\text{H}}_{2}\left(g\right)\rightleftharpoons2{\text{NH}}_{3}\left(g\right)[/latex]- How will the partial pressures of H
_{2}, N_{2}, and NH_{3}change if H_{2}is removed from the system? Will they increase, decrease, or remain the same? - Hydrogen is removed from the vessel until the partial pressure of nitrogen, at equilibrium, is 250 torr. Calculate the partial pressures of the other substances under the new conditions.

- How will the partial pressures of H
- The equilibrium constant (
*K*) for this reaction is 5.0 at a given temperature: [latex]\text{CO}\left(g\right)+{\text{H}}_{2}\text{O}\left(g\right)\rightleftharpoons{\text{CO}}_{2}\left(g\right)+{\text{H}}_{2}\left(g\right)[/latex]_{c}- On analysis, an equilibrium mixture of the substances present at the given temperature was found to contain 0.20 mol of CO, 0.30 mol of water vapor, and 0.90 mol of H
_{2}in a liter. How many moles of CO_{2}were there in the equilibrium mixture? - Maintaining the same temperature, additional H
_{2}was added to the system, and some water vapor was removed by drying. A new equilibrium mixture was thereby established containing 0.40 mol of CO, 0.30 mol of water vapor, and 1.2 mol of H_{2}in a liter. How many moles of CO_{2}were in the new equilibrium mixture? Compare this with the quantity in part (a), and discuss whether the second value is reasonable. Explain how it is possible for the water vapor concentration to be the same in the two equilibrium solutions even though some vapor was removed before the second equilibrium was established.

- On analysis, an equilibrium mixture of the substances present at the given temperature was found to contain 0.20 mol of CO, 0.30 mol of water vapor, and 0.90 mol of H
- Antimony pentachloride decomposes according to this equation: [latex]{\text{SbCl}}_{5}\left(g\right)\rightleftharpoons{\text{SbCl}}_{3}\left(g\right)+{\text{Cl}}_{2}\left(g\right)[/latex]

An equilibrium mixture in a 5.00-L flask at 448 °C contains 3.85 g of SbCl_{5}, 9.14 g of SbCl_{3}, and 2.84 g of Cl_{2}. How many grams of each will be found if the mixture is transferred into a 2.00-L flask at the same temperature? - Consider the reaction between H
_{2}and O_{2}at 1000 K

[latex]2H_{2}(g)+O_{2}(g)\rightleftharpoons{2H_{2}O(g)}[/latex] [latex]K_{P}=\frac{(P_{H_{2}O})^{2}}{(P_{O_{2}})(P_{H_{2}})^{3}}=1.33\times{10^{20}}[/latex]

If 0.500 atm of H2 and 0.500 atm of O2 are allowed to come to equilibrium at this temperature, what are the partial pressures of the components? - An equilibrium is established according to the following equation

[latex]{\text{Hg}}_{2}{}^{2+}\left(aq\right)+{\text{NO}}_{3}{}^{-}\left(aq\right)+3{\text{H}}^{+}\left(aq\right)\rightleftharpoons2{\text{Hg}}^{2+}\left(aq\right)+{\text{HNO}}_{2}\left(aq\right)+{\text{H}}_{2}\text{O}\left(l\right){K}_{c}=4.6[/latex]

What will happen in a solution that is 0.20*M*each in [latex]{\text{Hg}}_{2}{}^{2+}[/latex], [latex]{\text{NO}}_{3}{}^{-}[/latex], H^{+}, Hg^{2+}, and HNO_{2}?- [latex]{\text{Hg}}_{2}{}^{2+}[/latex] will be oxidized and [latex]{\text{NO}}_{3}{}^{-}[/latex] reduced.
- [latex]{\text{Hg}}_{2}{}^{2+}[/latex] will be reduced and [latex]{\text{NO}}_{3}{}^{-}[/latex] oxidized.
- Hg
^{2+}will be oxidized and HNO_{2}reduced. - Hg
^{2+}will be reduced and HNO_{2}oxidized. - There will be no change, because all reactants and products have an activity of 1.

- Consider the equilibrium: [latex]4{\text{NO}}_{2}\left(g\right)+6{\text{H}}_{2}\text{O}\left(g\right)\rightleftharpoons4{\text{NH}}_{3}\left(g\right)+7{\text{O}}_{2}\left(g\right)[/latex]
- What is the expression for the equilibrium constant (
*K*) of the reaction?_{c} - How must the concentration of NH
_{3}change to reach equilibrium if the reaction quotient is less than the equilibrium constant? - If the reaction were at equilibrium, how would a decrease in pressure (from an increase in the volume of the reaction vessel) affect the pressure of NO
_{2}? - If the change in the pressure of NO
_{2}is 28 torr as a mixture of the four gases reaches equilibrium, how much will the pressure of O_{2}change?

- What is the expression for the equilibrium constant (
- The binding of oxygen by hemoglobin (Hb), giving oxyhemoglobin (HbO
_{2}), is partially regulated by the concentration of H_{3}O^{+}and dissolved CO_{2}in the blood. Although the equilibrium is complicated, it can be summarized as

[latex]{\text{HbO}}_{2}\left(aq\right)+{\text{H}}_{3}{\text{O}}^{+}\left(aq\right)+{\text{CO}}_{2}\left(g\right)\rightleftharpoons{\text{CO}}_{2}-\text{Hb}-{\text{H}}^{+}+{\text{O}}_{2}\left(g\right)+{\text{H}}_{2}\text{O}\left(l\right)[/latex]- Write the equilibrium constant expression for this reaction.
- Explain why the production of lactic acid and CO
_{2}in a muscle during exertion stimulates release of O_{2}from the oxyhemoglobin in the blood passing through the muscle.

- The hydrolysis of the sugar sucrose to the sugars glucose and fructose follows a first-order rate equation for the disappearance of sucrose.

[latex]{\text{C}}_{12}{\text{H}}_{22}{\text{O}}_{11}\left(aq\right)+{\text{H}}_{2}\text{O}\left(l\right)\rightarrow{\text{C}}_{6}{\text{H}}_{12}{\text{O}}_{6}\left(aq\right)+{\text{C}}_{6}{\text{H}}_{12}{\text{O}}_{6}\left(aq\right)[/latex]

Rate =*k*[C_{12}H_{22}O_{11}]

In neutral solution,*k*= 2.1 × 10^{-11}/s at 27 °C. (As indicated by the rate constant, this is a very slow reaction. In the human body, the rate of this reaction is sped up by a type of catalyst called an enzyme.) (Note: That is not a mistake in the equation—the products of the reaction, glucose and fructose, have the same molecular formulas, C_{6}H_{12}O_{6}, but differ in the arrangement of the atoms in their molecules). The equilibrium constant for the reaction is 1.36 × 10^{5}at 27 °C. What are the concentrations of glucose, fructose, and sucrose after a 0.150*M*aqueous solution of sucrose has reached equilibrium? Remember that the activity of a solvent (the effective concentration) is 1). - The density of trifluoroacetic acid vapor was determined at 118.1 °C and 468.5 torr, and found to be 2.784 g/L. Calculate
*K*for the association of the acid._{c}

- Liquid N
_{2}O_{3}is dark blue at low temperatures, but the color fades and becomes greenish at higher temperatures as the compound decomposes to NO and NO_{2}. At 25 °C, a value of*K*= 1.91 has been established for this decomposition. If 0.236 moles of N_{P}_{2}O_{3}are placed in a 1.52-L vessel at 25 °C, calculate the equilibrium partial pressures of N_{2}O_{3}(*g*), NO_{2}(*g*), and NO(*g*). - A 1.00-L vessel at 400 °C contains the following equilibrium concentrations: N
_{2}, 1.00*M*; H_{2}, 0.50*M*; and NH_{3}, 0.25*M*. How many moles of hydrogen must be removed from the vessel to increase the concentration of nitrogen to 1.1*M*? - A 0.010
*M*solution of the weak acid HA has an osmotic pressure (see chapter on solutions and colloids) of 0.293 atm at 25 °C. A 0.010*M*solution of the weak acid HB has an osmotic pressure of 0.345 atm under the same conditions.- Which acid has the larger equilibrium constant for ionization

HA [latex]\left[\text{HA}\left(aq\right)\rightleftharpoons{\text{A}}^{-}\left(aq\right)+{\text{H}}^{+}\left(aq\right)\right][/latex] or HB [latex]\left[\text{HB}\left(aq\right)\rightleftharpoons{\text{H}}^{+}\left(aq\right)+{\text{B}}^{-}\left(aq\right)\right][/latex] ? - What are the equilibrium constants for the ionization of these acids?

(Hint: Remember that each solution contains three dissolved species: the weak acid (HA or HB), the conjugate base (A^{–}or B^{–}), and the hydrogen ion (H^{+}). Remember that osmotic pressure (like all colligative properties) is related to the total number of solute particles. Specifically for osmotic pressure, those concentrations are described by molarities.)

- Which acid has the larger equilibrium constant for ionization