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Chapter 13: Problem 19

For a typical equilibrium problem, the value of \(K\) and the initial reaction conditions are given for a specific reaction, and you are asked to calculate the equilibrium concentrations. Many of these calculations involve solving a quadratic or cubic equation. What can you do to avoid solving a quadratic or cubic equation and still come up with reasonable equilibrium concentrations?

Short Answer

Expert verified
To avoid solving quadratic or cubic equations when calculating equilibrium concentrations, you can make certain assumptions and simplifications. First, set up an ICE (Initial, Change, Equilibrium) table to organize the initial concentrations, changes, and equilibrium concentrations. Then, make a reasonable assumption, such as applying the 5% Rule, whereby you can ignore the change in concentration if it is less than 5% of the initial concentration. Next, substitute the approximated equilibrium concentrations into the equilibrium expression, resulting in a simpler linear equation. Finally, solve the equation for the variable, calculate the change in concentration, and verify that the assumption was reasonable. This method can provide reasonable equilibrium concentration estimates without the need to solve quadratic or cubic equations, as long as the assumptions made are valid.

Step by step solution

01

Understand the equilibrium expression

First, write down the balanced chemical equation for the reaction and the corresponding equilibrium expression in terms of the reaction quotient, Q. The reaction quotient is defined as the product of the concentrations of products divided by the product of the concentrations of reactants, each raised to their stoichiometric coefficients. When Q is equal to the equilibrium constant K, the system is at equilibrium.
02

Set up an ICE (Initial, Change, Equilibrium) table

An ICE table helps to organize the initial concentrations, changes, and equilibrium concentrations of the species involved in the reaction. To set up the ICE table, list the initial concentrations given, and then express the changes in concentration in terms of a variable, usually represented by "x". Finally, represent the equilibrium concentrations as the initial concentrations plus or minus the change in concentrations. Write the expressions for the equilibrium concentrations under the appropriate species.
03

Make a reasonable assumption

To avoid solving a quadratic or cubic equation while determining equilibrium concentrations, we can make certain assumptions. One common assumption is that the change in concentration (represented by 'x') is small compared to the initial concentrations. This is known as 'The 5% Rule,' where if the change in concentration is less than 5% of the initial concentration, we can assume that it is negligible. In such cases, we can approximate the equilibrium concentrations by ignoring the change due to the equilibrium constant.
04

Substitute the approximated equilibrium concentrations into the equilibrium expression

By making the assumption in Step 3, we can now substitute the approximated equilibrium concentrations from the ICE table into the equilibrium expression, replacing the initial concentrations with the approximated concentrations. This should result in a simpler, linear equation, which can be easily solved for the variable 'x'.
05

Calculate the change in concentration and verify the assumption

Solve the simplified linear equation from Step 4 to find the change in concentration, 'x'. Use the value of 'x' to calculate the equilibrium concentrations for each species involved in the reaction. Verify if the assumption made in Step 3 was reasonable by ensuring that the change in concentration is less than 5% of the initial concentration. If the assumption holds, the calculated equilibrium concentrations are reasonable approximations for the true values. By following these steps and making appropriate simplifications and assumptions, it is possible to estimate equilibrium concentrations without solving quadratic or cubic equations. However, it is essential to verify that the assumptions made are reasonable, and if not, it may be necessary to use more advanced techniques to solve for the equilibrium concentrations accurately.

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Most popular questions from this chapter

A sample of \(\mathrm{N}_{2} \mathrm{O}_{4}(g)\) is placed in an empty cylinder at \(25^{\circ} \mathrm{C}\). After equilibrium is reached the total pressure is \(1.5\) atm and \(16 \%\) (by moles) of the original \(\mathrm{N}_{2} \mathrm{O}_{4}(g)\) has dissociated to \(\mathrm{NO}_{2}(g)\). a. Calculate the value of \(K_{\mathrm{p}}\) for this dissociation reaction at \(25^{\circ} \mathrm{C}\). b. If the volume of the cylinder is increased until the total pressure is \(1.0 \mathrm{~atm}\) (the temperature of the system remains constant), calculate the equilibrium pressure of \(\mathrm{N}_{2} \mathrm{O}_{4}(g)\) and \(\mathrm{NO}_{2}(g)\). c. What percentage (by moles) of the original \(\mathrm{N}_{2} \mathrm{O}_{4}(g)\) is dissociated at the new equilibrium position (total pressure \(=1.00 \mathrm{~atm}\) )?

Novelty devices for predicting rain contain cobalt(II) chloride and are based on the following equilibrium: $$ \underset{\text { Purple }}{\mathrm{CoCl}_{2}(s)}+6 \mathrm{H}_{2} \mathrm{O}(g) \rightleftharpoons \underset{\text { Pink }}{\mathrm{CoCl}_{2} \cdot 6 \mathrm{H}_{2} \mathrm{O}(s)} $$ What color will such an indicator be if rain is imminent?

Consider the reaction $$ \mathrm{P}_{4}(g) \longrightarrow 2 \mathrm{P}_{2}(g) $$ where \(K_{\mathrm{p}}=1.00 \times 10^{-1}\) at \(1325 \mathrm{~K}\). In an experiment where \(\mathrm{P}_{4}(g)\) is placed into a container at \(1325 \mathrm{~K}\), the equilibrium mixture of \(\mathrm{P}_{4}(g)\) and \(\mathrm{P}_{2}(g)\) has a total pressure of \(1.00 \mathrm{~atm} .\) Calculate the equilibrium pressures of \(\mathrm{P}_{4}(g)\) and \(\mathrm{P}_{2}(g) .\) Calculate the fraction (by moles) of \(\mathrm{P}_{4}(g)\) that has dissociated to reach equilibrium.

At \(125^{\circ} \mathrm{C}, K_{\mathrm{p}}=0.25\) for the reaction $$ 2 \mathrm{NaHCO}_{3}(s) \rightleftharpoons \mathrm{Na}_{2} \mathrm{CO}_{3}(s)+\mathrm{CO}_{2}(g)+\mathrm{H}_{2} \mathrm{O}(g) $$ A 1.00-L flask containing \(10.0 \mathrm{~g} \mathrm{NaHCO}_{3}\) is evacuated and heated to \(125^{\circ} \mathrm{C}\). a. Calculate the partial pressures of \(\mathrm{CO}_{2}\) and \(\mathrm{H}_{2} \mathrm{O}\) after equilibrium is established. b. Calculate the masses of \(\mathrm{NaHCO}_{3}\) and \(\mathrm{Na}_{2} \mathrm{CO}_{3}\) present at equilibrium. c. Calculate the minimum container volume necessary for all of the \(\mathrm{NaHCO}_{3}\) to decompose.

Write the equilibrium expression \((K)\) for each of the following gas-phase reactions. a. \(\mathrm{N}_{2}(g)+\mathrm{O}_{2}(g) \rightleftharpoons 2 \mathrm{NO}(g)\) b. \(\mathrm{N}_{2} \mathrm{O}_{4}(g) \rightleftharpoons 2 \mathrm{NO}_{2}(g)\) c. \(\mathrm{SiH}_{4}(g)+2 \mathrm{Cl}_{2}(g) \rightleftharpoons \mathrm{SiCl}_{4}(g)+2 \mathrm{H}_{2}(g)\) d. \(2 \mathrm{PBr}_{3}(g)+3 \mathrm{Cl}_{2}(g) \rightleftharpoons 2 \mathrm{PCl}_{3}(g)+3 \mathrm{Br}_{2}(g)\)
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