Question

At \(35^{\circ} \mathrm{C}, K=1.6 \times 10^{-5}\) for the reaction $$2 \mathrm{NOCl}(g) \rightleftharpoons 2 \mathrm{NO}(g)+\mathrm{Cl}_{2}(g)$$ Calculate the concentrations of all species at equilibrium for each of the following original mixtures. a. 2.0 moles of pure \(\mathrm{NOCl}\) in a 2.0 \(\mathrm{L}\) flask b. 1.0 mole of NOCl and 1.0 mole of NO in a 1.0 - flask c. 2.0 moles of \(\mathrm{NOCl}\) and 1.0 mole of \(\mathrm{Cl}_{2}\) in a \(1.0-\mathrm{L}\) flask

Short answer

For scenario a, the equilibrium concentrations are \([\mathrm{NOCl}] \approx 0.974\, \mathrm{M}\), \([\mathrm{NO}] \approx 0.026\, \mathrm{M}\), and \([\mathrm{Cl_2}] \approx 1.3 \times 10^{-2}\, \mathrm{M}\).

Step-by-step solution

Set up an ICE table

For this scenario, consider the initial concentrations as follows: \[\mathrm{[NOCl]_0 = \frac{2.0\, moles}{2.0\, L} = 1.0\, M}\] \[\mathrm{[NO]_0 = [Cl_2]_0 = 0}\] Create an ICE table: | | \(\mathrm{NOCl}\) | \(\mathrm{NO}\) | \(\mathrm{Cl_2}\) | |--------|---------|-------|---------| | Initial| 1.0 M | 0 | 0 | | Change | -2x | +2x | +x | | Equil. | 1.0-2x | 2x | x |

Write the equilibrium expression

Write the equation for the equilibrium constant: \[K=\frac{[\mathrm{NO}]^2[\mathrm{Cl}_{2}]}{[\mathrm{NOCl}]^2}\]

Substitute values and solve for x

Substitute the values from the ICE table into the equilibrium expression: \[1.6 \times 10^{-5} = \frac{(2x)^2(x)}{(1.0-2x)^2}\] Solve the quadratic equation for x. Since the equilibrium constant is very small, we can assume that \(2x \ll 1\) and simplify to: \[ 1.6 \times 10^{-5} \approx \frac{4x^3}{1} \] Solving for x gives: \[ x \approx 1.3 \times 10^{-2} \]

Find equilibrium concentrations

Now calculate the equilibrium concentrations: \[ [\mathrm{NOCl}]_{eq} = 1.0 - 2x \approx 1.0 - 2(1.3 \times 10^{-2}) \approx 0.974\, \mathrm{M} \] \[ [\mathrm{NO}]_{eq} = 2x \approx 2(1.3 \times 10^{-2}) \approx 0.026\, \mathrm{M} \] \[ [\mathrm{Cl_2}]_{eq} = x \approx 1.3 \times 10^{-2}\, \mathrm{M} \] So, for scenario a, the equilibrium concentrations are approximately \([\mathrm{NOCl}] \approx 0.974\, \mathrm{M}\), \([\mathrm{NO}] \approx 0.026\, \mathrm{M}\), and \([\mathrm{Cl_2}] \approx 1.3 \times 10^{-2}\, \mathrm{M}\). Repeat the same process for scenarios b and c.

Key concepts

ICE Table

The ICE table method is a crucial tool in chemical equilibrium problems. This tool helps in identifying the change in concentration of different reactants and products over the course of a reaction. ICE stands for Initial, Change, and Equilibrium. The ICE table is set up by listing the balanced chemical reaction and organizing the concentrations of all species involved.
### How to Setup an ICE Table: - **Initial concentrations**: Start by determining the initial concentrations of each species. For gases and aqueous solutions, use molarity ( ext{Molarity} = rac{ ext{moles}}{ ext{volume}}). - **Change in concentrations**: Describe how the concentrations will change as the system moves toward equilibrium. Use variables (like -2x, +x) to signify changes for stoichiometrically related species. - **Equilibrium concentrations**: These are the concentrations after the reaction has gone to completion based on the initial data and changes. The ICE table simplifies the process of applying changes segmentally, making it easier to handle complex equilibrium calculations.

Equilibrium Expression

In chemical reactions, the equilibrium expression is fundamental to understanding how reactants and products convert into one another over time. The equilibrium expression connects them mathematically to the equilibrium constant (K).This expression is represented as the ratio of the concentrations of products to reactants, each raised to the power of their coefficients from the balanced equation.
### Writing an Equilibrium Expression:To write an equilibrium expression:- Take the balanced chemical equation.- Represent the concentration of each product and reactant in molarity.- For products, multiply their concentrations together, each term taken to the power of their stoichiometric coefficients.- Do the same for reactants.For example, for the reaction: \[ 2 \text{NOCl} \rightleftharpoons 2 \text{NO} + \text{Cl}_2 \]The equilibrium expression is represented as:\[ K = \frac{[\text{NO}]^2[\text{Cl}_2]}{[\text{NOCl}]^2} \]This formula is central in predicting and understanding the position of the equilibrium.

Reaction Quotient

The reaction quotient, symbolized by Q, serves as a tool to predict the direction in which a reaction mixture will proceed to reach equilibrium. **Understanding the Reaction Quotient (Q):** - **Purpose**: Evaluate the current state of a reaction relative to its equilibrium state, as represented by K. - **Calculation**: Substitute the concentrations at any given time into the equilibrium expression to calculate Q, just like K. - **Comparison with K**: - If **Q < K**, the reaction will proceed forward (toward products) to reach equilibrium. - If **Q > K**, the reaction will go in reverse (toward reactants), to reach equilibrium. - If **Q = K**, the system is already at equilibrium, and no net change will occur. Knowing how the reaction quotient behaves with respect to K helps in predicting how far or in what direction a reaction needs to go to achieve equilibrium.

Equilibrium Constant (K)

The equilibrium constant, denoted by K, is a fundamental concept in chemical equilibrium. It expresses the ratio of concentrations of products to reactants for a reaction at equilibrium.### Characteristics of K:- **Constants**: Dependent on temperature. Each equilibrium has a unique K value at a given temperature.- **Magnitude**: - A large K (>>1) indicates a reaction heavily favoring products. - A small K (<<1) means the reaction favors reactants.### Calculating K:To find K, use the concentrations of equilibrium species in the equilibrium expression:- Rearrange the balanced equation for products and reactants.- Insert their equilibrium concentrations to solve for K.For example, for the given reaction:\[ 2 \text{NOCl} \rightleftharpoons 2 \text{NO} + \text{Cl}_2 \]Given K at a specific temperature, concentrations can be calculated. It helps not only in finding the final equilibrium state but also in comparing how different initial concentrations reach equilibrium under the same conditions.