Question

Suppose that for the following 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: \([\mathrm{NO}(g)] 8.1 \times 10^{-3} M,\left[\mathrm{H}_{2}(g)\right]=4.1 \times 10^{-5} M,\left[\mathrm{N}_{2}(g)\right]\) \(=5.3 \times 10^{-2} \mathrm{M}\) 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.

Short answer

The equilibrium constant for the reaction at this temperature is approximately \(4 \times 10^6\).

Step-by-step solution

Write the equilibrium expression

To calculate the equilibrium constant, we need to set up the equilibrium expression using the balanced chemical equation and the Law of Mass Action. The reaction is given by: \[2 \mathrm{NO(g)} + 2 \mathrm{H_2(g)} \rightleftharpoons \mathrm{N_2(g)} + 2 \mathrm{H_2O(g)}\] The equilibrium constant expression can be written as: \[K = \frac{[\mathrm{N_2}] [\mathrm{H_2O}]^2}{[\mathrm{NO}]^2 [\mathrm{H_2}]^2}\]

Fill in the given concentrations

Now we have to use the given concentrations and substitute them into the equilibrium expression: \[[\mathrm{NO}] = 8.1 \times 10^{-3} \mathrm{M}\] \[[\mathrm{H_2}] = 4.1 \times 10^{-5} \mathrm{M}\] \[[\mathrm{N_2}] = 5.3 \times 10^{-2} \mathrm{M}\] \[[\mathrm{H_2O}] = 2.9 \times 10^{-3} \mathrm{M}\]

Calculate the value of K

Substitute the concentration values into the equilibrium expression and solve for K: \[K = \frac{(5.3 \times 10^{-2}) \times (2.9 \times 10^{-3})^2}{(8.1 \times 10^{-3})^2 \times (4.1 \times 10^{-5})^2}\] Now solve the equation to find the value of K: \[K = \frac{(5.3 \times 10^{-2}) \times (8.41 \times 10^{-6})}{(6.561 \times 10^{-5}) \times (1.681 \times 10^{-9})}\] Calculate K: \[K = \frac{4.4573 \times 10^{-7}}{1.1027 \times 10^{-13}}\] \[K \approx 4 \times 10^6\] The equilibrium constant for the reaction at this temperature is approximately \(4 \times 10^6\).

Key concepts

Law of Mass Action

The Law of Mass Action is a fundamental concept in chemistry that allows us to describe the relationship between the concentrations of reactants and products at equilibrium. This law states that for a reversible chemical reaction at equilibrium, the rate of the forward reaction equals the rate of the reverse reaction. This balance can be represented with an equilibrium constant expression, which is derived from the balanced chemical equation.
The equilibrium constant, often symbolized as \( K \), is calculated by multiplying the concentrations of the products, each raised to their respective stoichiometric coefficients, and dividing by the product of the reactants raised to their coefficients.

• The greater the value of \( K \), the more products are present at equilibrium compared to reactants, indicating that the reaction favors product formation.
• If \( K \) is less than one, it means that reactants are favored, and the equilibrium position lies to the left.

Chemical Equilibrium

Chemical equilibrium occurs when a chemical reaction takes place in such a manner that the rate at which the products are formed equals the rate at which the reactants are reformed. At this point, the concentrations of reactants and products remain constant over time.
Reaching equilibrium doesn't mean that the concentrations of reactants and products are equal, but rather that they reach a stable state. The direction in which the equilibrium position will lie depends on the equilibrium constant \( K \), as discussed above.
Characteristics of chemical equilibrium include:

• Dynamic nature - the reaction doesn't stop, but rather continues at an equal rate in both directions.
• Closed system - no substances are added or removed from the system.
• Dependence on temperature - altering the temperature can change the equilibrium position.

Reaction Concentration

Understanding reaction concentration is essential when calculating equilibrium constants. Reaction concentration refers to the amount of a substance in a given volume of a mixture. In chemical reactions, concentrations are usually expressed in molarity (M), which is moles per liter.
In the context of equilibrium, the concentrations of reactants and products need to be taken into account. As a reaction proceeds, the concentrations of the reactants and products change until they reach equilibrium.
Some key points to remember include:

• Initial concentrations change as the reaction approaches equilibrium.
• Knowing the equilibrium concentrations allows us to calculate the equilibrium constant \( K \) using the equilibrium expression.

Balanced Chemical Equation

A balanced chemical equation is crucial for understanding and working with chemical reactions. This equation gives us the exact proportions of reactants and products involved, ensuring that the law of conservation of mass is followed.
To balance a chemical equation, one must adjust the coefficients of each compound or element involved, ensuring that the number of atoms of each element in the reactants equals the number in the products.
Steps to balance a chemical equation include:

• Write down the unbalanced equation.
• Determine the number of atoms for each element in reactants and products.
• Add coefficients to make the number of atoms equal on both sides.
• Make sure that all coefficients are in the lowest possible ratio.

Our reaction example, \( 2 \mathrm{NO}(g) + 2 \mathrm{H}_{2}(g) \rightleftharpoons \mathrm{N}_{2}(g) + 2 \mathrm{H}_{2} \mathrm{O}(g) \), is balanced, providing the foundation for setting up the equilibrium expression.