Question

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.

Short answer

The value of the equilibrium constant (K) for the given reaction at this temperature is approximately \(K \approx 2.16 \times 10^{5}\).

Step-by-step solution

Write down the balanced chemical equation

Here's the balanced chemical equation for the given reaction: \[2 NO(g)+2 H_{2}(g) \rightleftharpoons N_{2}(g)+2 H_{2}O(g)\]

Write down the equilibrium constant expression

According to the law of mass action, the equilibrium constant expression for a given reaction is obtained by multiplying the concentration of the products raised to their stoichiometric coefficients and divided by the multiplication of the concentration of the reactants raised to their stoichiometric coefficients. For the given reaction, the expression for K is: \[K = \frac{[N_{2}]^{1} [H_{2}O]^{2}}{[NO]^{2} [H_{2}]^{2}}\]

Substitute the concentrations at equilibrium

We have the equilibrium concentrations as follows: \[[NO(g)] = 8.1 \times 10^{-3} M\] \[[H_{2}(g)] = 4.1 \times 10^{-5} M\] \[[N_{2}(g)] = 5.3 \times 10^{-2} M\] \[[H_{2}O(g)] = 2.9 \times 10^{-3} M\] Substitute these values into the K expression: \[K = \frac{(5.3 \times 10^{-2})^{1} (2.9 \times 10^{-3})^{2}}{(8.1 \times 10^{-3})^{2} (4.1 \times 10^{-5})^{2}}\]

Calculate K

Perform the calculations using a calculator to find the value of K at this temperature: \[K \approx \frac{(5.3 \times 10^{-2}) (2.9 \times 10^{-3})^{2}}{(8.1 \times 10^{-3})^{2} (4.1 \times 10^{-5})^{2}} = 2.16 \times 10^{5}\]

Write down the final result

The value of the equilibrium constant (K) for the given reaction at this temperature is approximately: \[K \approx 2.16 \times 10^{5}\]

Key concepts

Law of Mass Action

The Law of Mass Action is fundamental in the field of chemistry, especially when dealing with reactions at equilibrium. It helps us understand how concentrations of chemicals behave when the reaction reaches a balance. The law states that the rate of a chemical reaction is directly proportional to the product of the concentrations of the reactants, each raised to a power equal to its stoichiometric coefficient in the balanced reaction equation.

Here's why this is important: at equilibrium, the rate of the forward reaction equals the rate of the backward reaction. This results in constant concentrations of reactants and products. The equilibrium constant, often denoted as \( K \), thus can be expressed using the concentrations of the respective species involved in the chemical reaction. So, for a generic reaction, \( aA + bB \rightleftharpoons cC + dD \), the equilibrium constant expression would be \( K = \frac{[C]^c[D]^d}{[A]^a[B]^b} \).

In the example you have, the reaction involves nitrogen monoxide (\( NO \)) and hydrogen \(( H_2 )\) reacting to form nitrogen \(( N_2 )\) and water \(( H_2O )\). The equilibrium expression is derived following the Law of Mass Action principles, as seen in the reaction's step-by-step solution.

Chemical Equilibrium

Chemical equilibrium occurs when a reversible reaction ceases to change the populations of reactants and products over time. At this point, the reaction is said to be in a state of balance, even though the processes of conversion between reactants and products continue. It is an essential concept in chemistry, as it indicates the point at which a reaction has reached a condition where the rate of the forward reaction equals the rate of the reverse reaction.

Knowing when a system is at equilibrium tells you a lot about its stability and predictability. In our reaction involving \( NO \), \( H_2 \), \( N_2 \), and \( H_2O \), achieving equilibrium means that the concentrations provided reflect a balance where the formation of products and reactants occur at the same rate. This also influences various conditions like pressure, temperature, and concentration that could shift the equilibrium or change the position of equilibrium depending on external conditions.

The concept is applicable based on Le Chatelier's Principle, which predicts the effect of changes in concentration, pressure, and temperature on an equilibrium system. Thus, understanding chemical equilibrium enables predictions about how changes in a reaction environment can influence its balance.

Stoichiometry

Stoichiometry is the area of chemistry that focuses on the relative quantities of reactants and products in chemical reactions. It involves calculations that relate the moles and masses of reactants and products according to the coefficients found in a balanced chemical equation.

In the context of the given reaction, stoichiometry plays a critical role in determining the equilibrium constant. The coefficients in the balanced equation, such as the 2 in front of \( NO \) and \( H_2 \), indicate the proportions in which molecules react or form during the reaction. These coefficients become the exponents in the equilibrium constant expression according to the Law of Mass Action, where they help calculate how the equilibrium concentration values contribute to the overall equilibrium constant \( K \).

By knowing the stoichiometric relationships, you can understand the quantitative side of chemical reactions, predict the amounts of products formed, or determine the amounts of reactants needed to achieve a certain level of output. Stoichiometry is not just crucial for calculating equilibrium constants but also for performing any chemical calculation where precision is paramount.