Question

At \(373 \mathrm{~K}, K_{p}=0.416\) for the equilibrium \(2 \mathrm{NOBr}(g) \rightleftharpoons 2 \mathrm{NO}(g)+\mathrm{Br}_{2}(g)\) If the pressures of \(\mathrm{NOBr}(g)\) and \(\mathrm{NO}(g)\) are equal, what is the equilibrium pressure of \(\mathrm{Br}_{2}(\mathrm{~g})\) ?

Short answer

The equilibrium pressure of \(\mathrm{Br}_{2}(g)\) is \(0.416 \cdot P_{\mathrm{NO}}\) when the pressures of \(\mathrm{NOBr}(g)\) and \(\mathrm{NO}(g)\) are equal.

Step-by-step solution

Write down the equilibrium constant expression

For the reaction \(2 \mathrm{NOBr}(g) \rightleftharpoons 2 \mathrm{NO}(g) + \mathrm{Br}_{2}(g)\), the equilibrium constant expression is given by: \[K_p = \frac{(P_{\mathrm{NO}})^2 \cdot P_{\mathrm{Br_2}}}{(P_{\mathrm{NOBr}})^2}\] where \(P_{\mathrm{NO}}\), \(P_{\mathrm{Br_2}}\), and \(P_{\mathrm{NOBr}}\) are the equilibrium pressures of \(\mathrm{NO}(g)\), \(\mathrm{Br}_{2}(g)\), and \(\mathrm{NOBr}(g)\) respectively.

Given that the pressures of \(\mathrm{NOBr}(g)\) and \(\mathrm{NO}(g)\) are equal

It is given that \(P_{\mathrm{NOBr}} = P_{\mathrm{NO}}\). We can rewrite the equilibrium constant expression as follows: \[K_p = \frac{P_{\mathrm{Br_2}}}{P_{\mathrm{NOBr}}}\]

Substitute the given value of \(K_p\) and solve for the equilibrium pressure of \(\mathrm{Br}_{2}(g)\)

Now, substitute the given value of \(K_p = 0.416\) into the expression: \[\begin{aligned} 0.416 &= \frac{P_{\mathrm{Br_2}}}{P_{\mathrm{NOBr}}} \end{aligned}\] To find the equilibrium pressure of \(\mathrm{Br}_{2}(g)\), just multiply both sides by \(P_{\mathrm{NOBr}}\): \[\begin{aligned} P_{\mathrm{Br_2}} &= 0.416 \cdot P_{\mathrm{NOBr}} \end{aligned}\] Since \(P_{\mathrm{NOBr}} = P_{\mathrm{NO}}\), we can write: \[\begin{aligned} P_{\mathrm{Br_2}} &= 0.416 \cdot P_{\mathrm{NO}} \end{aligned}\] Thus, the equilibrium pressure of \(\mathrm{Br}_{2}(g)\) is \(0.416\) times the pressure of \(\mathrm{NO}(g)\) when the pressures of \(\mathrm{NOBr}(g)\) and \(\mathrm{NO}(g)\) are equal.

Key concepts

Chemical Equilibrium

Chemical equilibrium occurs when a chemical reaction and its reverse reaction proceed at the same rates, leading to the constant concentrations or pressures of the reactants and products over time. This state can be visually imagined as a balance scale where the rate of the forward reaction (reactants turning into products) is equal to the rate of the backward reaction (products turning back into reactants). However, the equilibrium does not mean that the reactants and products are present in equal amounts; rather, their ratios are fixed at a given temperature and don't change as time passes.

Understanding chemical equilibrium is crucial not only for academic studies but also for numerous practical applications. For instance, in industrial chemical processes, achieving a certain equilibrium can be essential for maximizing the yield of desired products while minimizing costs and unreacted materials.

Equilibrium Constant Expression

The equilibrium constant expression, denoted as K, is a mathematical representation of a chemical equilibrium state. For gaseous reactions, it is often expressed in terms of partial pressures and called the equilibrium constant for pressure, or Kp. The expression involves raising the partial pressures of the products and reactants to the power of their coefficients found in the balanced chemical equation and then taking their ratio.

To apply the equilibrium constant expression effectively, a step-by-step approach is beneficial. First, write the balanced chemical equation. Then, construct the expression by placing the products over the reactants, raising them to the power of their stoichiometric coefficients. Lastly, if necessary, substitute known values into the expression and solve for the unknown. The numerical value of Kp offers insights into the extent of the reaction, with higher values indicating a greater extent of product formation.

Partial Pressures

In the realm of gas-phase reactions, partial pressures refer to the pressure each gas in a mixture would exert if it were alone in the container. Dalton's Law of Partial Pressures asserts that the total pressure of a gas mixture is the sum of the partial pressures of each individual gas.

Partial pressures are a key component in determining the position of equilibrium in gaseous systems. When dealing with equilibrium problems, it's critical to understand that partial pressures can be used as a clue to assess the progress of a reaction or the degree of conversion of reactants to products. The ratio of these pressures is directly linked to the equilibrium constant expression for reactions involving gases. Thus, skill in calculating and understanding partial pressures is invaluable in solving equilibrium problems, predicting reaction behavior, and designing chemical systems in practical scenarios such as engineering and environmental science.