Question

A sample of gaseous nitrosyl bromide (NOBr) was placed in a container fitted with a frictionless, massless piston, where it decomposed at ${25}^{\circ }\mathrm{C}$ according to the following equation:$$2\mathrm{NOBr}(g)\rightleftharpoons 2\mathrm{NO}(g)+{\mathrm{Br}}_2(g)$$ The initial density of the system was recorded as $4.495\mathrm{g}/\mathrm{L}$ After equilibrium was reached, the density was noted to be $4.086\mathrm{g}/\mathrm{L}$ a. Determine the value of the equilibrium constant $K$ for the reaction. b. If $\mathrm{Ar}(g)$ is added to the system at equilibrium at constant temperature, what will happen to the equilibrium position? What happens to the value of $K?$ Explain each answer.

Short answer

The equilibrium constant K for the given reaction is 0.0145. When Ar(g) is added to the system, the equilibrium position does not shift and the value of K remains the same, as the partial pressures of the other gases do not change.

Step-by-step solution

Calculate initial moles of NOBr

First, we need to find the initial moles of NOBr. To do this, we'll use the initial density and the molar mass of NOBr. The molar mass of NOBr is: Molar mass of NOBr = $30.01\mathrm{g}/\mathrm{mol}+16.00\mathrm{g}/\mathrm{mol}+79.90\mathrm{g}/\mathrm{mol}=125.91\mathrm{g}/\mathrm{mol}$ Using the initial density, we can determine the initial moles of NOBr per liter: Initial moles of NOBr/L = $\frac{4.495\mathrm{g}/\mathrm{L}}{125.91\mathrm{g}/\mathrm{mol}}=0.0357\mathrm{mol}/\mathrm{L}$

Calculate equilibrium moles of NOBr

Next, we'll find the equilibrium moles of NOBr. This can be done using the equilibrium density and the molar mass of NOBr: Equilibrium moles of NOBr/L = $\frac{4.086\mathrm{g}/\mathrm{L}}{125.91\mathrm{g}/\mathrm{mol}}=0.0324\mathrm{mol}/\mathrm{L}$

Calculate moles of NO and Br2 at equilibrium

Since 2 moles of NOBr decompose to produce 2 moles of NO and 1 mole of Br2, we can use the change in NOBr concentration to determine the equilibrium concentrations of NO and Br2: $\mathrm{\Delta }$ NOBr = Initial moles of NOBr/L - Equilibrium moles of NOBr/L = 0.0357 - 0.0324 = 0.0033 mol/L Equilibrium moles of NO/L = 2 × $\mathrm{\Delta }$ NOBr = 2 × 0.0033 = 0.0066 mol/L Equilibrium moles of Br2/L = 1 × $\mathrm{\Delta }$ NOBr = 1 × 0.0033 = 0.0033 mol/L

Calculate the equilibrium constant K

Now we can use the equilibrium concentrations to calculate the equilibrium constant K for the reaction: $K=\frac{[\mathrm{NO}{]}^2[{\mathrm{Br}}_2]}{[\mathrm{NOBr}{]}^2}=\frac{(0.0066{)}^2\cdot (0.0033)}{(0.0324{)}^2}=0.0145$ So, the equilibrium constant K for the reaction is 0.0145.

Analyzing the effect of adding Ar(g) on the equilibrium position and the value of K

When Ar(g) is added to the system, the total pressure of the system will increase but the partial pressures of the other gases (NOBr, NO, and Br2) will not change. This is because Ar(g) is inert and does not participate in the reaction. As a result, the equilibrium position will not shift, and the value of K will remain constant. In conclusion, the equilibrium constant K for the given reaction is 0.0145. When Ar(g) is added to the system, the equilibrium position does not shift and the value of K remains the same, as the partial pressures of the other gases do not change.

Key concepts

Chemical Equilibrium

Chemical equilibrium is the state in a reversible reaction where the rates of the forward and reverse reactions are equal. This means that the concentrations of the reactants and products remain constant over time. For the decomposition of nitrosyl bromide ($2extNOBrightleftharpoons2extNO+ext{Br}_2$), establishing equilibrium means that the amount of $extNOBr$ being decomposed into $extNO$ and $ext{Br}_2$ is the same as the amount of $extNO$ and $ext{Br}_2$ recombining to form $extNOBr$.
- Reactants and products do not stop changing; they just do so at the same rate.- Over time, at equilibrium, the macroscopic properties (e.g., pressure, concentration, density) do not change.Remember, equilibrium does not mean that the reactants and products are in equal amounts; it just means their rates of formation are equal, leading to a dynamic yet stable state.

Reaction Kinetics

Reaction kinetics is the branch of chemistry that deals with the speed or rate of a chemical reaction and the factors that affect these rates. For the decomposition of nitrosyl bromide, reaction kinetics helps explain how quickly the reaction is proceeding towards equilibrium. - The reaction rate is influenced by factors such as temperature, concentration of reactants, and the presence of catalysts. - In our reaction, $2extNOBr$ decomposes to form $2extNO$ and $ext{Br}_2$, and the rate at which NOBr decomposes affects how quickly equilibrium is reached.
Understanding reaction kinetics is crucial because it provides insights into both the mechanism of the reaction and the conditions under which equilibrium is achieved. This helps in optimizing reactions for industrial or laboratory processes.

Le Chatelier's Principle

Le Chatelier's Principle predicts how an equilibrium system responds to changes in concentration, pressure, and temperature. According to this principle, if a dynamic equilibrium is disturbed by changing the conditions, the position of equilibrium moves to counteract the change.
In the case of adding argon gas (Ar), which is chemically inert, to our equilibrium system: - The total pressure in the container increases, but since Ar does not react, the partial pressures of the gases involved in the reaction ($extNOBr$, $extNO$, and $ext{Br}_2$) remain unchanged.- Therefore, the equilibrium position of the reaction doesn’t shift, as there is no change in the concentration of the reacting species.
Le Chatelier's Principle confirms that the value of the equilibrium constant, $K$, will remain unaffected by the addition of inert gases at constant temperature.

Gas Laws

Gas laws describe how gases behave with respect to volume, temperature, and pressure. They are crucial for understanding how changes in these conditions affect chemical reactions involving gases, such as the decomposition of nitrosyl bromide.
- **Ideal Gas Law:** $PV=nRT$, where $P$ is pressure, $V$ is volume, $n$ is moles, $R$ is the gas constant, and $T$ is temperature. It helps in relating changes in pressure and volume with molar amounts.- **Partial Pressure:** Each gas in a mixture exerts pressure as if it alone occupied the entire volume. Partial pressure affects the concentration of gaseous reactants and products.
In the presence of an inert gas like argon, the total pressure in the system increases, but since argon does not change the partial pressures of $extNOBr$, $extNO$, and $ext{Br}_2$, the dynamics of their reaction remain the same.
Understanding gas laws allows chemists to predict and explain how a reaction's behavior might change under different physical conditions.