Question

The equilibrium constant for the reaction $$ \mathrm{H}_{2}(g)+\mathrm{F}_{2}(g) \rightleftharpoons 2 \mathrm{HF}(g) $$ has the value \(2.1 \times 10^{3}\) at a particular temperature. When the system is analyzed at equilibrium at this temperature, the concentrations of both \(\mathrm{H}_{2}(g)\) and \(\mathrm{F}_{2}(g)\) are found to be \(0.0021 M\). What is the concentration of \(\mathrm{HF}(g)\) in the equilibrium system under these conditions?

Short answer

The equilibrium concentration of \(\mathrm{HF}(g)\) under the given conditions is approximately \(0.00304 M\).

Step-by-step solution

Write the equilibrium constant expression

For the given reaction, $$ \mathrm{H}_{2}(g) + \mathrm{F}_{2}(g) \rightleftharpoons 2 \mathrm{HF}(g), $$ the equilibrium constant expression would be: $$ K = \frac{[\mathrm{HF}]^2}{[\mathrm{H}_{2}] [\mathrm{F}_{2}]}, $$ where \([\mathrm{HF}]\), \([\mathrm{H}_{2}]\), and \([\mathrm{F}_{2}]\) represent the equilibrium concentrations of the species.

Substitute the given values and solve for the unknown

We are given that the value of \(K\) is \(2.1 \times 10^3\) and the concentrations of both \(\mathrm{H}_{2}(g)\) and \(\mathrm{F}_{2}(g)\) are \(0.0021 M\). We need to find the concentration of \(\mathrm{HF}(g)\), which we will denote as \(x\). Substitute the given values into the equilibrium constant expression: $$ 2.1 \times 10^3 = \frac{x^2}{(0.0021)(0.0021)}. $$ Now, solve for \(x\).

Solve for the equilibrium concentration of \(\mathrm{HF}(g)\)

We can rearrange the equation to isolate \(x\): $$ x^2 = (2.1 \times 10^3) \times (0.0021) \times (0.0021). $$ Calculate the value of \(x^2\): $$ x^2 = 9.261 \times 10^{-6}. $$ To find the value of \(x\), we need to take the square root of both sides of the equation: $$ x = \sqrt{9.261 \times 10^{-6}}. $$ Calculating the square root, we get \(x \approx 0.00304\).

State the final answer

The equilibrium concentration of \(\mathrm{HF}(g)\) under the given conditions is approximately \(0.00304 M\).

Key concepts

Chemical Equilibrium

Chemical equilibrium represents a state in a chemical reaction where the rate of the forward reaction equals the rate of the reverse reaction. In this balanced state, the concentrations of the reactants and products remain constant over time, although they do not necessarily have to be equal.

For a clearer understanding, imagine a see-saw with two children of equal weight on each end. They maintain the see-saw in a balanced position just as chemical species at equilibrium maintain the reaction, with no net change observed. This dynamic process continues indefinitely unless the system is disturbed by external factors, such as changes in temperature, pressure, or the addition of more reactants or products.

Equilibrium Constant Expression

The equilibrium constant expression (K) quantifies the balance of a chemical reaction at equilibrium. It is derived from the stoichiometry of the reaction.

Consider the general reaction where the lowercase letters represent the coefficients: $$aA + bB \rightleftharpoons cC + dD.$$For this reaction, the equilibrium constant expression would be: $$K = \frac{[C]^c[D]^d}{[A]^a[B]^b},$$with square brackets indicating the concentration of each species. The concentrations of products are raised to the power of their respective coefficients and multiplied together, and divided by the product of the reactants' concentrations also raised to the powers of their respective coefficients. The value of K provides insight into the position of equilibrium. A large K indicates a reaction that heavily favors the formation of products, while a small K indicates a reaction where reactants are predominantly present.

Chemical Reaction Equilibrium

A detailed understanding of chemical reaction equilibrium involves not just knowing that the reaction reaches a balance, but also understanding how equilibrium can shift in response to changes in conditions, known as Le Chatelier's Principle. This principle states that if an external condition is changed, the equilibrium will adjust to oppose that change.

For instance, increasing the pressure on a gaseous system will shift the equilibrium towards the side with fewer moles of gas. Similarly, a change in temperature can result in a shift of equilibrium; for example, an increase in temperature for an exothermic reaction shifts the equilibrium towards the reactants. These adjustments are important for chemists when designing processes to get the highest yield of a desired product. Monitoring concentration, volume, pressure, and temperature thus becomes crucial for managing a chemical equilibrium effectively.