Question

For the reaction: $$ 3 \mathrm{O}_{2}(g) \rightleftharpoons 2 \mathrm{O}_{3}(g) $$ \(K=1.8 \times 10^{-7}\) at a certain temperature. If at equilibrium \(\left[\mathrm{O}_{2}\right]=0.062 M\), calculate the equilibrium \(\mathrm{O}_{3}\) concentration.

Short answer

The equilibrium concentration of \(\mathrm{O}_3\) is approximately \(2.07 \times 10^{-6} M\).

Step-by-step solution

Write the equilibrium expression

Based on the given reaction, the equilibrium expression is as follows: \[K = \frac{[\mathrm{O}_3]^2}{[\mathrm{O}_2]^3}\] Now, we substitute the known values and solve for the unknown concentration.

Substitute given values into the equation

We know the equilibrium constant and the concentration of \(\mathrm{O}_2\). Substitute these values into the equation: \[ 1.8 \times 10^{-7} = \frac{[\mathrm{O}_3]^2}{(0.062)^3} \]

Solve for the equilibrium concentration of \(\mathrm{O}_3\)

To find the concentration of \(\mathrm{O}_3\), we first need to isolate it by multiplying both sides of the equation by \((0.062)^3\): \[ [\mathrm{O}_3]^2 = 1.8 \times 10^{-7} \times (0.062)^3 \] Calculate the right-hand side of the equation: \[ [\mathrm{O}_3]^2 \approx 4.29 \times 10^{-11} \] Now, take the square root of both sides to find the equilibrium concentration of \(\mathrm{O}_3\): \[ [\mathrm{O}_3] = \sqrt{4.29 \times 10^{-11}} \] \[ [\mathrm{O}_3] \approx 2.07 \times 10^{-6} M \] So, the equilibrium concentration of \(\mathrm{O}_3\) is approximately \(2.07 \times 10^{-6} M\).

Key concepts

Chemical Equilibrium

Understanding chemical equilibrium is fundamental when studying reactions and their behaviors. It's a state in which both reactants and products have no net change over time. This doesn't mean the reactants stop turning into products or vice versa; rather, the rate at which the reactants produce products is equal to the rate at which products revert back to reactants. This dynamic state might seem static, but it's actually a balance of ongoing processes.

Imagine a crowded dance floor where the number of people entering is the same as those leaving; while people are always moving in and out, the total number doesn't change – that's like chemical equilibrium.

Equilibrium Constant (K)

The equilibrium constant, symbolized by 'K', is a snapshot of a reaction's balance at a specific temperature. It tells us the ratio of product concentrations to reactant concentrations at equilibrium, raised to the power of their coefficients from the balanced equation. The value of 'K' is essential for chemists to understand how much product can form under given conditions and to predict the direction of the reaction.

For instance, a large 'K' suggests that at equilibrium, the reaction favors the formation of products, whereas a small 'K' implies that reactants are preferred. Remember, 'K' is temperature-dependent; if the temperature changes, so does the 'K' value. By knowing the equilibrium constant, chemists can forecast the composition of a reaction mixture when equilibrium is reached.

Equilibrium Expression

To converse with chemists, you need to speak the language, and the equilibrium expression is part of that lexicon. It mathematically conveys the ratio of products to reactants at equilibrium, in the form of concentrations and partial pressures, depending on the state of the reactants and products involved (aqueous or gaseous).

For a general reaction like \(aA + bB \rightleftharpoons cC + dD\), the equilibrium expression is \(K = \frac{[C]^c[D]^d}{[A]^a[B]^b}\), where each letter represents the substances involved, and the brackets denote their molarity. The coefficients from the balanced reaction equation become the exponents in this expression. Importantly, solids and pure liquids are left out of this equation, as their concentrations don't change during the reaction.

Le Chatelier's Principle

Le Chatelier's principle is like the universe's way of maintaining balance. It states that if a dynamic equilibrium is disturbed, the system will adjust itself to counteract the disturbance and restore equilibrium. Picture a seesaw where kids are constantly adjusting their positions to keep it level; that's what reactions do, according to this principle.

Changes can be in the form of concentration, pressure, volume, or temperature. For example, increasing the concentration of a reactant will push the equilibrium toward making more products. If you increase the pressure, the equilibrium will shift to the side with fewer gas molecules. Heating or cooling a system will also cause shifts, depending on whether the reaction is endothermic or exothermic. Le Chatelier's principle helps to predict how changes will affect the position of equilibrium, enabling chemists to optimize reaction conditions for the desired outcome.