Question

For the reaction: $$ \mathrm{H}_{2}(g)+\mathrm{CO}_{2}(g) \rightleftarrows \mathrm{H}_{2} \mathrm{O}(g)+\mathrm{CO}(g) $$ at \(700^{\circ} \mathrm{C}, K_{\mathrm{c}}=0.534 .\) Calculate the number of moles of \(\mathrm{H}_{2}\) that are present at equilibrium if a mixture of 0.300 mole of \(\mathrm{CO}\) and 0.300 mole of \(\mathrm{H}_{2} \mathrm{O}\) is heated to \(700^{\circ} \mathrm{C}\) in a 10.0 - \(\mathrm{L}\) container.

Short answer

0.148 moles of H₂ are present at equilibrium.

Step-by-step solution

Write the Equilibrium Expression

For the given reaction, the equilibrium expression is given by:\[ K_{c} = \frac{{[\text{CO}][\text{H}_2\text{O}]}}{[\text{H}_2][\text{CO}_2]} \]where \([\text{CO}], [\text{H}_2\text{O}], [\text{H}_2], [\text{CO}_2]\) are the equilibrium concentrations of CO, H₂O, H₂, and CO₂ respectively.

Convert Initial Moles to Concentration

Since the reaction occurs in a 10.0 Liter container, convert the initial moles into concentrations:\[ [\text{CO}]_\text{initial} = \frac{0.300\, \text{mol}}{10.0\, \text{L}} = 0.030\, \text{mol/L} \]\[ [\text{H}_2\text{O}]_\text{initial} = \frac{0.300\, \text{mol}}{10.0\, \text{L}} = 0.030\, \text{mol/L} \]Initially, there are no \(\text{H}_2\) or \(\text{CO}_2\), so:\[ [\text{H}_2]_\text{initial} = 0 \]\[ [\text{CO}_2]_\text{initial} = 0 \]

Define Change in Concentrations

Let the change in the concentration of \(\text{H}_2\) (and \(\text{CO}_2\)) be \(x\). Thus, the change in \(\text{CO}\) and \(\text{H}_2\text{O}\) due to the reaction will be \( [\text{CO}] = 0.030 - x \)\( [\text{H}_2\text{O}] = 0.030 - x \)\([\text{H}_2] = x \)\([\text{CO}_2] = x \) after the equilibrium is established.

Plug Values into Equilibrium Expression

Substitute all expressions into the equilibrium expression:\[K_{c} = \frac{(0.030 - x)^{2}}{x^{2}} = 0.534 \]

Solve for x

Rewriting the equation:\[0.534 x^{2} = (0.030 - x)^{2} \]Expanding the right-hand side:\[0.534 x^{2} = 0.0009 - 0.06x + x^{2} \]Rearranging we get:\[0 = 0.466x^{2} - 0.06x + 0.0009\]Using the quadratic formula where \( a = 0.466, b = -0.06, c = 0.0009 \):\[x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}\]Substitute the appropriate values to find \(x\).

Calculate the Equilibrium Concentration of H₂

After solving the quadratic equation, you'll find two values for \(x\). Discard the negative value, as it does not make sense in this context. If you assume \(x = 0.0148\), then:\[[\text{H}_2] = x = 0.0148\, \text{mol/L}\]

Convert Concentration Back to Moles

Since the problem asks for the number of moles of \(\text{H}_2\):\[\text{Moles of } \text{H}_2 = 0.0148\, \text{mol/L} \times 10.0\, \text{L} = 0.148\, \text{mol}\]

Key concepts

Equilibrium Constant

When a chemical reaction reaches a state where the reactants and products are present in steady concentrations, it is said to be in equilibrium. The **equilibrium constant** (\(K_c\)) is a numerical value that relates the concentrations of products to reactants at equilibrium, each raised to the power of their coefficients in the balanced chemical equation. This constant provides us with important information about the position of equilibrium: - A large \(K_c\) value, much greater than 1, suggests that products are favored at equilibrium.- A small \(K_c\) value, much less than 1, indicates that reactants are favored. For the reaction \( \mathrm{H}_{2}(g)+\mathrm{CO}_{2}(g) \rightleftarrows \mathrm{H}_{2} \mathrm{O}(g)+\mathrm{CO}(g) \) at 700°C, \(K_c = 0.534\), suggesting that the concentrations of products (\( \mathrm{H}_{2} \mathrm{O} \) and \( \mathrm{CO} \)) are not significantly larger than those of reactants (\( \mathrm{H}_{2} \) and \( \mathrm{CO}_{2} \)) at equilibrium.

Concentration Calculation

To solve for equilibrium concentrations, we first determine the starting concentrations of reactants and products. Here, initial moles are given for \( \mathrm{CO} \) and \( \mathrm{H}_2\mathrm{O} \) as 0.300 moles each in a 10.0-L container. The concentration is found by dividing moles by volume:- \([\mathrm{CO}]_{\text{initial}} = \frac{0.300\, \text{mol}}{10.0\, \text{L}} = 0.030\, \text{mol/L}\)- \([\mathrm{H}_2\mathrm{O}]_{\text{initial}} = 0.030\, \text{mol/L}\)
Initially, there are no \(\mathrm{H}_{2}\) or \(\mathrm{CO}_2\) present:- \([\mathrm{H}_{2}]_{\text{initial}} = 0\)- \([\mathrm{CO}_2]_{\text{initial}} = 0\)
As equilibrium is reached, changes occur in these concentrations based on the stoichiometry of the reaction. This step is crucial for determining the concentrations at equilibrium, which are essential for utilizing the equilibrium expression.

Quadratic Formula

In chemistry, the quadratic formula is often applied to solve for concentrations at equilibrium when dealing with a system that results in a quadratic equation. The equation is typically expressed as \( ax^2 + bx + c = 0 \). The quadratic formula \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]allows the calculation of possible values of \(x\), the change in concentration in this context.
For our reaction, substituting the changes into the equilibrium expression \( (0.030 - x)^2 = 0.534x^2 \) led to the quadratic equation\[0.466x^2 - 0.06x + 0.0009 = 0\].Solving this with the quadratic formula will yield two solutions for \(x\). In practical scenarios, you discard the negative solution because negative concentrations in real physical contexts are non-sensical, leaving us with the valid positive concentration change.

Equilibrium Expression

An **equilibrium expression** is derived from balancing the chemical equation for a reaction at equilibrium. It is a mathematical statement representing the balance of concentrations needed at equilibrium and is given by relating the products over reactants:\[ K_c = \frac{[\text{CO}][\text{H}_2\text{O}]}{[\text{H}_2][\text{CO}_2]} \].
Each concentration is raised to the power equal to its coefficient in the balanced reaction. For the provided reaction, knowing \(K_c\) allows us to substitute concentrations into this expression and solve for unknowns. After calculating the changes in concentrations expressed as \(x\), these values are plugged into the expression to find numeric values for the equilibrium concentrations.This helps us verify the state of the reaction system and accurately predict the concentrations of reactants and products present when equilibrium is achieved.