Question

1. ethyne, c₂h₂, is used in the process of gas welding. at 500.0 k, ( k_{\text{eq}} ) is 702. if each gas has the following concentrations, ( \text{ch}_4 = 4.76 , \text{mol/l} ), ( \text{c}_2\text{h}_2 = 15.74 , \text{mol/l} ) and ( \text{h}_2 = 4.75 , \text{mol/l} ), determine ( q_{\text{eq}} ) for the system and indicate the direction toward which the system will move to reach equilibrium.

( 2 , \text{ch}_4 , _{(\text{g})} leftrightarrow 3 , \text{h}_2 , _{(\text{g})} + \text{c}_2\text{h}_2 , _{(\text{g})} )

Explanation:

Step1: Recall the formula for \( Q_{eq} \)

For the reaction \( 2 \text{CH}_4 \,_{(g)}
ightleftharpoons 3 \text{H}_2 \,_{(g)} + \text{C}_2\text{H}_2 \,_{(g)} \), the reaction quotient \( Q_{eq} \) is given by the formula:
\[ Q_{eq} = \frac{[\text{H}_2]^3 [\text{C}_2\text{H}_2]}{[\text{CH}_4]^2} \]

Step2: Substitute the given concentrations

We are given \( [\text{CH}_4] = 4.76 \, \text{mol/L} \), \( [\text{C}_2\text{H}_2] = 15.74 \, \text{mol/L} \), and \( [\text{H}_2] = 4.75 \, \text{mol/L} \). Substitute these values into the formula:
\[ Q_{eq} = \frac{(4.75)^3 \times 15.74}{(4.76)^2} \]
First, calculate \( (4.75)^3 \):
\[ (4.75)^3 = 4.75 \times 4.75 \times 4.75 = 107.171875 \]
Then, calculate \( (4.76)^2 \):
\[ (4.76)^2 = 4.76 \times 4.76 = 22.6576 \]
Now, substitute these back into the \( Q_{eq} \) formula:
\[ Q_{eq} = \frac{107.171875 \times 15.74}{22.6576} \]
Calculate the numerator:
\[ 107.171875 \times 15.74 \approx 107.171875 \times 15 + 107.171875 \times 0.74 = 1607.578125 + 79.3071875 = 1686.8853125 \]
Then divide by the denominator:
\[ Q_{eq} \approx \frac{1686.8853125}{22.6576} \approx 74.45 \]

Step3: Compare \( Q_{eq} \) with \( K_{eq} \)

We know that \( K_{eq} = 702 \). Since \( Q_{eq} (74.45) < K_{eq} (702) \), the system will shift to the right (toward the products) to reach equilibrium because when \( Q < K \), the reaction proceeds in the forward direction to form more products and increase \( Q \) until it equals \( K \).

Answer:

\( Q_{eq} \approx 74.45 \), and the system will shift to the right (toward the products) to reach equilibrium.