To describe what happens within the system when the given chemical reaction is at equilibrium, we need to consider several key aspects of chemical equilibrium:
### Given Chemical Reaction
The chemical reaction is:
[tex]\[ 2 CH_4(g) \leftrightarrow C_2H_2(g) + 3 H_2(g) \][/tex]
### Equilibrium Concept
At chemical equilibrium, the rates of the forward and reverse reactions are equal. This means that the rate at which methane ([tex]\(CH_4\)[/tex]) is converting to acetylene ([tex]\(C_2H_2\)[/tex]) and hydrogen gas ([tex]\(H_2\)[/tex]) is the same as the rate at which acetylene and hydrogen gas are converting back to methane.
### Reaction Rates
For the forward reaction:
[tex]\[ 2 CH_4 \rightarrow C_2H_2 + 3 H_2 \][/tex]
For the reverse reaction:
[tex]\[ C_2H_2 + 3 H_2 \rightarrow 2 CH_4 \][/tex]
At equilibrium:
1. The rate of the forward reaction is given by:
[tex]\[ \text{rate}_{\text{forward}} = k_{\text{forward}} [CH_4]^2 \][/tex]
2. The rate of the reverse reaction is given by:
[tex]\[ \text{rate}_{\text{reverse}} = k_{\text{reverse}} [C_2H_2] [H_2]^3 \][/tex]
### Equilibrium Condition
At equilibrium, the forward and reverse reaction rates are equal:
[tex]\[ \text{rate}_{\text{forward}} = \text{rate}_{\text{reverse}} \][/tex]
[tex]\[ k_{\text{forward}} [CH_4]^2 = k_{\text{reverse}} [C_2H_2] [H_2]^3 \][/tex]
### Equilibrium Constant (K_eq)
The equilibrium constant ([tex]\(K_{eq}\)[/tex]) expresses the ratio of the concentration of products to reactants at equilibrium. Therefore:
[tex]\[ K_{eq} = \frac{ [C_2H_2] [H_2]^3 }{ [CH_4]^2 } \][/tex]
The equilibrium constant is a fixed value at a given temperature, which indicates the extent to which reactants convert into products at equilibrium.
### Concentrations at Equilibrium
At equilibrium:
- The concentrations of [tex]\(CH_4\)[/tex], [tex]\(C_2H_2\)[/tex], and [tex]\(H_2\)[/tex] remain constant over time.
- This constancy happens because the forward and reverse reactions are occurring at equal rates. Consequently, there's no net change in the concentrations of the reactants and products.
### Summary
To summarize, at equilibrium in the reaction [tex]\(2 CH_4 \leftrightarrow C_2H_2 + 3 H_2\)[/tex]:
- The concentrations of [tex]\(CH_4\)[/tex], [tex]\(C_2H_2\)[/tex], and [tex]\(H_2\)[/tex] do not change over time.
- The forward reaction (converting methane to acetylene and hydrogen gas) happens at the same rate as the reverse reaction (converting acetylene and hydrogen gas back to methane).
- This dynamic balance is maintained as long as the system is closed and at a constant temperature.
- The equilibrium constant ([tex]\(K_{eq}\)[/tex]) quantitatively represents the relationship between the concentrations of products and reactants at equilibrium.
