To find the concentration of CH4 in the equilibrium mixture, we can use the equilibrium constant expression for the reaction given. The reaction is:
CO(g) + 3H2(g) ⇌ CH4(g) + H2O(g)
The equilibrium constant expression, Kc, for this reaction is defined as:
Kc Expression
The equilibrium constant Kc is given by the formula:
Kc = [CH4][H2O] / ([CO][H2]^3)
Where [X] represents the molar concentration of species X at equilibrium. We know that Kc = 3.90 at 1300 K.
Initial Concentrations
We have the following initial amounts in a 1L flask:
- CO: 0.30 mol
- H2: 0.10 mol
- H2O: 0.02 mol
- CH4: unknown amount (let's denote it as x)
Since the volume of the flask is 1L, the concentrations are equal to the number of moles:
- [CO] = 0.30 M
- [H2] = 0.10 M
- [H2O] = 0.02 M
- [CH4] = x M
Setting Up the Equation
At equilibrium, we can express the concentrations in terms of x. The change in concentration for each reactant and product can be represented as follows:
- CO decreases by y: 0.30 - y
- H2 decreases by 3y: 0.10 - 3y
- H2O increases by y: 0.02 + y
- CH4 increases by y: x + y
Since we need to find the equilibrium concentrations, we can assume that the change in concentration for CH4 and H2O is equal to y, and we will express everything in terms of y. However, we can also directly substitute the known values into the Kc expression.
Substituting Values into Kc
We can rearrange the Kc expression to solve for x:
3.90 = (x)(0.02 + y) / ((0.30)(0.10 - 3y)^3)
Since we are looking for the concentration of CH4, we can simplify our calculations by assuming that the change in concentration of H2 is small compared to its initial concentration. This means we can approximate:
[H2] ≈ 0.10 M
Calculating Concentrations
Now we can substitute the known values into the Kc expression:
3.90 = (x)(0.02) / (0.30)(0.10)^3
Calculating the denominator:
(0.30)(0.10)^3 = 0.30 * 0.001 = 0.0003
Now substituting back into the equation:
3.90 = (x)(0.02) / 0.0003
Multiplying both sides by 0.0003:
3.90 * 0.0003 = x * 0.02
Calculating the left side:
0.00117 = x * 0.02
Now, solving for x:
x = 0.00117 / 0.02
x = 0.0585 M
Final Concentration of CH4
The concentration of CH4 in the equilibrium mixture is approximately 0.0585 M. This means that at equilibrium, there are about 0.0585 moles of CH4 per liter in the flask. This calculation illustrates how we can use the equilibrium constant and initial concentrations to find unknown concentrations in a chemical reaction.
