C(s) + H20(g)↔ CO(s) + H2(s)
the equilibrium constant for this reaction is 4.251 x 10^-2 at 800 K. if the equilibrium concentration of H2O(g)is 0.1990 M, what are the concentrations of CO, H2, and C?
To find the concentrations of CO, H2, and C at equilibrium, we need to use the equilibrium constant expression and the given information.
The equilibrium constant expression for the reaction is:
Kc = [CO][H2]/[C][H₂O]
Given:
Kc = 4.251 x 10^-2
[H2O] = 0.1990 M
Let's assume the initial concentrations of CO, H2, and C are represented as [CO]₀, [H2]₀, and [C]₀, respectively.
At equilibrium, the concentrations can be represented as:
[CO] = concentration of CO at equilibrium
[H2] = concentration of H2 at equilibrium
[C] = concentration of C at equilibrium
At equilibrium, the concentrations can be related to the initial concentrations by using the following relationships:
[CO] = [CO]₀ - x (change in concentration)
[H2] = [H2]₀ - x (change in concentration)
[C] = [C]₀ + x (change in concentration)
[H2O] = [H2O]₀ - x (since it is a product, its concentration decreases)
Here, 'x' represents the change in concentration. The change in concentration is typically small compared to the initial concentration, so we can assume that [H2O]₀ ≈ [H2O].
Now, substitute the values into the equilibrium constant expression:
Kc = [CO][H2]/[C][H2O]
4.251 x 10^-2 = ([CO]₀ - x)([H2]₀ - x)/([C]₀ + x)([H2O])
Since the value of Kc is given, substitute the known values into the equation:
4.251 x 10^-2 = ([CO]₀ - x)([H2]₀ - x)/([C]₀ + x)(0.1990)
Next, solve this equation for x. To do so, rearrange the equation:
4.251 x 10^-2 * ([C]₀ + x)(0.1990) = ([CO]₀ - x)([H2]₀ - x)
Expand and rearrange the equation further:
0.08474 * [C]₀ + 0.08474x = 4.251 x 10^-2 * [CO]₀ - 4.251 x 10^-2x - 4.251 x 10^-2[H2]₀ + 4.251 x 10^-2x²
Combine like terms:
4.251 x 10^-2x² + 0.08474x - 4.251 x 10^-2[H2]₀ = 4.251 x 10^-2 * [CO]₀ - 0.08474 * [C]₀
Rearrange the equation:
4.251 x 10^-2x² + (0.08474 + 0.08474 * [C]₀) x - (4.251 x 10^-2 * [CO]₀ - 4.251 x 10^-2[H2]₀) = 0
Now, we can solve this quadratic equation to find the value of x using numerical methods or a quadratic equation solver. Once we find the value of x, we can substitute it back into the equilibrium expressions to find the concentrations of CO, H2, and C at equilibrium.