Methane reacts with water to form CO and H2 as follows CH4 (g) + H2O (g) >< CO (g) + 3 H2 (g) The value of Kp at 298 K for the reaction is 7.7 x 1024, calculate Kc.
Kc = KpRT^-delta n
delta n = nproducts-nreactants.
Substitute and solve.
To calculate Kc from Kp, we need to use the ideal gas equation and the relationship between Kc and Kp.
The ideal gas equation is given by:
PV = nRT
where P is the pressure, V is the volume, n is the number of moles, R is the ideal gas constant, and T is the temperature.
In the given reaction, we have gases involved, so we can express the concentrations in terms of partial pressures.
According to the stoichiometry of the reaction, the molar ratio of CO to CH4 is 1:1, and the molar ratio of H2 to CH4 is 3:1.
Let's assume that at equilibrium, the pressure of CH4 is P1, the pressure of CO is P2, and the pressure of H2 is P3.
Using the ideal gas equation, we can express the partial pressures in terms of the number of moles:
P1 = n1RT/V
P2 = n2RT/V
P3 = n3RT/V
where n1, n2, and n3 are the number of moles of CH4, CO, and H2, respectively, and V is the volume.
Since we know the molar ratios, we can express the number of moles in terms of a common variable x:
n1 = n2 = x
n3 = 3x
Hence, the partial pressures become:
P1 = xRT/V
P2 = xRT/V
P3 = 3xRT/V
Now, we can substitute these expressions into the Kp expression:
Kp = (P2 * P3^3) / (P1 * P3)
= (xRT/V * (3xRT/V)^3) / (xRT/V * 3xRT/V)
= (27x^4 * RT^4) / (9x^2 * RT^2)
= 3x^2 * RT^2
Given that Kp = 7.7 x 10^24 and T = 298 K, we can rearrange the equation to solve for x:
7.7 x 10^24 = 3x^2 * (0.0821 L atm mol^-1 K^-1 * 298 K)^2
Simplifying the equation:
7.7 x 10^24 = 3x^2 * (0.0821 * 298)^2
Now, solve for x by taking the square root of both sides:
x^2 = (7.7 x 10^24) / (3 * (0.0821 * 298)^2)
x = √((7.7 x 10^24) / (3 * (0.0821 * 298)^2))
Finally, we can use the value of x to calculate Kc. Since Kc is defined as the ratio of product concentrations to reactant concentrations, we can substitute the value of x in the equation:
Kc = [CO] * [H2]^3 / [CH4] * [H2O]
Given that the initial pressure of H2O is 1 atm, we can assume that it remains constant during the reaction, so [H2O] = 1.
Substitute the values:
Kc = (x * 3x^3) / (x * 1)
Kc = 3x^4
Substitute the value of x obtained earlier to find Kc.