The value of Kp for the reaction,

CH4(g) + CO2(g) ⇌ 2CO(g) + 2H2(g)

is 4.5
×
102 atm2 at 825 K. What is Kc for the reaction at this temperature?

Question

The value of Kp for the reaction,

CH4(g) + CO2(g) ⇌ 2CO(g) + 2H2(g)

is 4.5 
×
 102 atm2 at 825 K. What is Kc for the reaction at this temperature?

BlackTom AI · Accepted Answer

We start by restating the problem to ensure clarity: the reaction CH4(g) + CO2(g) ⇌ 2 CO(g) + 2 H2(g) has Kp = 4.5 × 10^2 atm^2 at 825 K. We are asked for Kc at the same temperature.

First, recall the relationship between Kp and Kc for gas-phase reactions: Kp = Kc (RT)^(Δn). Here Δn is the change in moles of gas on the product side minus the reactant side. For this reaction, acquire the counts: reactants have 1 CH4 + 1 CO2 = 2 moles of gas; products have 2 CO + 2 H2 = 4 moles of gas. Therefore Δn = 4 − 2 = 2.

Next, compute RT at 825 K using R = 0.082057 L·atm/(mol·K): RT = 0.082057 × 825 ≈ 67.7 L·atm/mol. Then (RT)^2 ≈ (67.7)^2 ≈ 4580 (L^2·atm^2)/(mol^2).

Now solve for Kc: Kc = Kp / (RT)^(Δn) = (4.5 × 10^2 atm^2) / 4580 L^2·atm^2/mol^2 ≈ 450 / 4580 mol^2/L^2 ≈ 0.098 mol^2/L^2, i.e., 0.098 M^2.

Examine each option:
- Option: 2.1 × 10^6 M^2. This is vastly larger than the calculated 0.098 M^2; it would imply an enormous equilibrium concentration of products, which contradicts the given Kp value and the small Kc we computed. It shows a misunderstanding of units or the relation Kp = Kc (RT)^(Δn).
- Option: 4.5 × 10^−2 M^2 (i.e., 0.045 M^2). This is of the same order of magnitude as the correct result but off by about a factor of 2–3; the precise (RT)^2 factor at 825 K does not yield 0.045 M^2 when Kp = 450 atm^2, so this is too small.
- Option: 6.6 M^2. This value lacks the proper scale given Kp and RT; it’s much larger than the correct result and does not reflect the (RT)^2 division by Kp.
- Option: 0.098 M^2. This matches the calculation Kc = Kp / (RT)^2 with Δn = 2 at T = 825 K using R = 0.082057 and the provided Kp. The units M^2 are consistent with a reaction where Δn ≠ 0 and the stoichiometric changes lead to squared concentration terms in Kc.

In summary, the correct reasoning applies the Δn = 2 adjustment to relate Kp and Kc, uses the appropriate R and T to compute (RT)^2, and yields Kc ≈ 0.098 M^2. The other options deviate either by incorrect magnitude or by ignoring the (RT)^(Δn) factor required to convert Kp to Kc at the given temperature.

The value of Kp for the reaction, CH4(g) + CO2(g) ⇌ 2CO(g) + 2H2(g) is 4.5 × 102 atm2 at 825 K. What is Kc for the reaction at this temperature?

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