Which of the following reactions has K_p equal to K_c at all temperatures?

• A. 2NO_2(g) ightarrow 2NO(g) + O_2(g)
• B. N_2(g) + 3H_2(g) ightarrow 2NH_3(g)
• C. 2SO_2(g) + O_2(g) ightarrow 2SO_3(g)
• D. H_2(g) + I_2(g) ightarrow 2HI(g)

Answer: (D)

Verified step by step guidance

1. Recall the relationship between the equilibrium constants K_p and K_c for gaseous reactions: \[ K_p = K_c (RT)^{\Delta n} \] where \(\Delta n\) is the change in moles of gas (moles of gaseous products minus moles of gaseous reactants), \(R\) is the gas constant, and \(T\) is the temperature in Kelvin.
2. Understand that for \(K_p\) to be equal to \(K_c\) at all temperatures, the term \((RT)^{\Delta n}\) must be equal to 1 regardless of \(T\). This happens only if \(\Delta n = 0\), meaning the total number of moles of gaseous products equals the total number of moles of gaseous reactants.
3. For each reaction, calculate \(\Delta n\) by subtracting the total moles of gaseous reactants from the total moles of gaseous products:
4. - For \$2NO_2(g) \rightarrow 2NO(g) + O_2(g)\(: Products moles = 2 + 1 = 3, Reactants moles = 2, so \)\Delta n = 3 - 2 = 1$.
5. - For \(N_2(g) + 3H_2(g) \rightarrow 2NH_3(g)\): Products moles = 2, Reactants moles = 1 + 3 = 4, so \(\Delta n = 2 - 4 = -2\).
6. - For \$2SO_2(g) + O_2(g) \rightarrow 2SO_3(g)\(: Products moles = 2, Reactants moles = 2 + 1 = 3, so \)\Delta n = 2 - 3 = -1$.
7. - For \(H_2(g) + I_2(g) \rightarrow 2HI(g)\): Products moles = 2, Reactants moles = 1 + 1 = 2, so \(\Delta n = 2 - 2 = 0\).
8. Since only the reaction with \(\Delta n = 0\) has \(K_p = K_c\) at all temperatures, identify the reaction \(H_2(g) + I_2(g) \rightarrow 2HI(g)\) as the correct answer.