Chapter 16.6, Problem 1P

Write three different K_P relations for reacting ideal-gas mixtures, and state when each relation should be used.

To determine

The three different relations of $K_p$.

Answer to Problem 1P

The three relations of $K_p$ are $\frac{P_C^{v_C}{P}_D^{v_D}}{P_A^{v_A}{P}_B^{v_B}},e^{-\Delta G^{*}\left(T\right)/R_{u}T}\text{ and }\frac{N_C^{v_C}{N}_D^{v_D}}{N_A^{v_A}{N}_B^{v_B}}{\left(\frac{P}{N_{\text{total}}}\right)}^{\Delta v}$.

Explanation of Solution

Write the relation of equilibrium constant $\left(K_p\right)$ using partial pressure calculation.

$K_p=\frac{P_C^{v_C}{P}_D^{v_D}}{P_A^{v_A}{P}_B^{v_B}}$

Here, stoichiometric coefficient of element i is $v_i$ and partial pressure of element iis $P_i$.

Write the relation of equilibrium constant using Gibbs function $\left(\Delta G^{*}\left(T\right)\right)$.

$K_p=e^{-\Delta G^{*}\left(T\right)/R_{u}T}$

Here, universal gas constant is $R_u$ and temperature is T.

Write the relation of equilibrium constant $\left(K_p\right)$ using equilibrium composition calculation.

$K_p=\frac{N_C^{v_C}{N}_D^{v_D}}{N_A^{v_A}{N}_B^{v_B}}{\left(\frac{P}{N_{\text{total}}}\right)}^{\Delta v}$

Here, partial pressure is $P$, number of moles of element i is $N_i$, and summation of number of moles is $N_{\text{total}}$.

Thus, the three relations of $K_p$ are $\frac{P_C^{v_C}{P}_D^{v_D}}{P_A^{v_A}{P}_B^{v_B}},e^{-\Delta G^{*}\left(T\right)/R_{u}T}\text{ and }\frac{N_C^{v_C}{N}_D^{v_D}}{N_A^{v_A}{N}_B^{v_B}}{\left(\frac{P}{N_{\text{total}}}\right)}^{\Delta v}$.