0.0487 mol of nitrosyl chloride was injected into an otherwise empty 1.00-L reaction vessel at 500 K. The nitrosyl chloride decomposed into nitrogen monoxide and chlorine:

2 NOCl(g)

2 NO(g) + Cl

_{2}(g)

When the system reached equilibrium at 500 K, the total pressure within the reaction vessel was found to be 2.63 atm. What is K

_{p} for the above reaction at 500 K?

First, I converted moles of nitrosyl chloride into pressure. Using the ideal gas law PV=nRT and solving for P using 0.08206 L·atm·mol

^{-1}·K

^{-1} as the gas constant, I obtained 2.00 atm as the initial pressure of NOCl.

Setting up the ICE table:

Usually in other problems I've seen you would be given partial pressure values for one or more of the compounds that appear in the balanced equation, making it a simple task to solve for x. In this case, the total pressure is given. So Dalton's law of partial pressure should be applicable, as the "E" terms in the ICE table are the equilibrium partial pressures of each gas, and the sum of the partial pressures at equilibrium should be equal to the total pressure of the system at equilibrium, as per Dalton's law.

Hence

(2.00 atm - 2x) + 2x + x = 2.63 atm

and x = 0.63 atm.

Substituting in the newly found equilibrium partial pressure values into the expression for K

_{p}, I obtain

1.83 as the K

_{p} (sorry for using Microsoft Word; I will make an effort to learn LaTeX in the future).

Here is the issue:

I was told that an alternative way to solve this problem is to observe that the stoichiometric coefficients represent the fraction that each compound in the chemical equation contributes to the total pressure at equilibrium.

So the partial pressure of NOCl should be 2/5 of the total pressure, which is

(2/5)2.63 atm = 1.052 atm

Similarly, the partial pressure at equilibrium of NO should be 2/5 of the total pressure, and that of chlorine gas is 1/5 of the total pressure, or 0.526 atm.

This means that the expression for K

_{p} is

and the K

_{p} would be 0.526.

Clearly one of these methods is incorrect as they result in different answers. So which of these methods is correct? Or are they both wrong and there is yet another way to solve this problem? Also, for the incorrect method, can you please explain which particular step(s) and/or assumption(s) used is flawed?