Animated Solution for Chemistry - Chemical Equilibrium: Phosphorus pentachloride dissociates as follows, in a closed reaction vessel, ${\text{PCl}}_5(g)\rightleftharpoons {\text{PCl}}_3(g)+{\text{Cl}}_2(g)$ If total pressure, at equilibrium, of the reaction mixture is $p$ and degree of dissociation of ${\text{PCl}}_5$ is $x$, the partial pressure of ${\text{PCl}}_3$ will be

Imagine a closed reaction vessel where a dynamic chemical dance is taking place. We have Phosphorus pentachloride (${\text{PCl}}_5$) gas, which is unstable at higher temperatures, breaking down into Phosphorus trichloride (${\text{PCl}}_3$) and Chlorine (${\text{Cl}}_2$) gas.

The balanced chemical equation for this reversible process is:

In chemical equilibrium problems, when the initial amount of the reactant is not explicitly given, it is a standard and highly effective practice to assume we start with exactly $1$ mole of the reactant. This simplifies our algebra immensely. So, at time $t=0$, we have $1$ mole of ${\text{PCl}}_5$ and $0$ moles of both products.

Degree of Dissociation

As the reaction proceeds towards equilibrium, a certain fraction of the initial ${\text{PCl}}_5$ molecules will dissociate. This fraction is called the degree of dissociation, denoted by $x$.

Since we started with $1$ mole, exactly $x$ moles of ${\text{PCl}}_5$ will break apart. According to the stoichiometry of our balanced equation ($1:1:1$ ratio), the dissociation of $x$ moles of ${\text{PCl}}_5$ will produce exactly $x$ moles of ${\text{PCl}}_3$ and $x$ moles of ${\text{Cl}}_2$.

Therefore, at equilibrium, the number of moles of each species will be: - Moles of ${\text{PCl}}_5=1-x$ - Moles of ${\text{PCl}}_3=x$ - Moles of ${\text{Cl}}_2=x$

Dalton's Law

To find the partial pressure of any gas in a mixture, we must invoke Dalton's Law of Partial Pressures. This law states that the partial pressure of a gas is equal to its mole fraction multiplied by the total pressure of the mixture ($p$).

First, we need the total number of moles at equilibrium:

Next, we find the mole fraction of ${\text{PCl}}_3$ (${\chi }_{{\text{PCl}}_3}$), which is the ratio of its moles to the total moles:

Finally, we multiply this mole fraction by the total equilibrium pressure $p$ to get the partial pressure of ${\text{PCl}}_3$:

And there we have it! A beautifully elegant expression derived purely from stoichiometry and Dalton's Law.