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$\ce{PCl5 <=> PCl3 +Cl2} $

It's stated in my book that addition of $\ce{PCl5}$ to the equilibrium mixture increases the rate of forward reaction but no reason is mentioned for it.

However, before this the author derived $K_c = \dfrac{x^2}{(1-x)V}$ (using law of mass action at equilibrium ) where V is the volume of the container and $x$ is the number of $\ce{PCl5}$ molecules dissociated from $a$ number of moles initially. I couldn't relate the "rate of forward reaction" statement to this equation while earlier in case of the reaction:

$\ce{H2 + I2 <=> HI}$

I was able to relate the equation I obtained through law of mass action with the effect of concentration on rate of forward or backward reaction.

Could someone please explain why addition of $\ce{PCl5}$ to the equilibrium mixture increases the rate of forward reaction?

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Imagine a overpopulated kingdom suddenly gaining a large area of land (Start of Reaction). Immediately, groups of people start migrating to the area of land (Forward Reaction). However, as people migrate to this new land, some decide that, for whatever reason, they don't like this area (Reverse Reaction). As more and more people move there, more start coming back until there is an equal rate of people moving to the territory and people moving back (Dynamic Equilibrium).

To your question: To model this system using my example of immigration, we suddenly add an enourmous influx of refugees into the old kingdom (Increase in $\ce{PCl5}$).

So the land in the outer territory seems more valuable now (less crowding, more open space, better air, etc.). This causes a temporary increase in the forward direction until the reverse reaction increases to compensate. This is called Le Chatelier's principle in chemistry.

References: Le Chatelier's Principle, My Handy Old 2011 Introductory Chemistry Textbook.

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