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I'm confused with the terms "position of equilibrium" and "equilibrium shifting towards left/right". I do have a vague picture that if a reaction shifts to the right, for example, it basically means that the reaction is now biased towards the products. But I'm not thorough with it. Does it mean that the rate of forward reaction is now faster than the rate of backward reaction, which would mean that the reaction isn't at equilibrium anymore?

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The term "equilibrium shifting" is mostly used for a reaction that was already at equilibrium, but we changed something (in the simplest case adding one reactant or product to the equilibrium mixture so the reaction is no longer at equilibrium). The "equilibrium is shifting" is a shortcut for saying there is a net reaction to re-establish equilibrium.

The term is used both in cases where the equilibrium constant does not change (change in one of the concentrations) and in cases where the equilibrium constant does change (change in temperature, for example).

Does it mean that the rate of forward reaction is now faster than the rate of backward reaction, which would mean that the reaction isn't at equilibrium anymore?

Yes, for some reason the reaction is out of equilibrium, otherwise there would be no net reaction (no "shift").

I'm confused with the terms "position of equilibrium" and "equilibrium shifting towards left/right".

"Position of equilibrium" could be roughly equated with the reaction quotient Q or more generally with the set of all concentrations at equilibrium.

Taking a change in temperature as an example, you start with one set of equilibrium concentrations ("old position of equilibrium" corresponding to some reaction quotient $Q_1$). As you change the temperature, the reaction is no longer at equilibrium because the equilibrium constant changes (lets say from $K_{T_1}$ to $K_{T_2}$), so $Q_1$ is equal to the former equilibrium constant $K_{T_1}$, but not to the current equilibrium constant $K_{T_2}$). There is a net reaction until you reach the new equilibrium ("new position of equilibrium" corresponding to some other reaction quotient $Q_2$). At the first temperature $T_1$ after equilibrium is established, $Q_1 = K_{T_1}$. At the second temperature, $Q_2 = K_{T_2}$ once the equilibrium is re-established.

For adding or removing one of the species, you start with the old position of equilibrium, then the addition or removal of a species changes Q, which no longer equals K. As a consequence, there is a net reaction that changes Q back to K. Now, you have a new set of concentrations, but the reaction quotient is back to what it was at equilibrium. I'm not sure whether to call this a new position of equilibrium, or to say the reaction went back to equilibrium (this might be confusing because the equilibrium corresponds to a third set of concentrations - first set: original equilibrium concentrations; second set: concentrations right after the manipulation; third set: concentrations after the net reaction re-establishes equilibrium).

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