Theoretical plates in Plate theory of chromatography

In "Chromatography theory" by Scott and Cazes, The plate volume is given as ($$v_m+Kv_s$$) where $$v_m$$ and $$v_s$$ are the volume of stationary and mobile phase in the column and $$K$$ is the distribution coefficient (basically the equilibrium constant of the solute concentration in stationary and mobile phase). They give the definition of plate volume as

The plate volume is defined as that volume of mobile phase that can contain all the solute in the plate at the equilibrium concentration of the solute in the mobile phase.

The plate concept is introduced to explain the equilibrium attained by the solute in distributing between stationary and mobile phase. We can't assume this equilibrium in the whole column as mobile phase continuously flows during the process and disturbs the equilibrium, so column is considered to be made of number of theoretical plates in which equilibrium is attained for some finite time.

Basically, I am not able to understand the definition of plate volume. Because as the name suggests one can interpret as the column to be made of number of plates which is made of the volume of mobile phase and stationary phase in which the solute is distributed between the two phases. So, I initially misinterpreted the formula of plate volume as $$v_m+v_s$$.

Please explain the definition of the plate volume. It seems somewhat confusing.

I shall take the example of the mixture ethanol-water, first without chromatography. If I start distilling 250 mL of a mixture 10% ethanol and 90% water, the solution boils at 96°C, and the vapor contains 55% water + 45% ethanol. This vapor is condensed by passing through the cold condenser. So the first fraction of maybe 10 mL condensed vapor contains 45% ethanol. Suppose 250 mL of this first mixture (45% ethanol) is heated in another distilling apparatus. At maybe 85°C, the liquid boils and the first vapor will contain 70% ethanol and 25% water. If this vapor is condensed, the liquid so obtained contains 70% ethanol. These distillation - condensation operations can be repeated, with the following results.

1st distillation. Original liquid : 10% ethanol. Distilled liquid : 45% ethanol

2nd distillation. Original liquid : 45% ethanol. Distilled liquid : 70% ethanol

3rd distillation. Original liquid : 70% ethanol. Distilled liquid : 83% ethanol

4th distillation. Original liquid : 83% ethanol. Distilled Liquid : 90% ethanol.

etc.

nth distillation. Original liquid : 96% ethanol. Distilled liquid : 96% ethanol.

Now suppose all these distillations are carried out in the same apparatus. Instead of having the vapor go through a cold tube and be recovered in a flask out of the condenser, imagine that the vapor is sent into a long vertical tubes situated just above the first container half filled with the mixture 10% ethanol. Suppose this vertical tube contains a lot of successive floors or plates, punched with one or two small holes. In the beginning the vapor made of 45% ethanol will pass through the holes of the first floor or plate, and will be condensed just above this level and may partly go down to the bottom of the setup. But a fraction of this liquid will now boil again and produce a vapor made of 70% ethanol. This vapor passes through the second floor and be condensed just above the second floor. On this second floor the liquid (containing 70% ethanol) will later on be evaporated and produce a vapor containing 83% ethanol. And so on. After n floors the vapor and the liquid will contain 96% ethanol. As a result, the separation ethanol-water improves if the number of floors or plates increases. And you may imagine that the height of the room between two floors is critical. If it is too small, it will be immediately filled. If it is too big, some secondary condensation will happen on the wall of each room.

Now ethanol is not the best example for my purpose because the successive distillations cannot get to pure ethanol. Suppose for the rest of my description that after n distillations, a pure amount of a solvant X can be obtained. Suppose that to get a better that 99% separation you need 8 floors, which are all 5 mm high.

Suppose now that instead of having used a distillation apparatus, you are using of chromatographic column. In order to get a better than 99% separation of pure compounds you need a column measuring 60 cm. The plate volume is : 60 cm / 8 plates = 7.5 cm. Of course, if you use a longer column, the separation might be 99.9% or better.

Of course this explanation is an over-simplification of the reality. But it may help you understand what is a plate volume.

• Thanks for the explanation. Basically you use the analogy of fractional distillation. This helps a lot. But one thing which I am not able to understand is that if we consider 1 theoretical plate of the chromatographic column then it has stationary and mobile phase with solute distributed between the two phases, so shouldn't the plate volume be $v_m+v_s$ instead of $v_m+Kv_s$. In derivation of the plate theory, there comes an expression of $v_m+Kv_s$ and they defined it to be plate volume. But I am not able to link this expression to plate volume?