The column pressure under any conditions can be predicted using the following equation:

P = (150 η L F) / (dparticle2 dcolumn2)

where P is the pressure, η is the solvent viscosity, L is the column length, F is the flow rate, dparticle is the particle diameter, and dcolumn is the column inner diameter. If only the particle size is varied (same method, same column dimensions), then the factor (150 η L F) / dcolumn2 becomes a constant k:

P2µm = d2µm2
P4µm = d4µm2

The constant will be the same for 4µm and 2µm columns so we can solve each equation for k

P2µm d2µm2 = k
P4µm d4µmk

…and then set the two expressions equal to each other:

P2µm d2µm2 = P4µm d4µmk

Note that we don’t need to know the actual numerical value of k. Now we want to know how the pressures compare, so we will rearrange the equation to show the pressure ratio of 2µm vs. 4µm columns as a function of particle size:

(P2µm P4µm) d4µmd2µm2

(P2µm P4µm) = 42 / 22 = 16 / 4 = 4

The equation shows the pressure will be theoretically 4 times higher on the 2µm column than the 4µm column using equivalent conditions. In practice you may not use the exact same method conditions when transferring a 4µm method to a 2µm column. A lower flow rate or shorter column may be warranted, for example. You can still predict the pressure difference using different conditions but the comparison becomes more complicated if more than one variable is changed because k is no longer the same between the two methods.