Determine dimensions and si units of constants?

Question

I am am trying to work out the dimensions and si units of the constants a and b in the following equation.

(P + a/V^2) (V - b) = RT

Where:

P=Pressure

V=Volume

T=Temperature

R=Gas Constant

Thanks!

Answer 1

Favorite Answer

Multiplying out the LHS using FOIL yields

PV - Pb + a/V^3 - ab/V^2

For this addition of quantities to be meaningful, we must have that all four carry the same dimensions.

This suggests that b carries the same dimensions as volume, m^3 in SI units.

Also a/V^2 must have the same dimensions as P,

So, [a/V^2] = [P], or [a]/[V^2] = [P], so [a] = [P][V^2].

Note that pressure is force divided by area, or mass times acceleration divided by Area.

So [P] = [M][L][T]^2/[L]^2 = [M][T]^2/[L], which in SI units is the kg s^2/m

And [V]^2 = ([L]3)^2 = [L]^6 which in SI units is m^6

So [a] = [P][V]^2

= [M][T]^2[L]^6/[L]

= [M][L]^5[T]^2

which in SI units is kg m^5 s^2.