Dimensional Analysis

Question

In the van der Waals equation of state, written as (P + a / V^2)(V - b) = R T for a real gas, what is the dimensional formula of the constant a?

RankGuru · Accepted Answer

In any sum, each term must share the dimensions of the others, so the corrective term a/V^2 must carry the same dimensions as pressure P, with which it is added. Pressure is force per unit area, giving [P] = [M L T^-2]/[L^2] = [M L^-1 T^-2]. Volume has dimensions [L^3], so V^2 has dimensions [L^6]. Setting [a]/[V^2] equal to [P] yields [a] = [P][V^2] = [M L^-1 T^-2][L^6] = [M L^5 T^-2]. The choice [M L^-1 T^-2] is merely the dimension of pressure itself and ignores the multiplication by V^2. The choice [M L^3 T^-2] would arise from multiplying pressure by V to the first power rather than its square. The choice [M L^5 T^-1] has the wrong time exponent and cannot match a pressure-based product. This style of analysis, isolating an additive term and equating its dimensions, is standard in JEE Advanced problems on real-gas equations. As a check, the constant b is added to V and must have dimension [L^3], a volume, which is mutually consistent with the structure of the equation.

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