Source: Extra Practice

In the Van der Waals equation of state for a real gas, (P+aV2)(Vb)=RT\left( P + \frac{a}{V^2} \right) (V - b) = RT, where PP is pressure, VV is volume, TT is temperature, and RR is the universal gas constant. What are the dimensions of the constant aa?

Options

Option A is correct

[ML5T2][ML^5T^{-2}]

Option B

[ML2T2][ML^2T^{-2}]

Option C

[L3][L^3]

Option D

[ML1T2][ML^{-1}T^{-2}]

Explanation

According to the principle of dimensional homogeneity, terms added or subtracted must have the same dimensions. In the first parenthesis, PP is added to aV2\frac{a}{V^2}. Therefore, the dimension of aV2\frac{a}{V^2} must be the same as the dimension of pressure (PP). Dimensions of pressure P=ForceArea=[MLT2][L2]=[ML1T2]P = \frac{\text{Force}}{\text{Area}} = \frac{[MLT^{-2}]}{[L^2]} = [ML^{-1}T^{-2}] . Dimensions of volume V=[L3]V = [L^3] , so V2V^2 has dimensions [L6][L^6]. Equating the dimensions: [aV2]=[P][\frac{a}{V^2}] = [P] [a]=[P][V2][a] = [P][V^2] [a]=[ML1T2][L6]=[ML(1+6)T2]=[ML5T2][a] = [ML^{-1}T^{-2}][L^6] = [ML^{(-1+6)}T^{-2}] = [ML^5T^{-2}].