Dimensional Analysis
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?
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Solution
[ML5T−2]
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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About This Question
- Subject
- physics
- Chapter
- physics and measurement
- Topic
- dimensional analysis
- Difficulty
- Hard
- Year
- 2025
Solution
Correct Answer:
[ML5T−2]
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.
This hard difficulty physics question is from the chapter physics and measurement, covering the topic of dimensional analysis. It appeared in the 2025 exam.
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