dimensional consistency

Question

The equation of state for a real gas is (P + a/V²)(V - b) = RT, where a and b are constants. The dimensions of 'a' are:

RankGuru · Accepted Answer

1. **Principle of Homogeneity**: Only physical quantities with identical dimensions can be added or subtracted.
2. **Analysis of Term $(P + a/V^2)$**:
   - Dimension of $a/V^2$ must equal dimensions of Pressure $P$.
3. **Derive 'a' Dimensions**:
   - $[a] = [P] 	imes [V^2]$.
   - Dimensions of $P$ (Pressure): Force/Area $= [ML^{-1}T^{-2}]$.
   - Dimensions of $V^2$ (Volume squared): $([L^3])^2 = [L^6]$.
4. **Calculation**:
   - $[a] = [ML^{-1}T^{-2}] 	imes [L^6] = [ML^5T^{-2}]$.
5. **Result**: The dimensions of the Van der Waals constant 'a' are **$[ML^5T^{-2}]$**.

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