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An equation is given as : \( \left( P + \frac{a}{v^2} \right) = \frac{b}{v} \) where P = Pressure, V = Volume & θ = Absolute temperature. If a and b are constants, then dimensions of a will be

1. [ML⁵T⁻²]
2. [M⁻¹L⁵T²]
3. [ML⁵T⁻¹]
4. [ML⁵T⁻¹]

Correct answer: [ML⁵T⁻²]

Solution

The term \( \frac{a}{v^2} \) must have the same dimensions as pressure \( P \), which is \([ML^{-1}T^{-2}]\). Since \( v \) (volume) has dimensions \([L^3]\), \( v^2 \) has dimensions \([L^6]\). Therefore, \( a \) must have dimensions \([ML^5T^{-2}]\) to make \( \frac{a}{v^2} \) dimensionally consistent with pressure.

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