(P+a/V²)(V-b)=R T represents the equation of state of some gases. Where P is the pressure, V is the volume, T is the temperature and a, b, R are the constants. The physical quantity, which has dimensional formula as that of b²/a, will be:

Solution

Physics sir · 3 steps

You can only add like to like, so a/V² must be a pressure and b must be a volume.

1. Find a. Quantities can be added only if they have the same dimensions. So a/V² has the dimensions of P, which gives [a] = [P][V²].
2. Find b. In (V - b), b is subtracted from a volume, so [b] = [V].

   [b²/a] = [V²]/[P][V²] = 1/[P] = [M⁻¹LT²]

   Option	Defined as	Dimensions
   Energy density	energy / volume	[ML⁻¹T⁻²], same as pressure
   Bulk modulus	stress / volumetric strain	[ML⁻¹T⁻²], same as pressure
   Modulus of rigidity	shear stress / shear strain	[ML⁻¹T⁻²], same as pressure
   Compressibility	1 / bulk modulus	[M⁻¹LT²], inverse of pressure

3. So. b²/a has the dimensions of 1/pressure. Only compressibility matches, so the answer is (d).

The idea to keep. In any equation, every term added to or subtracted from another has the same dimensions. Use this to read off the dimensions of unknown constants like a and b in one line.

Answer from the NTA final answer key. Solution by PrepWiser.