Under isothermal condition, the pressure of a gas is given by P=a V⁻³, where a is a constant and V is the volume of the gas. The bulk modulus at constant temperature is equal to

Solution

Physics sir · 4 steps

Bulk modulus is just -V dP/dV, so differentiate P = aV⁻³ and the power 3 drops straight out.

1. Definition. Bulk modulus measures how much pressure it takes to squeeze a volume. The minus sign makes B positive, because volume falls when pressure rises.

   B = -VdP/dV

2. Differentiate. From P = aV⁻³, we get dP/dV = -3aV⁻⁴.
3. Put it together. B = -V × (-3aV⁻⁴) = 3aV⁻³. Since aV⁻³ = P, this is B = 3P.
4. So. The isothermal bulk modulus is 3P, option (c).

The idea to keep. For any gas law of the form PVⁿ = constant, the bulk modulus is nP. That is why it is P for an isothermal ideal gas and γ P for an adiabatic one.

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