Question Details

An ideal gas of mass m, volume V, pressure p and temperature T undergoes a small change in state at constant temperature. Its adiabatic exponent i.e., Cp/Cv is γ. The bulk modulus of the gas at the constant temperature process called isothermal process is

Options

A

p

B

γp

C

mγp/T

D

γpV/T

Correct Answer :

p

Solution :

The correct option is p.

To find the bulk modulus of the gas during an isothermal process, we start by understanding the definition of bulk modulus. The bulk modulus (K) of a substance measures its resistance to uniform compression and is defined as the ratio of the infinitesimal pressure increase to the resulting relative decrease of the volume:
K=-VdpdV

For an ideal gas undergoing an isothermal process, the temperature T is kept constant. According to the ideal gas law, the equation of state for an isothermal process is given by Boyle's Law:
pV=constant

To find the relationship between the change in pressure dp and change in volume dV, we differentiate both sides of the equation pV=constant with respect to V:
ddV(pV)=0
Applying the product rule of differentiation:
pdVdV+VdpdV=0
p+VdpdV=0
Rearranging the terms, we get:
-VdpdV=p

Comparing this with the definition of bulk modulus, we find:
Kisothermal=p

Thus, the bulk modulus of an ideal gas during an isothermal process is equal to its pressure p.

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