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
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 () 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:
For an ideal gas undergoing an isothermal process, the temperature is kept constant. According to the ideal gas law, the equation of state for an isothermal process is given by Boyle's Law:
To find the relationship between the change in pressure and change in volume , we differentiate both sides of the equation with respect to :
Applying the product rule of differentiation:
Rearranging the terms, we get:
Comparing this with the definition of bulk modulus, we find:
Thus, the bulk modulus of an ideal gas during an isothermal process is equal to its pressure .
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