In the isothermal condition, the isothermal bulk modulus of an ideal gas is equal to:
Try it first: pick an option, then read the explanation.
Asked in SSC JE 25-01-2018 (Evening Shift)
Detailed Explanation
Bulk modulus depends on the thermodynamic process of compression. For an ideal gas undergoing slow compression at constant temperature, Boyle's law governs the relationship between pressure and volume.
For an isothermal process of an ideal gas: \(P V = \text{constant}\).
Differentiating both sides:
\[ P\,dV + V\,dP = 0 \;\implies\; \frac{dP}{dV} = -\frac{P}{V} \]
By definition of bulk modulus \(K\):
\[ K = -V \frac{dP}{dV} = -V \left(-\frac{P}{V}\right) = P \]
(For an adiabatic process, \(P V^\gamma = \text{constant}\), which gives \(K_{\text{adiabatic}} = \gamma P\).)
Why “pressure” Is Correct
Differentiating the isothermal gas relation PV = const gives K = P: the isothermal bulk modulus equals the absolute pressure.
Why the Other Options Are Wrong
- A. gas constant: The characteristic gas constant R has different units (J/(kg·K)) and does not equal K.
- C. temperature: Temperature affects absolute pressure via PV = mRT, but K equals the pressure P itself.
- D. viscosity: Viscosity governs shear resistance, not volumetric elastic resistance.
Key Point
Isothermal bulk modulus of ideal gas: K = P. Adiabatic bulk modulus: K = γP.
Key Concepts for Students
- Bulk modulus of a gas is directly proportional to its operating pressure.
- Adiabatic bulk modulus is larger by the heat capacity ratio γ = Cp/Cv (γ ≈ 1.4 for air).
- Speed of sound in a gas is c = √(K_adiabatic / ρ) = √(γRT).
Practice more: this is question 39 in Properties of Fluid MCQs – Part 3. Try the whole set and check every answer instantly.