Question:
Consider a closed system of one mole of an ideal gas undergoing a polytropic process. The process follows \( PV^n = \text{constant} \), where P is the pressure, V is the volume and n is a constant. If the process is isobaric, which one of the following is the value of n?
Consider a closed system of one mole of an ideal gas undergoing a polytropic process. The process follows \( PV^n = \text{constant} \), where P is the pressure, V is the volume and n is a constant. If the process is isobaric, which one of the following is the value of n?
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Set the exponent on volume to zero so pressure alone remains constant in PV^n = C.
Updated On: Aug 10, 2026
- 0
- 1
- \( \infty \)
- the ratio of specific heat capacity at constant pressure to specific heat capacity at constant volume
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The Correct Option is A
Solution and Explanation
Step 1: State the general polytropic relation and what "isobaric" means.
A polytropic process is described by PV^n = constant. An isobaric process is, by definition, a process carried out at constant pressure, meaning P does not change as V changes.
Step 2: Substitute the isobaric condition into the polytropic relation.
Write the polytropic relation as P = C/V^n = C V^(-n). For P to remain independent of V, the exponent on V must vanish, that is -n = 0, giving n = 0. Substituting back confirms: with n=0, PV^0 = P = constant, exactly the isobaric condition.
Step 3: Verify the other listed special cases.
With n=1, PV = constant, the isothermal process for an ideal gas, not isobaric. As n approaches infinity, the relation collapses to V = constant, the isochoric process. When n equals the ratio of specific heats, the process becomes the reversible adiabatic process.
Step 4: Conclude.
Only n = 0 reduces PV^n = constant to P = constant, the definition of an isobaric process. \[ \boxed{n = 0} \]
A polytropic process is described by PV^n = constant. An isobaric process is, by definition, a process carried out at constant pressure, meaning P does not change as V changes.
Step 2: Substitute the isobaric condition into the polytropic relation.
Write the polytropic relation as P = C/V^n = C V^(-n). For P to remain independent of V, the exponent on V must vanish, that is -n = 0, giving n = 0. Substituting back confirms: with n=0, PV^0 = P = constant, exactly the isobaric condition.
Step 3: Verify the other listed special cases.
With n=1, PV = constant, the isothermal process for an ideal gas, not isobaric. As n approaches infinity, the relation collapses to V = constant, the isochoric process. When n equals the ratio of specific heats, the process becomes the reversible adiabatic process.
Step 4: Conclude.
Only n = 0 reduces PV^n = constant to P = constant, the definition of an isobaric process. \[ \boxed{n = 0} \]
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