Understanding Work Done in Adiabatic Expansion

Derive an expression for the work done during adiabatic process is an essential derivation in JEE Main Physics, connecting thermodynamics, gas laws, and energy considerations. In an adiabatic process, a system undergoes expansion or compression with no heat exchange (Q = 0) between the system and its surroundings, making the derivation distinct from the isothermal case.

Physical Principles Behind Adiabatic Work Derivation

During an adiabatic process, all energy changes in a gas manifest as work or changes in internal energy since no heat enters or leaves. This process is governed by the First Law Of Thermodynamics, which states that the change in internal energy equals the work done by or on the system. Understanding how work during adiabatic expansion or compression differs from isothermal work is critical for JEE Main.

Stepwise Derivation: Work Done in an Adiabatic Process

Let us systematically derive the expression for work done during an adiabatic process for an ideal gas. We proceed step by step to ensure clarity and completeness, using standard NCERT symbols and SI units.

Step 1: The First Law of Thermodynamics for a quasi-static process:
dQ = dU + dW
Since the process is adiabatic, dQ = 0:
0 = dU + dW
Thus,
dW = – dU

Step 2: For an ideal gas, the change in internal energy is given by:
dU = nC_v dT
where n = number of moles, C_v = molar specific heat at constant volume, dT = change in temperature.
Therefore,
dW = – nC_v dT

Step 3: Also, work done (by the gas) in a small expansion dV at pressure P:
dW = P dV
From previous step, equate both expressions for dW:
P dV = – nC_v dT

Step 4: For an adiabatic process in an ideal gas, the relation between P, V, and T is:
P V^γ = constant
where γ = C_p / C_v (ratio of specific heats).
Let the constant be K:
P = K V^–γ

Step 5: Substitute for P into the work integral:
Work done as volume changes from V₁ to V₂:
W = ∫_V₁^V₂ P dV = ∫_V₁^V₂ K V^–γ dV
= K ∫_V₁^V₂ V^–γ dV

Step 6: Integrate:
∫ V^–γ dV = (1/(1–γ)) V^1–γ
Therefore,
W = K [V^1–γ / (1–γ)]_V₁^V₂
= K/(1–γ)[V₂^1–γ – V₁^1–γ]

Step 7: Express K in terms of initial state:
Since K = P₁ V₁^γ = P₂ V₂^γ,
W = [P₂ V₂ – P₁ V₁]/(γ–1)

Final Expression:
W = [P₂ V₂ – P₁ V₁]/(γ–1)

Analysis and Physical Meaning

The derived work done during adiabatic process expression depends on the initial and final states (P₁, V₁, P₂, V₂) of the gas and the ratio of specific heats (γ). This form clearly distinguishes it from isothermal work, which depends on the logarithm of volume ratio. The negative sign in expansion indicates that work is done by the gas, reducing its internal energy.

Comparing Adiabatic Work with Other Thermodynamic Processes

Unlike the isothermal process, where temperature is constant and the work depends on heat absorption, adiabatic work arises purely from internal energy changes. For further clarity, the topic of Thermodynamics offers deeper insight into process-specific energy transformations relevant for JEE Main.

• Adiabatic process: No heat exchange, temperature changes with volume change
• Isothermal process: Temperature constant, heat exchange balances work done
• Isochoric process: No work done since volume remains constant
• Isobaric process: Work done is simply PΔV for constant pressure

Key Points and Common Pitfalls for JEE Main Aspirants

Mastering the derivation of work during adiabatic expansion or compression is crucial for scoring in thermodynamics questions. Always differentiate between the work during adiabatic and isothermal changes. Remember to use the correct values for γ for monoatomic (5/3), diatomic (7/5), or polyatomic ideal gases depending on the question. Revisiting Work Energy And Power helps connect these ideas with broader mechanics concepts.

Vedantu’s experts offer stepwise derivations and interactive problem-solving to help students internalise concepts like the work done during adiabatic process. These foundational principles support higher-level understanding in both JEE Main and future physics studies.