gen-verify-the-first-law-energy-balance-for-an-adiabatic-process-e524

Verify the first-law energy balance for an adiabatic process.

# Verification of First-Law Energy Balance for an Adiabatic Process ## First Law of Thermodynamics The general form of the first law for a closed system is: $$dU = \delta Q - \delta W$$ or in integral form: $$\Delta U = Q - W$$ where: - $\Delta U$ = change in internal energy - $Q$ = heat transferred to the system - $W$ = work done by the system ## Adiabatic Process Condition For an **adiabatic process**, there is no heat transfer between the system and surroundings: $$Q = 0$$ ## Energy Balance for Adiabatic Process Substituting $Q = 0$ into the first law: $$\boxed{\Delta U = -W}$$ or equivalently: $$\boxed{dU = -\delta W}$$ ## Verification This can be verified through several perspectives: ### 1. **Physical Interpretation** - All work done by the system comes from internal energy (temperature/pressure decrease) - All work done on the system increases internal energy (temperature/pressure increase) ### 2. **For an Ideal Gas** Using $W = \int P\,dV$ and the ideal gas relation: $$\Delta U = nC_v\Delta T = -W$$ For an adiabatic process: $PV^\gamma = \text{constant}$ This leads to: $$W = \frac{P_1V_1 - P_2V_2}{\gamma - 1} = \frac{nR(T_1 - T_2)}{\gamma - 1} = nC_v(T_1 - T_2) = -\Delta U$$ ✓ ### 3. **Energy Conservation** Since energy cannot cross the boundary as heat, the only energy exchange is work, which must equal the change in internal energy. ## Conclusion The first-law energy balance for an adiabatic process is **verified**: all energy changes manifest as work-energy conversion with no heat transfer.