Lecture 39 The First Law of Thermodynamics and various kinds of processes Quasistatic processes The relation of heat and work First Law of Thermodynamics Important types of thermodynamic processes

Quasistatic processes A quasistatic process is a thermodynamic process that happens infinitely slowly. During the process, it is always in thermal equilibrium. No real process is quasistatic, but such processes can be approximated by performing them very slowly.

PV Diagrams for quasi-static process A thermodynamic process can be visualized by graphically plotting the changes to the system's state variables. Thermodynamic processes which make up the Stirling cycle

Work When the volume of a system changes, the system does work 𝛿𝑊=𝑝𝑑𝑉 The total work done in a process is 𝑊= 𝑉 𝑖 𝑉 𝑓 𝑝𝑑𝑉 It’s the area under the 𝑝-𝑉 curve

𝑝-𝑉 work can be positive or negative

Does the path of the PV change matter? The start, the finish, and the shape of the curve are all significant.

Heat and heat capacity The heat 𝑄 that the object absorbs or loses and the resulting temperature change Δ𝑇 is 𝑄=𝐶Δ𝑇=𝐶( 𝑇 𝑓 − 𝑇 𝑖 ) where 𝐶 is the heat capacity, which depends on the process. During a thermodynamic process, energy may be transferred into the system from the thermal reservoir (positive heat) or vice versa (negative heat). Also, work can be done by the system to raise the loaded piston (positive work) or lower it (negative work).

The First Law of Thermodynamics When a system changes from a given initial state to a given final state, both the work 𝑊 and the heat 𝑄 depend on the nature of the process. Experimentally, however, we find the quantity 𝑄 −𝑊 is the same for all processes. It depends only on the initial and final states and does not depend at all on how the system gets from one to the other. Define internal energy 𝐸 𝑖𝑛𝑡 (𝑃,𝑉) and Δ 𝐸 𝑖𝑛𝑡 = 𝐸 𝑖𝑛𝑡,𝑓 − 𝐸 𝑖𝑛𝑡,𝑖 =𝑄−𝑊 For differential change 𝑑 𝐸 𝑖𝑛𝑡 =𝛿𝑄−𝛿𝑊 It is the first law of thermodynamics.

Isochoric process: constant volume Δ𝑉=0 → Δ𝑊=𝑝Δ𝑉=0 For 𝑑 𝐸 𝑖𝑛𝑡 =𝛿𝑄−𝑝𝑑𝑉=𝛿𝑄 Heat capacity at constant volume 𝐶 𝑉 = 𝛿𝑄 𝑑𝑇 = 𝑑 𝐸 𝑖𝑛𝑡 𝑑𝑇 The heat absorbed or lose is 𝑄= 𝑇 𝑖 𝑇 𝑓 𝐶 𝑉 𝑑𝑇 = 𝐶 𝑉 ( 𝑇 𝑓 − 𝑇 𝑖 )

Isobaric process: constant pressure 𝑊= 𝑝𝑑𝑉 =𝑝Δ𝑉 Ideal gas equation 𝑝𝑉=𝑛𝑅𝑇 𝑑 𝑝𝑉 =𝑝𝑑𝑉=𝑛𝑅𝑑𝑇 So 𝑊=𝑛𝑅Δ𝑇 Notice the internal energy is a state function Δ 𝐸 𝑖𝑛𝑡 = 𝐶 𝑉 Δ𝑇 For Δ 𝐸 𝑖𝑛𝑡 =𝑄−𝑊 𝑄=Δ 𝐸 𝑖𝑛𝑡 +𝑊= 𝐶 𝑉 Δ𝑇+𝑛𝑅Δ𝑇= 𝐶 𝑝 Δ𝑇 So the heat capacity at constant pressure is 𝐶 𝑝 = 𝐶 𝑉 +𝑛𝑅

Isothermal process: constant temperature 𝑝= 𝑛𝑅𝑇 𝑉 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝑉 𝑊= 𝑉 𝑖 𝑉 𝑓 𝑝𝑑𝑉 =𝑛𝑅𝑇ln V 𝑓 𝑉 𝑖 For the temperature is hold constant Δ 𝐸 𝑖𝑛𝑡 =0 So 𝑄=𝑊 How about the heat capacity?

Adiabatic process: without loss or gain of energy as heat 𝑄=0 → Δ 𝐸 𝑖𝑛𝑡 =−𝑊 𝑑 𝐸 𝑖𝑛𝑡 = 𝐶 𝑉 𝑑𝑇 → 𝛿𝑊=− 𝐶 𝑉 𝑑𝑇 For Quasistatic adiabatic process 𝑝𝑉=𝑛𝑅𝑇 So 𝑝𝑑𝑉+𝑉𝑑𝑝=𝑛𝑅𝑑𝑇 And 𝛿𝑊=𝑝𝑑𝑉=− 𝐶 𝑉 𝑑𝑇 From above we get 𝑝𝑑𝑉+𝑉𝑑𝑝=− 𝑛𝑅 𝐶 𝑉 𝑝𝑑𝑉 𝐶 𝑉 +𝑛𝑅 𝐶 𝑉 𝑝𝑑𝑉=−𝑉𝑑𝑝 → 𝐶 𝑝 𝐶 𝑉 𝑑𝑉 𝑉 =− 𝑑𝑝 𝑝

𝐶 𝑝 𝐶 𝑉 𝑑𝑉 𝑉 =− 𝑑𝑝 𝑝 Define 𝛾= 𝐶 𝑝 𝐶 𝑉 , we get ln𝑝+𝛾ln𝑉=𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 Or 𝑝 𝑉 𝛾 =𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 For 𝑝𝑉=𝑛𝑅𝑇, we have 𝑇 𝑉 𝛾−1 =𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝑇 𝛾 𝑝 𝛾−1 =𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡

Polytropic process A polytropic process obeys the relation 𝑝 𝑉 𝑛 =𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 So 𝑉 𝑛 𝑑𝑝+𝑛𝑝 𝑉 𝑛−1 𝑑𝑉=0 or 𝑝𝑑𝑉=− 1 𝑛 𝑉𝑑𝑝 For 𝑉𝑑𝑝+𝑝𝑑𝑉=𝜈𝑅𝑑𝑇 We get 𝑝𝑑𝑉= 1 𝑛 𝑝𝑑𝑉−𝜈𝑅𝑑𝑇 and 𝑝𝑑𝑉=− 𝜈 𝑛−1 𝑅𝑑𝑇 The work done 𝑊= 𝜈𝑅 𝑛−1 ( 𝑇 𝑖 − 𝑇 𝑓 )

The internal energy Quasistatic adiabatic process 𝑝 𝑉 𝛾 =𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝑉 𝛾 𝑑𝑝+𝛾𝑝 𝑉 𝛾−1 𝑑𝑉=0 or 𝑉𝑑𝑝+𝛾𝑝𝑑𝑉=0 𝑉𝑑𝑝+𝑝𝑑𝑉= 1−𝛾 𝑝𝑑𝑉→ 𝑑 𝑝𝑉 𝛾−1 =−𝑝𝑑𝑉 𝑑 𝐸 𝑖𝑛𝑡 =−𝛿𝑊=−𝑝𝑑𝑉= 𝑑 𝑝𝑉 𝛾−1 i.e. 𝑝𝑉= 𝛾−1 𝐸 𝑖𝑛𝑡 For 𝑝𝑉=𝑛𝑅𝑇 𝐸 𝑖𝑛𝑡 = 1 𝛾−1 𝑛𝑅𝑇