Thermodynamics

Physics · Class 11

Lesson 13 of 13 · 18 min

Chapter review

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Must-know facts

21 facts

  1. 1Rumford's cannon boring showed heat is energy produced by work, not a fluid (caloric).
  2. 2Adiabatic wall: no heat passes; diathermic wall: heat passes.
  3. 3Zeroth law: two systems each in equilibrium with a third are in equilibrium with each other; it defines temperature.
  4. 4Internal energy U excludes the kinetic energy of the system moving as a whole.
  5. 5U is a state variable; heat and work are not, they are modes of energy transfer.
  6. 6First law: ΔQ = ΔU + ΔW; ΔQ − ΔW is path-independent.
  7. 7Boiling 1 g of water: ΔQ = 2256 J, ΔW = 169.2 J, ΔU = 2086.8 J.
  8. 8Solids: molar heat ≈ 3R (Table 11.1), carbon excepted.
  9. 91 cal = 4.186 J; water's specific heat 4186 J kg⁻¹ K⁻¹.
  10. 10C_p − C_v = R for an ideal gas.
  11. 11Extensive: U, V, M. Intensive: P, T, ρ.
  12. 12Quasi-static: infinitely slow, always in equilibrium with the surroundings.
  13. 13Isothermal: PV = constant, ΔU = 0, W = Q = μRT ln(V₂/V₁).
  14. 14Adiabatic: Q = 0, PV^γ = constant, W = μR(T₁ − T₂)/(γ − 1); expansion cools, compression heats.
  15. 15Adiabat is steeper than isotherm on a P–V diagram.
  16. 16Isochoric: W = 0, Q = ΔU. Isobaric: W = PΔV = μRΔT.
  17. 17Cyclic: ΔU = 0, net Q = net W = area of the loop.
  18. 18Kelvin–Planck: no perfect heat engine. Clausius: no perfect refrigerator. They are equivalent.
  19. 19Reversible needs quasi-static and no dissipation; natural processes are irreversible.
  20. 20Carnot cycle: two isothermals joined by two adiabatics; η = 1 − T₂/T₁.
  21. 21No engine between T₁ and T₂ beats Carnot, whatever the working substance.

Common traps

Where marks are lost

Writing ΔQ = ΔU − ΔW with work by the system.

With ΔW done by the system, ΔQ = ΔU + ΔW; work done on the gas enters as negative ΔW.

Saying a hot body contains a lot of heat.

A body has internal energy; heat is only energy in transit.

Taking ΔU = 0 whenever heat is supplied.

ΔU = 0 only if the temperature of an ideal gas is unchanged (isothermal) or the process is a complete cycle.

Assuming no heat flows in an isothermal process.

Heat does flow (Q = W); it is the temperature that stays fixed. No heat flows in an adiabatic process.

Using PV = constant for a sudden, insulated compression.

Insulated or rapid changes are adiabatic: use PV^γ = constant, and the gas heats up.

Using W = PΔV for an isothermal or adiabatic change.

Pressure varies there; use μRT ln(V₂/V₁) or μR(T₁ − T₂)/(γ − 1).

Putting Celsius temperatures into 1 − T₂/T₁.

Carnot efficiency needs kelvin: 27 °C is 300 K.

Believing a good enough engine could reach 100% efficiency.

The second law forbids η = 1; even a Carnot engine reaches only 1 − T₂/T₁.

Counting a moving container's kinetic energy in U.

U is measured in the frame where the centre of mass is at rest.

Calling temperature or pressure extensive.

Halving a system leaves P, T and ρ unchanged: they are intensive.

Formulas

12 to know

First law

ΔQ = ΔU + ΔW

ΔQ into the system, ΔW by the system.

Work at constant pressure

ΔW = P ΔV

Area under the P–V curve in general.

Heat capacities

S = ΔQ/ΔT; s = (1/m) ΔQ/ΔT; C = (1/μ) ΔQ/ΔT

J K⁻¹, J kg⁻¹ K⁻¹, J mol⁻¹ K⁻¹.

Solids (equipartition)

U = 3RT per mole; C = 3R

Fits at ordinary temperatures; carbon is an exception.

Specific heats of an ideal gas

C_p − C_v = R

Ideal gas, molar values.

Ideal-gas equation of state

PV = μRT

R = 8.31 J mol⁻¹ K⁻¹.

Isothermal work

W = Q = μRT ln(V₂/V₁)

ΔU = 0 for an ideal gas.

Adiabatic relation

PV^γ = constant; P₁V₁^γ = P₂V₂^γ

γ = C_p/C_v; also TV^(γ−1) = constant.

Adiabatic work

W = (P₁V₁ − P₂V₂)/(γ − 1) = μR(T₁ − T₂)/(γ − 1)

Q = 0, so W = −ΔU.

Isobaric work

W = P(V₂ − V₁) = μR(T₂ − T₁)

Heat goes to both ΔU and W.

Cyclic process

ΔU = 0; Q_net = W_net

W_net = area of the loop.

Carnot efficiency

η = W/Q₁ = 1 − Q₂/Q₁ = 1 − T₂/T₁

T in kelvin; Q₂/Q₁ = T₂/T₁.

Key terms

15 terms

Caloric
The discarded idea of heat as an invisible fluid flowing from hot to cold bodies.
Thermal equilibrium
The state in which a system's macroscopic variables no longer change in time.
Adiabatic wall
An insulating wall that lets no heat through.
Diathermic wall
A conducting wall that lets heat flow between systems.
Zeroth law
Two systems each in thermal equilibrium with a third are in thermal equilibrium with each other.
Internal energy
Sum of the molecular kinetic and potential energies, measured in the centre-of-mass frame.
State variable
A quantity fixed by the present equilibrium state alone, not by the path taken, e.g. P, V, T, U.
Equation of state
The relation between the state variables, e.g. PV = μRT for an ideal gas.
Quasi-static process
An infinitely slow process in which the system stays in equilibrium with its surroundings.
Isothermal process
A process at constant temperature.
Adiabatic process
A process with no heat exchange between system and surroundings.
Isochoric process
A process at constant volume.
Isobaric process
A process at constant pressure.
Reversible process
A process that can be undone so that system and surroundings both return to their initial states with no other change.
Carnot engine
A reversible engine between two temperatures, working on two isothermal and two adiabatic steps.
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