Quick Revision | Thermodynamics

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Question

What is the First Law of Thermodynamics? Write the mathematical form.

Answer

Energy is conserved.

ΔU = Q − W

ΔU = change in internal energy
Q = heat absorbed (+ve in)
W = work done by gas (+ve out)

Question

State the efficiency formula for Carnot engine. What must the temperatures be in?

Answer

η = 1 − T_C / T_H

MUST be in Kelvin!

27°C = 300 K (add 273)
Never use Celsius!

Question

What is γ (gamma) for monoatomic, diatomic, and polyatomic gases?

Answer

Monoatomic (He, Ar): γ = 5/3 ≈ 1.67

Diatomic (H₂, N₂, O₂): γ = 7/5 = 1.4

Polyatomic (H₂O): γ = 4/3 ≈ 1.33

Formula: γ = (f+2)/f

Question

State Mayer's Relation. How do you derive it?

Answer

Cp − Cv = R

Derivation:
Isobaric: Q = nCpΔT = ΔU + W
= nCvΔT + nRΔT
∴ Cp = Cv + R

Question

What happens to ΔU, Q, W in adiabatic expansion?

Answer

Q = 0 (definition of adiabatic)

W > 0 (gas expands)

ΔU = Q − W = −W < 0

Temperature DECREASES

Question

The adiabatic curve is steeper/shallower than isothermal on PV diagram? Why?

Answer

STEEPER!

Slope_adiabatic = −γP/V
Slope_isothermal = −P/V

Since γ > 1 → adiabatic slope is γ times steeper

Question

What is the work done in a cyclic process? When is it positive?

Answer

W_net = Area enclosed in PV diagram

Positive (CW) → Engine (converts heat to work)

Negative (CCW) → Refrigerator (work done on gas)

Question

What is COP of a refrigerator? How does it relate to Carnot efficiency?

Answer

COP = Q_cold / W = T_C / (T_H − T_C)

COP can be > 1!

Relation: COP = T_C / (T_H − T_C)
η = 1 − T_C/T_H
COP = (1−η)/η

Quick Reference

Formula Dump

🔑 Must-Know Formulas

First LawΔU = Q − W

Ideal GasPV = nRT

Internal EnergyU = f/2 · nRT

Mayer's RelationCp − Cv = R

γ (ratio)γ = (f+2)/f = Cp/Cv

Carnot Efficiencyη = 1 − T_C/T_H

COP RefrigeratorCOP = T_C/(T_H−T_C)

AdiabaticPVᵞ = const

Isothermal WorkW = nRT·ln(V₂/V₁)

Entropy ChangeΔS = Q_rev/T

📊 γ & DoF Reference

Gas	f	Cv	Cp	γ
Monoatomic	3	3R/2	5R/2	5/3
Diatomic	5	5R/2	7R/2	7/5
Polyatomic	6	3R	4R	4/3

Constants

R (Gas Constant)8.314 J/mol·K

k_B (Boltzmann)1.38 × 10⁻²³ J/K

N_A (Avogadro)6.022 × 10²³

Mnemonics

Memory Tricks

🧠 Mnemonic 1: Process Types

T-P-V-Q

IsoThermal (T=const) → IsoBaric (P=const) → IsoChoric (V=const) → AdiaBatic (Q=0). "Top Bankers Can't Buy"

🧠 Mnemonic 2: γ Values

Mono=5/3, Di=7/5, Poly=4/3

Think: 5→7→4 (odd-odd-even). Or: one atom needs 3 DoF, two atoms need 5, many atoms need 6 → γ = (f+2)/f.

🧠 Mnemonic 3: Sign of Work

EXPAND = +W, COMPRESS = -W

Gas expands → does positive work ON surroundings. Gas compressed → surroundings do positive work ON gas. First Law sign: W is work BY gas (positive when gas expands).

🧠 Mnemonic 4: Carnot Efficiency

η = 1 − (COLD/HOT)

Cold over Hot. Always less than 1. Higher COLD temp → lower efficiency. η → 100% only if COLD → 0 K (impossible by Third Law). Hot reservoir temperature is the NUMERATOR in T_C/T_H check: NO, T_C is numerator = COLD reservoir.

🧠 Mnemonic 5: Isochoric vs Isobaric Heat

Cv = "V stays, less heat" | Cp = "P stays, more heat"

At constant V: all heat → ΔU (efficient). At constant P: heat goes to ΔU + work (inefficient). So Cp > Cv. The EXTRA heat in Cp goes to doing work PΔV = nRΔT, hence Cp − Cv = R.

🧠 Mnemonic 6: PV Diagram Area

AREA = WORK

Work done by gas = area under PV curve. For CYCLE: net work = area enclosed. Clockwise = positive work (engine). Counterclockwise = negative work (refrigerator). Rectangle: W = ΔP × ΔV. Triangle: W = ½ × base × height.

Last Minute

20 Key One-Liners

1.For ideal gas, internal energy depends only on temperature — not on P or V separately.

2.Isothermal ≠ adiabatic. Isothermal = T constant (heat exchange occurs). Adiabatic = Q = 0 (no heat exchange).

3.Temperature in Kelvin always. T(K) = T(°C) + 273.

4.Carnot efficiency = 1 − T_C/T_H — the maximum possible efficiency between two temperatures.

5.Adiabatic curve is steeper than isothermal on PV diagram (slope ratio = γ).

6.In isochoric process: W = 0, Q = ΔU = nCvΔT.

7.In isobaric process: Q = nCpΔT, W = nRΔT, ΔU = nCvΔT.

8.For cyclic process: ΔU_net = 0, W_net = Q_net = Area enclosed in PV.

9.Cp − Cv = R (Mayer's Relation). Cp > Cv always.

10.γ = 5/3 (monoatomic), 7/5 (diatomic), 4/3 (polyatomic).

11.Work done in isothermal process: W = nRT ln(V₂/V₁).

12.Adiabatic: PVᵞ = const, TV^(γ−1) = const.

13.In free expansion: Q = 0, W = 0, ΔU = 0, but ΔS > 0 (irreversible).

14.Entropy is a state function. ΔS depends only on initial and final states.

15.ΔS_universe ≥ 0 always (= 0 for reversible, > 0 for irreversible).

16.COP of refrigerator = T_C/(T_H − T_C). COP can be > 1.

17.Internal energy: U = f/2 · nRT. Monoatomic: 3/2 nRT. Diatomic: 5/2 nRT.

18.PV diagram work = area under curve. Clockwise = engine (+W), Anticlockwise = refrigerator (−W).

19.Two Carnot engines in series with equal efficiency: T = √(T_H × T_C) (geometric mean).

20.Speed of sound in gas: v = √(γRT/M). Proportional to √T. Independent of pressure at constant T.

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