First Law & Work
Most-tested category in all examsFirst Law
Change in Internal Energy
ΔU = Q − W
ΔU = change in internal energy, Q = heat absorbed, W = work done by system
Work Done
Isobaric Work
W = PΔV = P(V₂ − V₁)
Work done by gas in isobaric (constant pressure) process
Work Done
Isothermal Work
W = nRT ln(V₂/V₁)
Work done in isothermal process. Since PV = nRT and T is constant.
Work Done
Isochoric Work
W = 0 (V = constant)
No work is done at constant volume. All heat goes to changing internal energy.
Work Done
Adiabatic Work
W = −ΔU = nCv(T₁ − T₂)
No heat exchange. Work comes entirely from internal energy change.
Work Done
Cyclic Process Work
W_net = Area of PV loop
Net work = area enclosed by cycle. Positive (clockwise). ΔU = 0 for full cycle.
Ideal Gas & Kinetic Theory
Foundation for all process calculationsIdeal Gas Law
Ideal Gas Equation
PV = nRT
P = pressure (Pa), V = volume (m³), n = moles, R = 8.314 J/mol·K, T = temperature (K)
Ideal Gas Law
Gas Law (Boltzmann)
PV = Nk_BT
N = number of molecules, k_B = 1.38 × 10⁻²³ J/K (Boltzmann constant)
Internal Energy
Internal Energy (Ideal Gas)
U = f/2 · nRT
f = degrees of freedom. Monoatomic: f=3, Diatomic: f=5, Polyatomic: f=6
Heat Capacity
Mayer's Relation
Cp − Cv = R
Difference between molar heat capacities at constant pressure and volume equals R (gas constant)
Heat Capacity
Ratio of Heat Capacities
γ = Cp/Cv = (f+2)/f
γ = 5/3 (monoatomic), 7/5 (diatomic), 4/3 (polyatomic)
Heat Transfer
Heat at Constant Volume
Q = nCvΔT (isochoric)
At constant volume, W = 0, so all heat goes to internal energy change
Thermodynamic Processes
Process-specific equations tested heavily in JEEAdiabatic
Adiabatic Process Laws
PVᵞ = const
Also: TV^(γ-1) = const and T^γP^(1-γ) = const. No heat exchange (Q = 0)
Adiabatic
Adiabatic T-V Relation
TV^(γ−1) = constant
Temperature-volume relation in adiabatic process. Derived from PVᵞ = const and PV = nRT
Adiabatic
Adiabatic T-P Relation
T^γ P^(1−γ) = constant
Temperature-pressure relation for adiabatic process
Isothermal
Boyle's Law
PV = constant (T = const)
P₁V₁ = P₂V₂. ΔU = 0 for ideal gas. Work done = heat absorbed.
Slope
PV Diagram Slopes
|Slope_adi| = γ × |Slope_iso|
Adiabatic curve is steeper than isothermal. Critical graph-based question fact.
Polytropic
Polytropic Process
PVⁿ = constant
General process: n=0 isobaric, n=1 isothermal, n=γ adiabatic, n=∞ isochoric
Carnot Engine & Efficiency
Always in NEET and JEE MainEfficiency
Carnot Efficiency
η = 1 − T_C/T_H
T_C and T_H must be in Kelvin. Maximum possible efficiency for given temperatures.
Efficiency
Heat Engine Efficiency
η = W/Q₁ = 1 − Q₂/Q₁
W = Q₁ − Q₂ (net work done), Q₁ = heat absorbed from hot source
Refrigerator
COP of Refrigerator
COP = Q₂/W = T_C/(T_H−T_C)
Coefficient of Performance. Higher COP = more efficient refrigerator.
Entropy
Entropy Change
ΔS = Q_rev / T
For reversible process. Entropy is a state function. ΔS = 0 for reversible adiabatic.
Efficiency
Efficiency Improvement
Δη ≈ ΔT/T_H
Increasing T_H by ΔT increases efficiency more than decreasing T_C by same ΔT if T_H > T_C
Heat Engine
Work-Heat Relation
Q₁/T_H = Q₂/T_C (Carnot)
For Carnot cycle (reversible engine). Heat ratios equal temperature ratios.
Dimensional Analysis & SI Units
CBSE derivation marks + JEE verification| Quantity | SI Unit | Dimensions | Notes |
|---|---|---|---|
| Internal Energy (U) | J (Joule) | [ML²T⁻²] | Same as work and heat |
| Heat (Q) | J (Joule) | [ML²T⁻²] | Not a state function |
| Work (W) | J (Joule) | [ML²T⁻²] | W = ∫PdV |
| Entropy (S) | J/K | [ML²T⁻²Θ⁻¹] | State function |
| Temperature (T) | K (Kelvin) | [Θ] | Always in Kelvin for formulas |
| Pressure (P) | Pa = N/m² | [ML⁻¹T⁻²] | 1 atm = 101325 Pa |
| Specific Heat (c) | J/kg·K | [L²T⁻²Θ⁻¹] | Per unit mass |
| Molar Heat Cap. (C) | J/mol·K | [ML²T⁻²Θ⁻¹mol⁻¹] | Per mole |
| Gas Constant (R) | J/mol·K | [ML²T⁻²Θ⁻¹mol⁻¹] | R = 8.314 J/mol·K |
| Boltzmann Const. (k_B) | J/K | [ML²T⁻²Θ⁻¹] | k_B = 1.38×10⁻²³ J/K |
| Efficiency (η) | Dimensionless | [1] | 0 ≤ η < 1 |
| COP | Dimensionless | [1] | Can be > 1 |
Exam Insight — Dimensional Analysis
In JEE, dimensional analysis is often used to verify formula correctness in multi-step problems. Key check: R = Cp − Cv. Both have units J/mol·K = [ML²T⁻²Θ⁻¹mol⁻¹]. The ratio γ = Cp/Cv is dimensionless — it cancels all units. If a formula gives you γ with units, something is wrong.
Key Constants & Values
Memorize these — they appear repeatedlyGas Constant
R = 8.314 J/mol·K
≈ 2 cal/mol·K
Boltzmann Constant
k_B = 1.38×10⁻²³ J/K
R = N_A × k_B
Avogadro Number
N_A = 6.022×10²³
molecules per mole
1 Calorie
1 cal = 4.186 J
Mechanical equivalent of heat