Class 11 · Chapter 12

Thermodynamics

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Thermodynamics Formula Sheet

12 formulas across 3 topics in Thermodynamics.

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Updated 2026-07-04 · v1.0.1

First Law & Work Done by a Gas

Energy bookkeeping: heat in = internal energy change + work out. Signs are everything in this chapter.

Q = ΔU + W

First law of thermodynamics

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Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
Q	heat SUPPLIED to the gas (positive in)	J	[ML²T⁻²]
ΔU	change in internal energy	J	[ML²T⁻²]
W	work done BY the gas (positive out)	J	[ML²T⁻²]

Valid when

Sign convention above is the NCERT/exam standard — chemistry uses W done ON the gas

Common mistakes

ΔU depends ONLY on temperature change for an ideal gas — same ΔT means same ΔU regardless of the process

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W = ∫ P dV (area under the P–V curve)

Work done by a gas

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Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
P	gas pressure	N/m²	[ML⁻¹T⁻²]
dV	infinitesimal volume change	m³	[L³]

Valid when

Expansion → W positive; compression → W negative
For a CYCLE, net work = area enclosed by the loop (positive if clockwise on a P–V diagram)

Common mistakes

Work is PATH-dependent — two processes between the same states generally do different work
Anticlockwise P–V loops have NEGATIVE net work — the cycle is a refrigerator, not an engine

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ΔU = n Cᵥ ΔT (any process, ideal gas)

Internal energy change of an ideal gas

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Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
n	number of moles	mol	[N]
Cᵥ	molar heat capacity at constant volume	J/(mol·K)	[ML²T⁻²Θ⁻¹N⁻¹]

Valid when

Holds for EVERY process of an ideal gas, not only isochoric — U is a state function of T alone

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Cₚ − Cᵥ = R; γ = Cₚ/Cᵥ

Mayer's relation & adiabatic exponent

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First law applied to an isobaric process of an ideal gas

Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
Cₚ, Cᵥ	molar heat capacities at constant pressure and volumemonatomic: Cᵥ = 3R/2, γ = 5/3; diatomic: Cᵥ = 5R/2, γ = 7/5	J/(mol·K)	[ML²T⁻²Θ⁻¹N⁻¹]
R	universal gas constantR = 8.314 J/(mol·K)	J/(mol·K)	[ML²T⁻²Θ⁻¹N⁻¹]

Valid when

Cₚ > Cᵥ always: at constant pressure some heat leaves as expansion work

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Worth remembering

Free expansion into vacuum: W = 0, Q = 0, so ΔU = 0 and temperature is unchanged (ideal gas) — no formula, pure first law

Thermodynamic Processes

Each named process kills one variable: isothermal fixes T, isochoric fixes V, isobaric fixes P, adiabatic blocks heat. Know W and Q for each.

W = nRT ln(V₂/V₁) (isothermal)

Work in an isothermal process

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W = ∫P dV with P = nRT/V

Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
V₁, V₂	initial and final volumes	m³	[L³]
T	constant temperature	K	[Θ]

Valid when

ΔU = 0, so Q = W — all heat becomes work
Slow (quasi-static) process in a conducting vessel

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PV^γ = constant; TV^(γ−1) = constant

Adiabatic process relations

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First law with Q = 0 combined with the ideal gas law

Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
γ	adiabatic exponent Cₚ/Cᵥ	dimensionless	[M⁰L⁰T⁰]

Valid when

No heat exchange (fast process or insulated walls)
Adiabatic curve is STEEPER than the isothermal through the same point (slope ratio = γ)

Common mistakes

Adiabatic expansion COOLS the gas — work is paid from internal energy

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

Work in an adiabatic process

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W = −ΔU = −nCᵥΔT with Cᵥ = R/(γ−1)

Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
T₁, T₂	initial and final temperatures	K	[Θ]

Valid when

Positive for expansion (T₂ < T₁)

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C = Cᵥ + R/(1 − x) for PV^x = constant

Molar heat capacity of a polytropic process

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Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
x	polytropic indexx = 0 isobaric, 1 isothermal, γ adiabatic, ∞ isochoric	dimensionless	[M⁰L⁰T⁰]
C	molar heat capacity of the process	J/(mol·K)	[ML²T⁻²Θ⁻¹N⁻¹]

Valid when

C can be NEGATIVE for 1 < x < γ — the gas cools while absorbing heat

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Worth remembering

Process-spotting on a P–V diagram: hyperbola = isothermal, steeper falling curve = adiabatic, vertical = isochoric, horizontal = isobaric

NEET asks W/Q/ΔU signs per process; JEE Advanced chains processes into cycles and asks for net quantities — tabulate each leg first.

Heat Engines, Refrigerators & Second Law

No engine beats Carnot between the same two reservoirs. Efficiency and COP are just ratios of the same three energy flows.

η = W/Q₁ = 1 − Q₂/Q₁

Efficiency of a heat engine

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Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
Q₁	heat absorbed from the hot reservoir	J	[ML²T⁻²]
Q₂	heat rejected to the cold reservoir	J	[ML²T⁻²]
W	net work output (= Q₁ − Q₂)	J	[ML²T⁻²]

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η(cycle) = W(net)/Q(in) = (area enclosed on P–V) / (heat absorbed in the positive-Q legs)

Efficiency of a general P–V cycle

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Over a full cycle ΔU = 0, so W(net) = Q(net) = Q(in) − Q(out)

Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
W(net)	net work per cycle (enclosed area, clockwise positive)	J	[ML²T⁻²]
Q(in)	heat ABSORBED (sum over legs where Q > 0 only)	J	[ML²T⁻²]

Valid when

Compute Q leg-by-leg first (nCᵥΔT, nCₚΔT, or nRT ln for isothermal), THEN sum only the positive ones for Q(in)

Common mistakes

Dividing by total heat exchanged |Q(in)| + |Q(out)| instead of Q(in) alone — the definition uses only absorbed heat

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η(Carnot) = 1 − T₂/T₁

Carnot efficiency

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For the reversible Carnot cycle, Q₂/Q₁ = T₂/T₁

Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
T₁, T₂	hot and cold reservoir temperatures	K	[Θ]

Valid when

Temperatures in KELVIN, always
This is the maximum possible efficiency between T₁ and T₂ — no real engine reaches it
η = 1 only if T₂ = 0 K, which is unattainable

Common mistakes

Plugging Celsius values is the single most common thermodynamics error in NEET

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COP = Q₂/W = T₂/(T₁ − T₂) (Carnot refrigerator)

Coefficient of performance of a refrigerator

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Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
COP	coefficient of performancecan exceed 1 — and usually does	dimensionless	[M⁰L⁰T⁰]
Q₂	heat extracted from the cold space	J	[ML²T⁻²]

Valid when

Relation to the same-reservoir engine: COP = (1 − η)/η

Common mistakes

A refrigerator does not 'create cold' — it pumps Q₂ + W into the room; running one with the door open WARMS the room

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Worth remembering

Second law in one line: heat cannot flow spontaneously from cold to hot, and no cyclic engine can convert heat entirely into work

NEET recycles Carnot-with-changed-reservoirs numericals; JEE Main likes 'efficiency becomes double when T₂ is lowered by x' algebra.

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