Heat & Thermo formulas

Every formula JEE Advanced expects you to know for Heat & Thermo, with the variables spelled out. Free, no sign-up. Read it before a mock, not during one.

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Formulas: 19
Subject: Physics
Exam: JEE Advanced
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Heat & Thermo, every formula

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Linear, superficial, cubic expansion
$L = L_{0} (1 + α Δ T), A = A_{0} (1 + β Δ T), V = V_{0} (1 + γ Δ T)$
$\alpha,\beta,\gamma$ = coefficients of linear, area, volume expansion; $\beta = 2\alpha$, $\gamma = 3\alpha$
First Law of Thermodynamics
$Δ Q = Δ U + Δ W$
$\Delta Q$ = heat supplied, $\Delta U$ = change in internal energy, $\Delta W = \int P\,dV$ = work done by the gas
Thermal stress in a rod
$Stress = Y α Δ T, F = Y A α Δ T$
$Y$ = Young's modulus; arises when a rod's expansion/contraction is fully constrained (both ends fixed)
Work done in various processes
$W = \int_{V_{1}}^{V_{2}} P d V; W_{isobaric} = P Δ V; W_{isothermal} = n R T ln \frac{V _{2}}{V _{1}}; W_{adiabatic} = \frac{P _{1} V _{1} - P _{2} V _{2}}{γ - 1}$
Area under P-V curve; choose the correct expression based on the process
Heat, specific & molar heat capacity
$Q = m s Δ T = n C Δ T$
$s$ = specific heat capacity, $C$ = molar heat capacity, $n$ = number of moles ($C=Ms$, $M$ = molar mass)
Molar specific heats & Mayer's relation
$C_{P} - C_{V} = R, γ = \frac{C _{P}}{C _{V}}, C_{V} = \frac{f}{2} R, C_{P} = \frac{f + 2}{2} R$
$f$ = degrees of freedom (3 monoatomic, 5 diatomic); $\Delta U = nC_V\Delta T$ always (for ideal gas)
Latent heat (phase change)
$Q = m L$
$L$ = latent heat of fusion/vaporization; heat exchanged occurs at constant temperature during phase change
Adiabatic process relations
$P V^{γ} = constant, T V^{γ - 1} = constant, T^{γ} P^{1 - γ} = constant$
No heat exchange, $\Delta Q = 0$; slope of adiabatic curve is steeper than isothermal
Principle of calorimetry
$\sum Q_{lost} = \sum Q_{gained}$
Heat lost by hotter body equals heat gained by cooler body in a thermally isolated system (conservation of heat energy)
Efficiency of a heat engine
$η = \frac{W}{Q _{1}} = 1 - \frac{Q _{2}}{Q _{1}}$
$Q_1$ = heat absorbed from source, $Q_2$ = heat rejected to sink, $W$ = net work done by engine
Conduction: Fourier's law & thermal resistance
$\frac{d Q}{d t} = \frac{K A Δ T}{L}, R_{t h} = \frac{L}{K A}$
$K$ = thermal conductivity; series/parallel combination of rods uses $R_{th}$ analogous to electrical resistance
Carnot engine efficiency
$η_{Carnot} = 1 - \frac{T _{2}}{T _{1}}$
Maximum possible efficiency between temperatures $T_1$ (source) and $T_2$ (sink), in Kelvin
Stefan-Boltzmann law
$\frac{d Q}{d t} = e σ A T^{4}$
$e$ = emissivity (0 to 1), $\sigma = 5.67\times10^{-8}\ \text{W m}^{-2}\text{K}^{-4}$; for a body in surroundings at $T_0$: $\dfrac{dQ}{dt}=e\sigma A (T^4-T_0^4)$
Coefficient of performance (refrigerator)
$COP = \frac{Q _{2}}{W} = \frac{Q _{2}}{Q _{1} - Q _{2}} = \frac{T _{2}}{T _{1} - T _{2}}$
Ratio of heat extracted from cold reservoir to work input; last form for Carnot refrigerator
Wien's displacement law
$λ_{m} T = b$
$b$ $\approx 2.9\times10^{-3}\ \text{m K}$; wavelength of peak emission $\lambda_m$ is inversely proportional to absolute temperature
Entropy change (reversible process)
$Δ S = \int \frac{δ Q}{T}$
State function measuring disorder; $\Delta S_{\text{universe}} \geq 0$ for any process (2nd law)
Newton's law of cooling
$\frac{d T}{d t} = - k (T - T_{0}), T = T_{0} + (T_{i} - T_{0}) e^{- k t}$
Valid for small temperature differences with surroundings at $T_0$; used for approximate cooling-curve problems ($T_i$ = initial temperature)
Polytropic process & molar heat capacity
$P V^{n} = constant, C = C_{V} + \frac{R}{1 - n}$
General process index $n$; $n=0$ isobaric, $n=1$ isothermal, $n=\gamma$ adiabatic, $n=\infty$ isochoric
Ideal gas equation & Mayer's relation
$P V = n R T, C_{p} - C_{v} = R$
Links thermal expansion of gases with pressure-volume-temperature behaviour; $C_p, C_v$ = molar heat capacities at constant pressure/volume, foundational for later thermodynamics problems

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Heat & Thermo formulas

Heat & Thermo, every formula

Linear, superficial, cubic expansion

First Law of Thermodynamics

Thermal stress in a rod

Work done in various processes

Heat, specific & molar heat capacity

Molar specific heats & Mayer's relation

Latent heat (phase change)

Adiabatic process relations

Principle of calorimetry

Efficiency of a heat engine

Conduction: Fourier's law & thermal resistance

Carnot engine efficiency

Stefan-Boltzmann law

Coefficient of performance (refrigerator)

Wien's displacement law

Entropy change (reversible process)

Newton's law of cooling

Polytropic process & molar heat capacity

Ideal gas equation & Mayer's relation

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