Heat and Thermodynamics
Complete Formula Sheet & Shortcut Bible · BITSAT 2026
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Kinetic Theory & Gas Laws
Ideal Gas Equation
PV = nRT
PV = NkT
PV = NkT
n = moles, N = no. of molecules. R = 8.314 J/mol·K, k = 1.38x10⁻²³ J/K.
RMS Velocity
vᵣₘₛ = √(3RT/M)
= √(3kT/m)
= √(3kT/m)
M is molar mass (kg/mol), m is molecular mass (kg).
Average Kinetic Energy
KEₐᵥ₉ = (3/2)kT per molecule
KEₐᵥ₉ = (3/2)RT per mole
KEₐᵥ₉ = (3/2)RT per mole
Pressure & Density
P = (1/3)ρvᵣₘₛ²
Relates macroscopic pressure to microscopic motion.
Internal Energy (U)
U = (f/2)nRT
f = degrees of freedom. Mono: 3, Dia: 5, Poly: 6.
Degrees of Freedom (f)
γ = 1 + 2/f
Cᵥ = fR/2
Cₚ = (1+f/2)R
Cᵥ = fR/2
Cₚ = (1+f/2)R
γ is the adiabatic index or specific heat ratio.
Thermal Properties & Heat Transfer
Specific & Latent Heat
Q = mcΔT (Specific Heat)
Q = mL (Latent Heat)
Q = mL (Latent Heat)
c is specific heat capacity, L is latent heat of fusion/vaporization.
Thermal Conduction
dQ/dt = H = KA(T₁ - T₂)/L
K is thermal conductivity. For series, R_eq = R₁+R₂. For parallel, 1/R_eq = 1/R₁+1/R₂.
Stefan-Boltzmann Law
P = σεAT⁴
Radiated power. σ is Stefan's constant, ε is emissivity (1 for blackbody).
Newton's Law of Cooling
dT/dt = -k(T - Tₛ)
Rate of cooling is proportional to temperature difference with surroundings (Tₛ).
Laws of Thermodynamics
First Law (FLOT)
ΔQ = ΔU + ΔW
Heat supplied = Change in Internal Energy + Work Done by system.
Work Done by Gas
W = ∫ P dV
Work is the area under the P-V diagram.
Mayer's Formula
Cₚ - Cᵥ = R
Relates molar specific heats for an ideal gas.
Second Law Statements
Kelvin-Planck & Clausius
No engine can be 100% efficient. Heat flows from hot to cold spontaneously.
BITSAT Speed Hacks
Adiabatic vs Isothermal Slope
(dP/dV)ₐₔᵢₐ = γ (dP/dV)ᵢₛₒ
The adiabatic curve is always steeper than the isothermal curve.
Efficiency Shortcut
η = 1 - T₂/T₁ = (T₁ - T₂)/T₁
For small changes, if T₁ is increased by ΔT, η increases more than if T₂ is decreased by ΔT.
Exam Day Strategy
Always convert temperature to Kelvin (K) for gas laws, efficiency, and radiation. Celsius is a common trap.
Remember f (degrees of freedom): Monoatomic (He, Ar) = 3, Diatomic (O₂, N₂) = 5. This is key for U, Cᵥ, and γ.
For P-V graphs, work done is area under the curve. Clockwise cycle = +W (engine), Anti-clockwise = -W (refrigerator).
Newton's Cooling questions often compare time taken to cool through two equal temperature intervals. The first interval is always faster.
Efficiency (η) is always < 1. Coefficient of Performance (COP) can be > 1. Don't mix them up.
In conduction, thermal resistance is R = L/KA. Treat it like electrical resistance for series/parallel combinations.
Carnot Engine & Refrigerator
Carnot Engine Efficiency (η)
η = W/Q₁ = 1 - Q₂/Q₁
η = 1 - T₂/T₁
η = 1 - T₂/T₁
T₁ = Source Temp (K), T₂ = Sink Temp (K). This is the maximum possible efficiency.
Refrigerator COP (β)
COP = Q₂/W = Q₂/(Q₁-Q₂)
COP = T₂/(T₁-T₂)
COP = T₂/(T₁-T₂)
Coefficient of Performance. For a heat pump, COP = Q₁/W = T₁/(T₁-T₂).
Thermodynamic Processes at a Glance
| Process | Condition | Work Done (W) | Internal Energy (ΔU) | First Law (ΔQ) |
|---|---|---|---|---|
Isochoric | V = const | 0 | nCᵥΔT | ΔQ = ΔU |
Isobaric | P = const | PΔV | nCᵥΔT | ΔQ = nCₚΔT |
Isothermal | T = const | nRT ln(V₂/V₁) | 0 | ΔQ = W |
Adiabatic | ΔQ = 0 | (P₁V₁-P₂V₂)/(γ-1) | nCᵥΔT | 0 = ΔU + W |