Formula sheet: Thermodynamics

Thermodynamics14 min readLesson 1 of 5

1.1 The first law and internal energy Heat and work are the two ways in which energy crosses the boundary of a thermodynamic system. The energy stored inside the system by the motion and interaction of its molecules is its internal energy, written U. The first law of thermodynamics says that the heat added to a system appears either as a rise of internal energy or as work done by the system, so for a small step dQ = dU + dW. For a gas held at constant pressure, the work it does is P times the increase in volume.

Δ Q = Δ U + Δ W, Δ W = P Δ V (isobaric), Δ U = n C_{V} Δ T

First law of thermodynamics, work at constant pressure and the internal-energy change of an ideal gas

1.2 Specific heats and the gas laws For gases the heat needed to raise the temperature depends on the process. At constant volume all the heat goes into internal energy, so C_V = dQ/dT at constant volume. At constant pressure the gas also does work, so C_P is larger. The difference is R for one mole, and the ratio gamma = C_P/C_V enters the adiabatic relations.

C_{P} - C_{V} = R, γ = \frac{C _{P}}{C _{V}}, C_{V} = \frac{R}{γ - 1}, C_{P} = \frac{γ R}{γ - 1}

Molar specific heats of an ideal gas and their relation through R and gamma

isothermal: P V = constant, adiabatic: P V^{γ} = constant, isobaric: P = constant, isochoric: V = constant

The four standard quasi-static processes for an ideal gas

1.3 Work in various processes The work done by a gas is the area under its P-V curve. For an isothermal process the curve is a hyperbola and the work is nRT ln(V2/V1). For an adiabatic process there is no heat exchange, and the work done by the gas is (P1V1 - P2V2)/(gamma - 1). In the adiabatic relations, temperature follows TV^(gamma-1) = constant and pressure follows T^gamma P^(1-gamma) = constant.

W_{i so} = n R T ln \frac{V _{2}}{V _{1}}, W_{a d i} = \frac{P _{1} V _{1} - P _{2} V _{2}}{γ - 1}, T V^{γ - 1} = constant

Work done in isothermal and adiabatic processes, and the adiabatic temperature relation

1.4 Second law and heat engines The second law limits how much heat can be converted into work. A heat engine takes heat Q1 from a hot reservoir, converts part into work W and rejects Q2 to a cold reservoir, with efficiency eta = W/Q1 = 1 - Q2/Q1. No engine operating between two fixed temperatures can beat the reversible Carnot engine, whose efficiency depends only on the two absolute temperatures. A refrigerator is the reverse device, and its coefficient of performance is the heat removed divided by the work input.

η = 1 - \frac{Q _{2}}{Q _{1}} = 1 - \frac{T _{2}}{T _{1}}, β = \frac{Q _{2}}{W} = \frac{T _{2}}{T _{1} - T _{2}}

Carnot efficiency and the coefficient of performance of a refrigerator

Exam

Use absolute temperatures in every efficiency and coefficient-of-performance formula. A Celsius value in the ratio T2/T1 gives a wrong answer that looks plausible.

in hinglish

Thermodynamics me pehla concept first law hai: dQ = dU + dW . Heat do tarah se use hoti hai, internal energy badhane me aur work karne me. Gas ke do specific heats hote hain: Cv constant volume par aur Cp constant pressure par, aur difference R hota hai. Isothermal process me PV constant, adiabatic me PV^gamma constant, aur work P-V curve ke area se aata hai. Second law kehta hai ki heat ko poora work me convert nahi kar sakte. Carnot efficiency 1 - T2/T1 hai, aur refrigerator ka COP T2/(T1 - T2) . Effeciency ke sawaalon me temperature kelvin me daalo, warna answer galat aayega. Aur yaad rakho: first law ke sawaalon me sign convention fix karo, warna heat aur work ka addition hi galat ho jayega. Unit conversion ka step kabhi skip mat karo, especially litre ko metre cube me badalna. Formula sheet ko revise karte samay yahi check karo ki har symbol ki unit kya hai.