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Thermodynamics · one page

Formula sheet

Notation: Q (or ΔQ) · W (or ΔW) · U (ΔU for change) · C_v · C_p · γ = C_p/C_v · f · n (in PVⁿ = constant) · n (context makes it clear vs polytropic exponent) · R. Q > 0: heat into the gas; W > 0: work done by the gas; U depends only on T.

Learning path · Formula sheet · Practice set

Print and revise: Revision sheet (PDF) · Formula sheet (PDF, one page)

Part 2 · First Law of Thermodynamics

d Q = d U + d W

Statement and equation of the first law

Δ Q = Δ U + Δ W

Statement and equation of the first law

d Q = n C d T

Specific Heat Capacities of Gases

Q = n C (T_{2} - T_{1})

Specific Heat Capacities of Gases

d W = P d V

Heating of a Gaseous System and Work done by a Gas

W = \int_{V_{1}}^{V_{2}} P d V

Heating of a Gaseous System and Work done by a Gas

d U = \frac{f}{2} n R d T

Change in Internal Energy of a Gas on Heating

Δ U = \frac{f}{2} n R (T_{2} - T_{1})

Change in Internal Energy of a Gas on Heating

Part 3 · Indicator Diagrams, State Variables and Cycles

W = \int_{V_{1}}^{V_{2}} f (V) d V

Properties of an Indicator Diagram

Δ U_{12} = Q - W (any path)

State Variable and Path Variables

Δ U_{cycle} = 0

Cyclic Processes

Q_{net} = W_{net} = loop area

Cyclic Processes

Part 4 · Molar Specific Heats: Cv, Cp and γ

C_{v} = \frac{f}{2} R

Molar Heat Capacity at Constant Volume (C_v)

C_{p} = C_{v} + R = \frac{f + 2}{2} R

Molar Heat Capacity at Constant Pressure (C_p)

γ = \frac{C _{p}}{C _{v}} = \frac{f + 2}{f}

Ratio of Heat Capacities of a Gas

\frac{n _{1}}{γ _{1} - 1} + \frac{n _{2}}{γ _{2} - 1} = \frac{n _{1} + n _{2}}{γ _{e q} - 1}

Ratio of Specific Heats for a Mixture of Gases

Part 5 · Isochoric, Isobaric and Isothermal Processes

isochoric: W = 0

Isochoric or Isometric Process

Q = Δ U = n C_{v} Δ T = \frac{n R Δ T}{γ - 1}

Isochoric or Isometric Process

isobaric: W = n R (T_{2} - T_{1})

Isobaric Process

Q = n C_{p} Δ T

Isobaric Process

Δ U = n C_{v} Δ T (any process)

Isobaric Process

isothermal: P V = const

Isothermal Process

W = n R T ln \frac{V _{2}}{V _{1}} = n R T ln \frac{P _{1}}{P _{2}}

Isothermal Process

Q = W

Isothermal Process

Part 6 · Adiabatic Process

adiabatic: P V^{γ} = const

Adiabatic Process

T V^{γ - 1} = const

Adiabatic Process

W = \frac{P _{1} V _{1} - P _{2} V _{2}}{γ - 1} = \frac{n R ( T _{1} - T _{2} )}{γ - 1}

Adiabatic Process

Part 7 · Free Expansion of a Gas

Q = 0, W = 0, Δ U = 0

Free Expansion of a Gas

P_{1} V_{1} = P_{2} V_{2}

Free Expansion of a Gas

Part 8 · Polytropic Process

P V^{n} = const

Standard Processes as a Special Case of Polytropic Process

C = \frac{R}{γ - 1} + \frac{R}{1 - n}

Standard Processes as a Special Case of Polytropic Process

\frac{d P}{d V} = - n \frac{P}{V}

Indicator Diagram for a Polytropic Process

Part 9 · Reversible Processes, Heat Engines and Refrigerators

W = Q_{1} - Q_{2}

Heat Engines

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

Thermal Efficiency of a Heat Engine

COP = \frac{Q _{2}}{W} = \frac{Q _{2}}{Q _{1} - Q _{2}}

Refrigerators

Part 10 · Second Law and the Carnot Cycle

η_{Carnot} = 1 - \frac{T _{2}}{T _{1}}

Carnot Cycle