Thermodynamics formula sheet

The formulas of a first thermodynamics course in one place, ready to print. Each row is a quantity and its formula, grouped by topic.

Ideal gas and properties

Quantity	Formula
Ideal gas	$p v = R T, p V = m R T$
Gas constant	$R = \frac{R _{u}}{M}, R_{u} = 8.314 \frac{kJ}{kmol\cdotK}$
Specific heats	$c_{p} - c_{v} = R, k = \frac{c _{p}}{c _{v}}$
Changes in u and h	$Δ u = c_{v} Δ T, Δ h = c_{p} Δ T$
Quality (saturated mixture)	$x = \frac{m _{g}}{m}, h = h_{f} + x h_{f g}$

First law

Quantity	Formula
First law, closed system	$Q - W = Δ U + Δ K E + Δ P E$
Boundary work	$W = \int p d V$
Isobaric process	$W = p (V_{2} - V_{1})$
Isothermal (ideal gas)	$W = m R T ln \frac{V _{2}}{V _{1}}$
Polytropic process	$p V^{n} = C, W = \frac{p _{2} V _{2} - p _{1} V _{1}}{1 - n}$
First law, steady flow	$\dot{Q} - \dot{W} = \overset{m}{˙} (h_{2} - h_{1} + \frac{V _{2}^{2} - V _{1}^{2}}{2} + g (z_{2} - z_{1}))$

Second law and efficiency

Quantity	Formula
Entropy	$d S = \frac{δ Q _{rev}}{T}$
Entropy change, ideal gas	$Δ s = c_{p} ln \frac{T _{2}}{T _{1}} - R ln \frac{p _{2}}{p _{1}}$
Entropy change, liquid or solid	$Δ s = c ln \frac{T _{2}}{T _{1}}$
Isentropic (ideal gas)	$\frac{T _{2}}{T _{1}} = (\frac{p _{2}}{p _{1}})^{\frac{k - 1}{k}} = (\frac{v _{1}}{v _{2}})^{k - 1}$
Heat engine efficiency	$η = \frac{W}{Q _{H}} = 1 - \frac{Q _{L}}{Q _{H}}$
Carnot efficiency	$η_{C} = 1 - \frac{T _{L}}{T _{H}}$
COP, refrigerator	$C O P_{R} = \frac{Q _{L}}{W}, C O P_{R, C} = \frac{T _{L}}{T _{H} - T _{L}}$
COP, heat pump	$C O P_{H P} = \frac{Q _{H}}{W} = C O P_{R} + 1$

Cycles

Quantity	Formula
Otto cycle	$η = 1 - \frac{1}{r ^{k - 1}}$
Diesel cycle	$η = 1 - \frac{1}{r ^{k - 1}} \cdot \frac{r _{c}^{k} - 1}{k ( r _{c} - 1 )}$
Brayton cycle	$η = 1 - \frac{1}{r _{p}^{(k - 1) / k}}$
Rankine cycle	$η = \frac{( h _{3} - h _{4} ) - ( h _{2} - h _{1} )}{h _{3} - h _{2}}$

Notes

Sign convention here: Q is positive into the system and W is positive when the system does work. Some textbooks use the opposite sign for work.

Every Carnot and entropy formula uses temperature in kelvin. In the Rankine cycle, state 1 leaves the condenser, 2 the pump, 3 the boiler and 4 the turbine.

Before an exam, check your course syllabus for whether a formula sheet is allowed and which.