Chapter 11 – Thermodynamics

Ideal gas equation = \large\displaystyle \mathrm{PV}=\mathrm{nRT}

First law of thermodynamics: Heat supplied to a system equals the increase in internal energy plus the work done by the system.
Heat supplied (ΔQ) = \large\displaystyle \Delta\mathrm{U}+\Delta\mathrm{W}

Change in internal energy (ΔU) = \large\displaystyle \mathrm{nC}_{\mathrm{V}}\Delta\mathrm{T}

Work done by a gas (W) = \large\displaystyle \int_{\mathrm{V}_1}^{\mathrm{V}_2}\mathrm{P}\,\mathrm{dV}

Work done at constant pressure (W) = \large\displaystyle \mathrm{P}(\mathrm{V}_2-\mathrm{V}_1)

Heat supplied at constant pressure (ΔQ) = \large\displaystyle \mathrm{nC}_{\mathrm{P}}\Delta\mathrm{T}

For a constant-volume process, work done (W) = \large\displaystyle 0,\qquad \Delta\mathrm{Q}=\Delta\mathrm{U}

Work done in reversible isothermal expansion (W) = \large\displaystyle \mathrm{nRT}\ln\left(\frac{\mathrm{V}_2}{\mathrm{V}_1}\right)

For an ideal gas in an isothermal process = \large\displaystyle \Delta\mathrm{U}=0,\qquad \Delta\mathrm{Q}=\Delta\mathrm{W}

For a reversible adiabatic process = \large\displaystyle \mathrm{PV}^{\gamma}=\text{constant}

For a reversible adiabatic process = \large\displaystyle \mathrm{TV}^{\gamma-1}=\text{constant}

Work done in reversible adiabatic expansion (W) = \large\displaystyle \frac{\mathrm{P}_1\mathrm{V}_1-\mathrm{P}_2\mathrm{V}_2}{\gamma-1}

Mayer’s relation: Difference of molar heat capacities = \large\displaystyle \mathrm{C}_{\mathrm{P}}-\mathrm{C}_{\mathrm{V}}=\mathrm{R}

Heat-capacity ratio (γ) = \large\displaystyle \frac{\mathrm{C}_{\mathrm{P}}}{\mathrm{C}_{\mathrm{V}}}

Heat-engine efficiency (η) = \large\displaystyle \frac{\mathrm{W}}{\mathrm{Q}_{\mathrm{H}}}=1-\frac{\mathrm{Q}_{\mathrm{C}}}{\mathrm{Q}_{\mathrm{H}}}

Carnot engine efficiency (η) = \large\displaystyle 1-\frac{\mathrm{T}_{\mathrm{C}}}{\mathrm{T}_{\mathrm{H}}}

Refrigerator coefficient of performance (COP) = \large\displaystyle \frac{\mathrm{Q}_{\mathrm{C}}}{\mathrm{W}}

Carnot refrigerator coefficient of performance (COP) = \large\displaystyle \frac{\mathrm{T}_{\mathrm{C}}}{\mathrm{T}_{\mathrm{H}}-\mathrm{T}_{\mathrm{C}}}