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thermodynamics

Thermal transfers, blackbody radiation, greenhouse balance, kinetic theory, laws of thermodynamics, engines.

temperature & phase change · transfer mechanisms · greenhouse effect · gas laws · kinetic model · first law · engines & entropy

Temperature, internal energy & phase change

Temperature. \(T(\mathrm K)=\theta({}^\circ\mathrm C)+273.15\).

Internal energy. \(U=\sum K_{\mathrm{random}}+\sum U_{\mathrm{intermolecular}}\).

Thermal equilibrium. Net heat flow zero; heat flows hot → cold.

Heating & phase change. \(Q=mc\Delta T\), \(Q=mL\); \(P=Q/t\). Latent heat of fusion/vaporization; during a phase change \(T\) is constant.

Calorimetry. Mixture in an isolated system: \(\sum Q=0\).

Conduction, convection & radiation

Conduction. Rate \(\dfrac{Q}{t}=kA\dfrac{\Delta T}{L}\); thermal resistance \(R_{\mathrm{th}}=L/(kA)\), \(P=\Delta T/R_{\mathrm{th}}\).

Radiation. \(P=\epsilon\sigma AT^4\); net against surroundings at \(T_s\): \(P=\epsilon\sigma A(T^4-T_s^4)\).

Wien's law. \(\lambda_{\max}T=b\).

Intensity/flux. \(I=P/A\); inverse-square \(I=L/(4\pi r^2)\).

Emissivity. \(0\le\epsilon\le1\); blackbody \(\epsilon=1\); absorptivity \(\approx\epsilon\).

Greenhouse effect

Solar constant. At distance \(r\): \(S=L_\odot/(4\pi r^2)\).

Planetary balance. Intercepted \(P_{\mathrm{in}}=S\pi R^2(1-A)\), albedo \(A\); emitted \(P_{\mathrm{out}}=4\pi R^2\epsilon\sigma T^4\). Equilibrium temperature \[T=\left[\frac{S(1-A)}{4\epsilon\sigma}\right]^{1/4}.\]

Atmosphere. Shortwave mostly transmitted; longwave IR absorbed and re-emitted; greenhouse gases absorb bands in the terrestrial IR.

Radiative forcing. \(\Delta P/A\) shifts the equilibrium \(T\) until \(P_{\mathrm{in}}=P_{\mathrm{out}}\).

Ideal gas laws

Equation of state. \(n=N/N_A\); \(pV=nRT=NkT\); \(pV/T\) constant for fixed \(n\).

Named laws. Boyle: \(pV\) constant at fixed \(T\). Charles: \(V/T\) constant at fixed \(p\). Pressure law: \(p/T\) constant at fixed \(V\).

Densities. Number density \(N/V\); mass density \(\rho=Nm/V\).

Kinetic model

Pressure. \(pV=\tfrac13Nm\avg{c^2}\), \(p=\tfrac13\rho c_{\mathrm{rms}}^2\).

rms speed. \(c_{\mathrm{rms}}=\sqrt{\avg{c^2}}=\sqrt{3kT/m}=\sqrt{3RT/M}\).

Mean kinetic energy. Per molecule, translational \(\avg{E_k}=\tfrac32kT\); monatomic ideal gas internal energy \(U=\tfrac32NkT=\tfrac32nRT\).

First law & processes

First law. \(\Delta U=Q-W_{\mathrm{by}}=Q+W_{\mathrm{on}}\), with \(W_{\mathrm{by}}\gt0\) when the system does work. Work by gas \(W_{\mathrm{by}}=\int p\dd V\) = area under the \(p\)–\(V\) curve. Cyclic: \(\Delta U=0\), \(Q_{\mathrm{net}}=W_{\mathrm{net}}\).

Processes. Isochoric: \(W=0\). Isobaric: \(W=p\Delta V\). Isothermal (ideal gas): \(\Delta U=0\), \(Q=W=nRT\ln(V_f/V_i)\). Adiabatic: \(Q=0\), \(\Delta U=-W\); ideal gas \(pV^\Gamma\) constant, \(TV^{\Gamma-1}\) constant, \(\Gamma=C_p/C_v\).

Molar heat capacities. \(Q=nC\Delta T\); ideal gas \(C_p-C_v=R\); monatomic \(C_v=\tfrac32R\), \(C_p=\tfrac52R\).

Engines, refrigerators & entropy

Heat engine. \(W=Q_H-Q_C\), \(\eta=W/Q_H=1-Q_C/Q_H\); Carnot \(\eta_C=1-T_C/T_H\).

Refrigerator & heat pump. \(\mathrm{COP}_{\mathrm{ref}}=Q_C/W=Q_C/(Q_H-Q_C)\); \(\mathrm{COP}_{\mathrm{hp}}=Q_H/W\).

Second law. No engine has \(\eta=1\); spontaneous heat flow cold \(\nrightarrow\) hot.

Entropy. \(\Delta S=\int\dd Q_{\mathrm{rev}}/T\); isothermal \(\Delta S=Q_{\mathrm{rev}}/T\); \(\Delta S_{\mathrm{universe}}\ge0\).