《大学物理》卢德鑫 NJU: University Physics I - 卢德鑫

Temperature

Basic Concept

phase: homogeneous part of system
equilibrium state: a state of a system in which macroscopic variables have definite values that remain constant as long as the external conditions are unchanged
relaxation time: $\tau$, the time system needs to adjust itself to follow the change of the surrounding.

state avariables
- mechanical: $p$
- electromagenetic: $\vec E, \vec P, \vec B, \vec M$
- geometric: $V, L$
- chemical: $n, x_i$
extensive: F(n)=nF(1)
intensive: F(n)=F(1)
quasistatic process: time the process takes is much longer than relaxation time
isobaric, isochoric, isothermal, isentropic(or adiabatic)
The zeroth law of thermodynamics:
- Two systems, each in thermal equilibrium wirh a third (thermometer) are in thermal equilibrium with each other
the triple point: 273.16K

Empirical temperature scales: $T(X) = 273.16K*\frac{R}{R_{tr}}$ (assuemd $T(X)=aX$) X 温标
the ideal gas scale: $T=273.16K*lim_{p_{tr}\rightarrow 0}(\frac{p}{p_{tr}})$
Celsius scale: $T_C=T_{ideal\ gas}-273.15$
Fahrenheit scale: $T_F=32^\circ+\frac{9}{5}T_C$
thermodynamic scale (or Kelvin scale): In the range of temperature in which a gas thermometer may be used it is identical to ideal gas scale
International Practical Scale IPTS-68, NMP(normal melting point)of tungsten is 3695K
$510^{-8}K\sim 510^8K$

equation of state of solid an d liquid
- Coefficient of expansion
  - linear: $\alpha_l=\frac{1}{L}(\frac{\partial L}{\partial T})_p$
  - thermal(for isotropic material): $\alpha_V=\frac{1}{V}(\frac{\partial V}{\partial T})_p = 3\alpha_l$
  - $\frac{\Delta V}{V}\approx\alpha_V\Delta T\approx 10^{-5}\sim10^{-3}$
- Isothermal compressibility: $\kappa_T=-\frac{1}{V}(\frac{\partial V}{\partial p})_T$
  - $|\frac{\Delta V}{V}|\approx\kappa_T\Delta p\approx 10^{-5}$
- state of equation: $V=V_0(1+\alpha_V\Delta T-\kappa_T\Delta p)$
equation of state for ideal gas: $PV=nRT$
- gas constant: $R=8.31 J*K^{-1}*mol^{-1}$
- Boyle-Mariotte Law: $pV=C(T)$
- gas constant scale: $\frac{V}{T}=D(p)$
- Avogadro’s Law: under same temperature and pressure, gases of equal volume contain same number of molecules
equation of state for real gas:
- Virial expansion: $pV=nRT(1+B(T)p+C(T)p^2+\cdots)$, where $B(T),C(T)$ are called second, third Virial coefficient.
- Van der Waal’s equation: $p+a(\frac{n}{V})^2=nRT$

work done by external force: $dW=-pdV$
Generalization: $dW=YdX$
- generalized force Y: $-p, J, \sigma, \vec E,\vec H,\varepsilon, \mu$
- generalized displacement X: $V, L, A, \vec P,\vec M,Q,n$
Work is path-dependent: $W=-\int_{V_i}^{V_f}pdV$
difference of the internal energy function U between two states A,B interms of adiabatic work done: $\Delta U=U_B-U_A=W_{BA}$
The First Law of THermodynamics
- $\Delta U = W+Q$
- $dU=dQ+dW$
- The perpetual motion machines of the first kind are impossible

heat capacity: $C=lim_{\Delta T\rightarrow0}\frac{\Delta Q}{\Delta T}$
specific heat capacity: $C=\frac{C}{m}$
$C<0$: gravitational system
$C=\infty$: heat reservoir
$C=0$: adiabatic
$C_V=lim_{\Delta T\rightarrow0}(\frac{\Delta Q}{\Delta T})_V = (\frac{\partial U}{\partial T})_V$
$C_p=lim_{\Delta T\rightarrow0}(\frac{\Delta Q}{\Delta T})_p = (\frac{\Delta U+p\Delta V}{\Delta T})_p = (\frac{\partial H}{\partial T})_p$
ratio of specific heat: $\gamma=\frac{C_p}{C_V}$
Heat capacity of ideal gases:

the second law of thermodynamics
- Kelvin-Planck statement: No process is possible whose sole result is the absorption of heat from a reservoir and the conversion of this heat into work
- Clausius statement: No process is possible whose sole result is the transfer of heat from a cooler to a hotter body
- Perpetual motion machines of the second kind are impossible
Carnot Theorem: $\eta_A(T_1,T_2)=1+\frac{Q_2}{Q_1}\leq\eta_R(T_1,T_2)=1-\frac{T_2}{T_1}$, R reversible
Clausius inequality: $\oint\frac{dQ}{T}\leq0$
Entropy: $\frac{dQ}{T}$, $S_A-S_B=\int_A^B(\frac{dQ}{T})_R$
entropy principle: $dS\geq\frac{dQ}{T}$
- Known the expression for entropy: $\Delta S=S_f-S_i$
- Entropy change for a reservoir: $\Delta S=\frac{\Delta Q}{T}$
- Entropy change for two state connected by a quasi-static process: $\Delta S=\int^f_idS$
- Entropy change for states connected by an irreversible process: $\Delta S=\int dS$ in a resersible process
Boltzmann Relation: $S=k_BlnW$
Thermodynamic potentials:
- $dU=TdS-pdV+\mu dn$
- Enthalpy: $H=pV+U$, $dH=TdS+Vdp+\mu dn$
- Helmholtz free energy: $F=U-TS$, $dF=-SdT-pdV+\mu dn$
- Gibbs free energy: $G=\mu n$, $dG=-SdT+Vdp+\mu dn$
- Gibbs-Duheim equation: $d\mu = -S_mdT+V_mdp$
- Maxwell equation
  - $-(\frac{\partial S}{\partial p}){T,n} = (\frac{\partial V}{\partial T}){p,n}$
  - and more\

macroscopic description
- $pV=nRT$
- $C_V=\frac{3}{2}nR$ for monatomic gas
pressure: $p=\frac{1}{3}\rho\overline{v^2}$
mean square speed: $v_{rms} = \sqrt{\overline{v^2}}=\sqrt\frac{3p}{\rho}$
$\frac{1}{2}m\overline{v^2}=\frac{3}{2}\frac{nR}{N}T$
Boltzman constant: $k_B=\frac{nR}{N}=1.3805*10^{-23}J/K$
Maxwell velocity distribution: $dn(v_x,v_y,v_z)=n_0(\frac{m}{2\pi}k_BT)^{\frac{3}{2}}exp(-\frac{1}{2}m(v_x^2+v_y^2+v_z^2)/k_BT)dv_xdv_ydv_z$
the most probable speed: $v_m=\sqrt\frac{2k_BT}{m}$
Boltzmann distribution: $\frac{e^{-\beta E(p,q)dpdq}}{\int\int e^{-\beta E(p,q)dpdq}}$ where p and q are generalized momentum and displacement
Transport Phenomena
$Q=JA$
- Heat conduction
  - Fourier’s Law: $J_Q=-\kappa\frac{dT}{dx}$
  - thermal conductivity: $\kappa$
- convection
  - $Q=hA\Delta T$
- thermal radiation
  - Stefan-Boltzman Law: $J_B=\sigma T^4$
  - Stefan Boltzman constant: $\sigma=5.6703*10^{-8} W/(m^2K^4)$
  - Kirchhoff’s Law: $J_\alpha=\alpha J_B$
- diffusion
  - Fick’s Law: $J_m=-D\frac{\partial\rho}{\partial x}$
  - Diffusion coefficient: D
- viscosity
  - Newton’s Law: $J_p = -\eta\frac{\partial v_y}{\partial x}$
continuity equation

Van der Waals equation
- b: finite size
- a: force
phase: solid, liquid, vapor, fluid
Clapeyron’s equation: $(\frac{dp}{dT})_{co}=\frac{\Delta H_m}{T\Delta V_m}=\frac{l}{T\Delta V_m}$
- latent heat: l
- Trouton’s rule: $\frac{l_v}{RT_b}=9
a phase transition is nth order if nth order derivative of chemnical potential is the first discontinuous one