The 1st law of thermodynamics

$\Delta E_{int}=Q+W$

Internal Energy $E_{int}$

$\Delta E_{int}=E_{int,f}-E_{int,i}$

  • ideal gas
    $E_{int}=\frac{i}{2}nRT$ only related to $T$

$\Delta E_{int}=\frac{i}{2}nR\Delta T$
where $i$ is degree of freedom.


  • deviration omitted.
  1. gas $\Rightarrow E=K=i\cdot\frac{1}{2}kT$

Heat $Q$

$Q>0$,the system absord heat.

  • thermal Conductivity $H$
    $H:=\frac{Q}{\Delta t}=kA \frac{\Delta T}{\Delta x}$

$k$ is a constant related to material(W·m^-1^ ·K^-1^); $A$ is contact area;
$\Delta T$ is the temperature difference between 2 end plane; $\Delta x$ is thickness.

  • heat capacity(热容)
    $C:=\frac{Q}{\Delta T}$

Work $W$

$W>0$,the system is positive worked towards.

$W=\int_{x_i}^{x_f} F_x dx=\int_{x_i}^{x_f}-pAdx=-\int_{V_i}^{V_f}pdV$

isothermal process ($T$ is invariant)

  • ideal gas
  1. equation:$pV=C$, where $C$ is a constant.

$E_{int}=\frac{i}{2}nRT$,$T$ is invariant, $\Delta E_{int}=0$
$\Rightarrow Q=-W=nRT\ln{\frac{V_f}{V_i}}$

isovolumetric process ($V$ is invariant)

$V$ is invariant $\Rightarrow dV=0\Rightarrow W=0$
In this process,define constant volume specific heat $C_V:=\frac{Q}{n\Delta T}$
$\Delta E_{int}=Q=nC_V\Delta T$
$\mathbb{NOTICE}$:$C_V$is measured in isovolumetric process,
but $\Delta E_{int}=nC_V\Delta T$ holds in all process.

  • ideal gas

isobaric process ($p$ is invariant)

In this process,define constant volume specific heat $C_p:=\frac{Q}{n\Delta T}$
$\Delta E_{int}=Q+W \Rightarrow nC_V\Delta T=nC_p\Delta T - p(V_f-V_i)$

  • ideal gas
    $W=-p(V_f-V_i)=-nR\Delta T$


adiabatic process ($Q=0$)

  • ideal gas
  1. equation:$pV^\gamma=C$, where $C$ is a constant,

$\gamma=\frac{i+2}{i}>1$ is a constant related to molecule(monoatomic,diatomic,triatomic),
$\Delta E_{int}=W=-\int_{V_i}^{V_f}pdV=-\int_{V_i}^{V_f}\frac{p_iV_i^\gamma}{V^\gamma}dV=\frac{p_fV_f-p_iV_i}{\gamma-1}$

free expansion

external force $F=0\Rightarrow W=0$, and $Q=0,\Delta E=0$
what changes?! Entropy $S$ !
to be continued.