Equipartition and Adiabatic Processes

• Equipartition Theorem

  There is $\frac{1}{2}kT$ of energy per degree of freedom of a molecule in a system. A monoatomic gas has three degrees of freedom (three dimensions of kinetic energy). A diatomic gas typically has five (three translational degrees, two rotational degrees, skip vibration and the third rotation). A solid typically has six (three translational degrees, three vibrational degrees). The internal energy of $N$ molecules of a monoatomic gas are thus:
  

  $$E_{\rm tot} = \frac{3}{2} NkT$$ (231)

  
  
  and so forth.

• Molecular Model of Ideal Gas leads to $PV = NkT$ using the equipartition theorem.

• Molar Specific Heats

  At constant volume, no work is done and all heat that goes into a system increases its internal energy. At constant pressure, heat going into a system can both do work and increase internal energy and typically does both. We define:
  

  $$\Delta Q = n C_v \Delta T {\rm (constant volume)}$$ (232)

  $$\Delta Q = n C_p \Delta T {\rm (constant pressure)}$$ (233)

  
  
  where $C_v$ is the molar specific heat at constant volume and $C_p$ is the molar specific heat at constant pressure (and $n$ is the number of moles of gas). These specific heats are also often expressed per molecule with an obvious conversion based on $N_A$.

• $\bf C_v$
  

  $$C_v = \frac{1}{n} \frac{dE_{\rm tot}}{dT} = \frac{d_f}{2}R$$ (234)

  
  
  where $d_f$ is the number of degrees of freedom of a molecule in the system (e.g. 3 for monoatomic gas, 5 for diatomic gas, 6 for solid).

• $\bf C_p$ From the first law:
  

  $$dQ = dE + dW = \frac{d_f}{2} nRdT + PdV = \frac{d_f}{2} nRdT + nRdT$$ (235)

  
  
  (where the last follows because with $P$ constant, $dV$ must be proportional to $dT$) so
  

  $$C_p = \frac{1}{n} \frac{dQ}{dT} = \frac{d_f}{2}R + R = C_v + R$$ (236)

  
  

• $\bf\gamma$ We define:
  

  $$\gamma = \frac{C_p}{C_v}$$ (237)

  
  
  Thus $\gamma =$ 5/3 for monoatomic gases, 7/5 for diatomic gases.

• Adiabatic Processes are characterized by $\Delta Q = 0$. From the first law, this means that:
  

  $$dE = -dW$$ (238)

  
  
  or
  

  $$n C_v dT = nRdT = -PdV$$ (239)

  
  

  If we take the full derivative of the ideal gas law:
  

  $$d(PV) = PdV + VdP = nRdT$$ (240)

  
  
  and substitute in the first law and rearrange, we get:
  

  $$PdV + VdP = - \frac{R}{C_v} PdV$$ (241)

  
  

  Collecting terms, rearranging, and dividing by $PV$ we get:
  

  $$(1 +\frac{R}{C_v}) PdV + VdP = 0$$ (242)

  $$(\frac{C_v + R}{C_v}) PdV + VdP = 0$$ (243)

  $$(\frac{C_p}{C_v}) PdV + VdP = 0$$ (244)

  $$\gamma PdV + VdP = 0$$ (245)

  $$\gamma \frac{dV}{V} + \frac{dP}{P} = 0$$ (246)

  
  
  and integrating it leads us to
  

  $$\ln P + \gamma \ln V = {\rm constant}$$ (247)

  
  
  or
  

  $$PV^\gamma = {\rm constant}$$ (248)

  
  

  This is the equation for the $PV$ curve of an adiabatic process. There are lots of ways to manipulate this algebraically by e.g. combining it with the ideal gas law to eliminate $P$ or $V$ in favor of $T$ and the remaining one.

  We will need the $PV^\gamma = {\rm constant}$ result in order to solve problems involving adiabatic processes in cyclic heat engines, so learn it well.