What is the value of #gamma# for a polyatomic gas?

1 Answer
Jan 10, 2018

For linear polyatomic gases (such as #"CO"_2# or #"N"_2"O"#):

#gamma ~~ 1.40#

For nonlinear polyatomic gases (such as #"SO"_2# or #"NH"_3#):

#gamma ~~ 1.33#

Read below for general expressions and rationale.


DETERMINING A GENERAL EXPRESSION FOR GAMMA

Assuming you mean

#gamma = barC_P//barC_V#,

where

  • #barC_P = C_P/n# is the molar heat capacity at constant pressure,
  • #barC_V = C_V/n# is the molar heat capacity at constant volume,

then recall from the equipartition theorem that the average molar internal energy in the high temperature limit is given by:

#<< epsilon >> = N/2RT#

where:

  • #N# is the number of degrees of freedom (DOF) in terms of translation, rotation, and vibration (we ignore electronic DOFs).
  • #R = "8.314472 J/mol"cdot"K"# is the universal gas constant.
  • #T# is the temperature in #"K"#.

Also recall that #barC_P = barC_V + R#, and that by definition, the derivative of the internal energy w.r.t. temperature at constant volume is:

#((del << epsilon >>)/(del T))_V = barC_V#

From this, it follows that:

#color(green)(barC_V) = ((del << epsilon >>)/(del T))_V = color(green)(N/2R)#

#color(green)(barC_P = (N+2)/2R)#

So,

#gamma = barC_P//barC_V#

#= (N+2)/cancel2cancelR cdot cancel2/N 1/cancelR#

#= (N+2)/N#

And therefore:

#barul|stackrel(" ")(" "color(black)(gamma = 1 + 2/N)" ")|#

which implies that #gamma > 1# for all polyatomic gases.

APPROXIMATING GAMMA VIA EQUIPARTITION

Now, what we seek is a way to determine the value of #N# for a polyatomic gas.

In general, as it turns out, for most polyatomic gases at #"298.15 K"#:

  • Translational and rotational contributions are significant.
  • Vibrational contributions are minimal, and if we try to estimate #N# for vibration the usual way, we would usually way overestimate it. So instead, we choose to omit it.

For any gas at most temperatures, where #N = N_(tr) + N_(rot) + . . . #,

  • #N_(tr) = 3# for any gas in three dimensions of linear motion (#x,y,z#)

  • #N_(rot) = 2# for a linear polyatomic gas for rotational motion (#theta,phi# in spherical coordinates)

  • #N_(rot) = 3# for a nonlinear polyatomic gas for rotational motion (#theta,phi, alpha#, where #alpha# is some third angle of rotation in spherical coordinates)

Therefore, for linear polyatomic gases (such as #"CO"_2# or #"N"_2"O"#):

#color(blue)(gamma ~~) 1 + 2/(3+2) = color(blue)(1.40)#

For nonlinear polyatomic gases (such as #"SO"_2# or #"NH"_3#):

#color(blue)(gamma ~~) 1 + 2/(3 + 3) = color(blue)(1.33)#