What is the value of #gamma# for a polyatomic gas?
1 Answer
For linear polyatomic gases (such as
#gamma ~~ 1.40#
For nonlinear polyatomic gases (such as
#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
#((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
In general, as it turns out, for most polyatomic gases at
- 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_(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
#color(blue)(gamma ~~) 1 + 2/(3+2) = color(blue)(1.40)#
For nonlinear polyatomic gases (such as
#color(blue)(gamma ~~) 1 + 2/(3 + 3) = color(blue)(1.33)#