Question

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

Answer 1

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.

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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)`