AskiitianExpert Shine
Last Activity: 15 Years ago
Hi
At low pressures, all gases behave ideally, that is, they follow the Ideal Gas Law
pVm = RT
where p = pressure, Vm = molar volume, T = temperature and R is the Gas Constant.
However, at higher pressures gases deviate from the Ideal Gas Law, following a linear expansion of the form
pVm = RT{1 + B´p + C´p2 + ...}
sometimes stated as
pVm = RT{1 + B/Vm + C/Vm2 + ...}
The values of the virial coefficients B, C, ... depend on temperature, and many gases have a particular temperature at which B = B´ = 0. This temperature is called the Boyle Temperature, TB.
TB can be derived from the van der Waals equation of state for a real gas,
![p = [RT/(V(m)-b)]-a/V(m)^2](http://www.madsci.org/posts/archives/1998-10/908947075.Ch.1.gif)
In order to do that, we express the van der Waals equation as
![p = [RT/V(m)]{1/[1-b/V(m)] - a/RTV(m)}](http://www.madsci.org/posts/archives/1998-10/908947075.Ch.2.gif)
So long as b/Vm < 1, we can expand the first term inside the brackets according to
(1 – x)-1 = 1 + x + x2 + ...
which gives
![p = [RT/V(m)]{1 + (b - a/RT)/V(m) + ...}](http://www.madsci.org/posts/archives/1998-10/908947075.Ch.3.gif)
Obviously the virial coefficient B is given by

At the Boyle Temperature B = 0, and so

(Tc = 8a/27Rb is the critical temperature of the gas. For a derivation of the relationship of the critical constants to the van der Waals constants, see P.W. Atkins, Physical Chemistry, 3d Ed. page 31.)