MadSci Network: Chemistry

Re: What is the equation to determine Boyle Temperature?

Date: Tue Oct 20 16:34:25 1998
Posted By: Dan Berger, Faculty Chemistry/Science, Bluffton College
Area of science: Chemistry
ID: 906267389.Ch

What is the equation to determine Boyle Temperature?

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 + Bīp + Cīp2 + ...}

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 = Bī = 0. This temperature is called the Boyle Temperature, TB.

TB can be derived from the va n der Waals equation of state for a real gas,

p = [RT/(V(m)-b)]-a/V(m)^2

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)}

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) + ...}

Obviously the virial coefficient B is given by

B = b - a/RT

At the Boyle Temperature B = 0, and so

T(B) = a/bR = 27T(c)/8

(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.)
Thus, the Boyle temperature may be found for any gas for which the critical temperature has been determined.

  Dan Berger
  Bluffton College

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