In this project I shall be trying to determine the Boyle temperature for a set gas based around the Redlich and Kwong gas equation that has been written in a form analogous to the van der Waals equation.

The equation is:                

Which can be re-arranged to:        

First We Find The First Derivative Of This Equation:

From a standard result, and the fact that the derivative of a constant times a variable is the same as the constant times the derivative of the variable:

Next, treating the constant in the same way as above, and using the quotient rule we get:

So overall:

Next, We Find The Second Derivative:

Using the same method again we first have:

And then for the second term we do the same but we also use the product rule, giving:

So overall we have:

Next we want to find the critical constants, V_c, T_c, and P_c.

At the critical point are zero, so we can equate them to each other in order to find V_c:

Firstly:

Secondly:

Therefore:

This has 3 solutions, 2 are imaginary, and are therefore of no interest to us, the third is real however and gives us:            

V_c = (2^(⅔) + 2^(⅓) + 1)b = 3.8473b

Next we use this result to find T_c
^3/2:

We do this by rearranging one of the differential equations and substituting this value of V_c for V:

Now that we have an expression for T_c and V_c we can determine P_c, using the original equation:

As we only have an expression for T_c ^3/2, we can’t form an equation with just P_c, a and b in it, but we can derive the following:

If we now take units of atm for P, l mol^-1 for V and K for T, we have R = 0.08206 l atm K^-1. Also as I have chosen to study the H₂ molecule, we can look up the values of P_c and T_c.

We can use the values to find a, b and V_c as follows:

P_c = 12.8 atm, T_c = 33.23 K

b is a term that accounts for the discrepancy in volume due to the gas being non-ideal, and is directly related to the volume of the molecule in question – namely H₂. We can use b to give a better value for the volume of 1 mole of H₂ by noting that: V = V_m – b, where V_m is the ideal molar volume. Then once we have the true volume of 1 mole of H₂ molecules, we can calculate the effective spherical volume occupied by just 1 molecule.

The ideal molar volume for H₂ at 1atm and 300 K is: V = = 0.08206 x 300 = 24.618 l

This means that a better value for the true volume is = 24.618 – 0.018 = 24.6 l

So 24.6 l contains 6.02 x 10²³ molecules, which implies that one molecule has an effective spherical volume of 4.086 x 10^-23 l.

The next thing to do would be to expand P in powers of V^-1, and show that B, the second virial coefficient is given by:

The easiest way to do this is to apparently to expand the expression for a perfect gas, and compare this expression to it. The expression for a perfect gas is:

And our equation for P is:

So far I have obtained an expression for B that is almost true to this, but at the time of writing, I am having trouble getting an exact match. As a result I will leave this part of the analysis to be included as an appendix at the end of the project, and continue, assuming the above expression for B to be true.

We can use a plot B for different values of T to determine the Boyle temperature for the gas:

At T = Boyle Temperature (T_B), B = 0. As the graph is approximately linear around this point, a straight line fitted to this range of values yields the equation:

B = 0.0003T – 0.0316.

Using this equation we can predict T_B as follows:

B = 0.0003T – 0.0316

0 = 0.0003T – 0.0316 at T_B

Þ

Therefore the Boyle Temperature for H₂ is 105.33 K, however the book states that it is 110.0K my result is a little out, but this is probably due to both rounding errors, and approximating the region of the graph where the T axis is crossed as linear instead of fitting an exact curve to the data points.

Whilst doing this project I also tried a couple of other scenarios; one using Xe as the basis ‘molecule’, and the other using NH₃. For Xe the predicted value for TB, was approximately ¹/₇ of the value stated in the book, and the predicted value for NH₃ is 1350 K. There is no value for T_B of NH₃ in the book, this is most probably because NH₃ dissociates below this predicted temperature. Also the value for Xe stated in the book is about 7 times as much as I predicted (about 130 – 170 K). This is probably due to Xe being an atom and not considerable as a molecule, and therefore this expression doesn’t apply, or perhaps Xe could be considered as a molecule but in a way that I’m not yet familiar with.

Appendix:

Expanding P In Powers Of V^-1 To Show That The Second Virial Co-Efficient Is:

Well, I’ve reached the project deadline and I still can’t get this quite right, but rather than avoid the question as others have, I have included my best attempt at reaching the correct expression.

The perfect gas expansion is:

Our equation is:

So re-writing our expression in the form of the ideal gas expression gives:

However assuming that the first term is always 1 (which I believe it is) then the equation should be modified slightly, and so should B. However this just leads to an even messier expression, which no more correct than the first, as simplification still doesn’t remove the “V’s”:

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