Virial coefficients of the ideal gas equation.

[EricVT]

Homework Statement

Even at low density, real gases don't quite obey the ideal gas law. A systematic way to account for deviations from ideal behavior is the virial expansion. Just working with the first order virial expansion we have:

PV=nRT(1 + B(T)/(V/n))

B(T) is a virial coefficient. A table is given with some measured values of B at different temperatures for nitrogen at atmospheric pressure (temp on left, B value on right separated by "/"):

T(K) / B(cm^3/mol)
-------------------
100 / -160
200 / -35
300 / -4.2
400 / 9.0
500 / 16.9
600 / 21.3

The problem asks to find the value of B(T)/(V/n) for each value of T given in the table.

Homework Equations

PV=nRT (ideal gas equation)

The Attempt at a Solution

Since B(T) is given in the table, it seems that only (V/n) needs to be found and then a simple division of terms should yield the answer for each T that you insert into the equation.

First I divided both sides by n and P, and set (V/n)=x, and set the quantity R/P = C

==> x = CT(1+B/x)
==> x/(1+b/x) = CT
==> x^2 = CTx +CTB
==> x^2 - CTx - CTB = 0

This is a quadratic for x, which I solved to get:

x = (1/2)(CT + sqrt[(CT)^2 -4(-CTB)] ==> x = (1/2)(CT + sqrt[(CT)^2 + 4(CTB)]

or x = (1/2)(CT - sqrt[(CT)^2 + 4(CTB)]

However, (CT)^2 is a very small number and for negative B terms such as -160 (the first value in the given table) you can't get real solutions to this quadratic. This doesn't seem correct since these virial terms represent corrections to real intermolecular interactions, so I feel like this technique must not be suitable for this problem.

Is there some misunderstanding on my part about what is being asked? The problem gives B as a function of T alone, but it seems like you can manipulate the ideal gas law to show B as a function of P,V,n, and T. This makes me think there is something I am not understanding about what is given in this problem.

Homework Statement

Another equation of state is given by the van der Waals equation:

(P + a*n^2/V^2)(V - n*b) = n*R*T

where a and b are constants that depend on the type of gas. Calculate the second virial coefficient B for a gas obeying the van der Waal's equation, in terms of a and b.

Homework Equations

Hint: The binomial expansion says that (1 + x)^p ~ 1 + p*x +(1/2)(p)(p-1)(x^2) for |px|<<1. Apply this approximation to the quantity [1-(n*b/V)]^(-1).

The Attempt at a Solution

First of all, is the virial expansion for the van der Waal's equation just

(P + a*n^2/V^2)(V - n*b) = n*R*T(1 + B/(V/n)) like for the ideal gas law virial expansion? I assumed it was and worked from there. I basically just divided out terms until I got

[1-n*b/V]^(-1) = [n*R*T*(1+B/(V/n))]/[P*V+ a*n^2/V)

Then applied the given hinted approximation and followed through all the boring algebra to finally get:

B = [P*V^2 + a*n^2]/[R*T*n^2 + ((R*T*b*n^3)/V)(1+n*b/V)] - V/n

Yikes! Doesn't seem too convincing (the units don't even make sense!), plus I'm not sure if my answer was only supposed to be in terms of "a" and "b" or if the other variables are fine as well.

Any help on this would be appreciated. This looks like a great community, so hopefully I can fit right in.

 

[EricVT]

One thing I've just thought of.

Can we approximate B/(V/n) as B/(RT/P), since this is a first order approximation? That could make the first question a bit easier and avoid the troubles of non-real solutions to the quadratic. But I have to wonder how accurate this approximation would be.

 

[m2003]

I would like to first clarify that the virial coefficients are used to account for deviations from ideal gas behavior in the ideal gas equation. These coefficients represent corrections to the ideal gas law, taking into account real intermolecular interactions. In the first problem, the table provides values of the first virial coefficient, B, for nitrogen at different temperatures. The problem asks to find the value of B(T)/(V/n) for each temperature given in the table. This means that we need to find the value of (V/n) for each temperature and then divide it by the corresponding B value to get the desired ratio.

In order to find the value of (V/n), we can use the ideal gas equation PV=nRT. Rearranging this equation, we get (V/n)=RT/P. Substituting this into the expression for B(T)/(V/n), we get B(T)/(V/n)=B(T)*P/(RT). Plugging in the values of P, T, and B from the table, we can find the values of B(T)/(V/n) for each temperature.

In the second problem, we are asked to find the second virial coefficient, B, for a gas obeying the van der Waals equation. The van der Waals equation is given by (P + a*n^2/V^2)(V - n*b) = n*R*T. To find the value of B, we can use the definition of the second virial coefficient, which is given by B = -a*n^2/V^2. Substituting this into the van der Waals equation, we get the expression (P + B*V^2)(V - n*b) = n*R*T. This equation can be solved for B to get the desired expression in terms of a and b.

Regarding the hint given in the problem, it suggests using the binomial expansion to approximate the term [1-(n*b/V)]^(-1). This approximation can be used to simplify the algebra and make the solution easier to obtain. However, it is not necessary to use this approximation to solve the problem. In conclusion, the second virial coefficient for a gas obeying the van der Waals equation can be calculated using the definition of the second virial coefficient and solving the equation for B.