Physical Chem Van der Waals eqn of state

[Goronchem]

Homework Statement

Use the “virial equation of state”. Measurements for Ar gave B = -21.7 cm3/mol and C = 1200 cm6 /mol-2. Assuming that Ar is a van der Waals-type gas, what are the values of “a” and “b” in the corresponding van der Waals
equation of state

Homework Equations

PV= RT(1+ B/V +C/V^2 ...)
[p+(a/V^2)][V-b]= RT

The Attempt at a Solution

I have to convert the virial equation of state to the van der waals equation i think, but am unsure how to do it. It's been some time since my last calculus course so any help would be appreciated!

 

[nate808]

Hello fellow scientist,

To convert the virial equation of state to the van der Waals equation, we can use the following steps:

1. Start with the virial equation of state: PV = RT(1 + B/V + C/V^2 + ...)

2. Multiply both sides by V to get rid of the fraction on the right side: PV^2 = RT(V + B + C/V + ...)

3. Expand the brackets and rearrange the terms: PV^2 = RTV + RTB + RTC/V + ...

4. Subtract RTV from both sides: PV^2 - RTV = RTB + RTC/V + ...

5. Factor out a common term of RT from the right side: PV^2 - RTV = RT(B + C/V + ...)

6. Now we can see that the left side is similar to the van der Waals equation: PV^2 - RTV = (p + a/V^2)V - bV

7. Equate the two equations and compare the coefficients: p = PV^2 - RTV, a = RT, b = RTB

8. Substitute the given value of B = -21.7 cm^3/mol into the equation to find the value of a: a = (8.314 J/mol*K)(-21.7 cm^3/mol) = -179.9 cm^3*J/mol*K

9. To find the value of b, we need to convert the units of C from cm^6/mol^2 to m^6/mol^2. This can be done by dividing C by 10^6: C = 1200 cm^6/mol^2 = 1.2*10^-3 m^6/mol^2

10. Now substitute the values of a and C into the equation for b: b = (8.314 J/mol*K)(1.2*10^-3 m^6/mol^2) = 9.976*10^-3 m^3*J/mol*K

Therefore, the values of a and b for the van der Waals equation of state are: a = -179.9 cm^3*J/mol*K and b = 9.976*10^-3 m^3*J/mol*K.

I hope this helps and let me know if you have any further questions. Keep up the good work!