Evaluating a Derivative of the van der Waals Equation using the Cyclic Rule For

Question

Evaluating a Derivative of the van der Waals Equation using the Cyclic Rule

For the van der Waals equation of state:
V-b(P+ a/v^2)=RT

the derivative (dV/dT)p is difficult to obtain directly because finding an equation for V in terms of P or T requires solving the cubic equation:
PV^3 - (bP +RT)V^2 + a(V-b)=0

a and b are two parameters that take into account the size of the molecule and the strength of the attractive interaction.

Because P is linear in the van der Waals equation, it should be easier to find the partial derivatives:
(dP/dT)v and (dV/dP)T = 1/(dP/dV)T

needed to utilize the cyclic rule.

Using the cyclic rule, find
(dV/dT)P

Express your answer in terms of the parameters, constants, and variables in the van der Waals equation (P,V,R,T,a,and b).

Thanks for any help!

Explanation / Answer

The van der Waals equation can be written as: P = RT/(Vm-b) - a/Vm^2 If you take the derivative of P in terms of Vm, you get the first equation: dP/dVm = -RT/(Vm-b)^2 + 2a/Vm^3 As a recent chemistry major graduate, I should assume that you have taken a calculus course prior to taking a PChem course, but just in case you haven't: RT/(Vm-b) = RT*(Vm-b)^(-1) When you take the derivative of a power, you multiply the term by the power and decrease the power of the term by 1, giving you -RT/(Vm-b)^2 or -RT(Vm-b)^(-2) If you repeat this for -a/Vm^2, you should get 2a/Vm^3. The second equation is just the second derivative of the van der Waals equation. So you take the derivative of the first derivative (equation 1). Following the same instruction for the first derivative, you should get equation 2 (dP/dVm^2 = 2RT/(Vm-b)^3 - 6a/Vm^4). How to get Vc: From equation 1, you know that RT/(Vm-b)^2 = 2a/Vm^3. If you substitute this into equation 2, you get: 4a/(Vm^3 * (Vm-b)) - 6a/Vm^4 = 0 You can factor out 2a/Vm^3, leaving you with: (2a/Vm^3) * (2/(Vm-b) - 3/Vm) = 0 Since you can't divide by 0, you can forget about the (2a/Vm^3). Multiply out (2/(Vm-b) - 3/Vm) to get a common denominator. Then calculate the numerator (you should have gotten 2Vm - 3Vm + 3b) to equal 0. This should give you 3b. How to get Tc: Substitute in 3b for Vm in equation 1 to get: -RT/(4b^2) + 2a/(27b^3) = 0 If you simplify this and get T on one side, you will find that T = 8a/27Rb How to get Pc: Substitute in the values for Vc and Tc into the van der Waals equation P = RT/(Vm-b) - a/Vm^2: P = [8Ra]/[27Rb(2b)] - a/(9b^2) Simplifying this (including finding a common denominator) will give you P = a/27b^2

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