For the van der Waals equation of state ( V m b )( P + a V 2 m )= R T where V m

Question

For the van der Waals equation of state

(Vmb)(P+aV2m)=RT

where Vm is the molar volume, which is the volume divided by the moles of gas, n, the derivative

(VT)p

is difficult to obtain directly because finding an equation for Vm in terms of T or P requires solving the cubic equation

PV3m(bP+RT)V2m+a(Vmb)=0

where 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

(PT)V and (VP)T=1(PV)T

needed to utilize the cyclic rule. Keep in mind that the cyclic rule for the variables Vm, P, and T is

(VT)P(TP)V(PV)T=1

For example, the partial derivative (V/T)P can be found by evaluating

(VT)P=(PT)V(PV)T

I already put RVm2(Vmb)RTVm32a(Vmb)2 this but it was incorrect

please give me correct answer

Part C Evaluating a derivative of the van der Waals equation using the cyclic rule Find the partial derivatives (0P/OT) and (0P/ov)T, and apply them to the equation derived from the cyclic rule OT /V OT P to find (ov/OTP Express your answer in terms of the parameters, constants, and variables in the van der Waals equation (P, V, R, T, a, and b) AE OT

Explanation / Answer

SOLUTION:

van Der Waals equation is :

(P + n2a/V2) (V - nb) = nRT

solving for P

P + = (nRT/V - nb)

P = (nRT/V - nb) - n2a/V2

(P/V)T = 1/V[(nRT/V - nb) - n2a/V2]

(P/V)T = = nRT/ (V - nb)2 - 2n2a/V3

similarly

(P/T)V = 1/T [(nRT/V - nb) - n2a/V2] = (nR / V - nb)

(V/T)P = (P/V)T / (P/T)V = {nRT/ (V - nb)2 - 2n2a/V3} / (nR / V - nb) = {T / (V - nb) - 2na(V - nb)/RV2}

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