數學包含科學問題

Q1.
Ideal Gas Law states that PV= nRT
P = pressure, V = Volume, T = temperature n = no. of moles R = Gas Constant
P, V, T are variables
van der Waals equation accounts for the deviation from the ideal behaviour. a, b are selected for each gas to give the best possible agreement between the equation and actually observed behaviour.
The corrected pressure = (P + an^2/V^2)
The corrected volume = (V – nb)
(P + an^2/V^2) (V – nb) = nRT

(i) P + an^2/V^2 = nRT/(V – nb)
P = nRT/(V – nb) - an^2/V^2

(ii) T = [(P + an^2/V^2) (V – nb)]/nR

(iii) Expand the Van der Waals equation
-nbP –n^3ab/V^2 +n^2a/V +PV = nRT
–n^3ab/V^2 +n^2a/V +PV = (nRT+ nbP)
Multiply both sides by V^2
–n^3ab +n^2aV +PV^3 = (nRT+ nbP)V^2
+PV^3 - (nRT+ nbP)V^2 +n^2aV – n^3ab
Compare a cubic equation which has the form ax^3 + bx^2 + cx + d
(P) V^3 - (nRT+ nbP)V^2 +(n^2a)V – (n^3ab)

(iii) Since van der Waals equation is a multivariable equation (P, V, T)
You have to keep one variable constant in order to find the partial derivative.
The first one (dV/dT) , pressure is kept constant.
V can’t be expressed in terms of P and T (~ a cubic equation). We have to do it by implicit differentiation.
PV^3 - (nRT+ nbP)V^2 + (n^2a)V – (n^3ab) = 0
PV^3 - nRTV^2 - nbPV^2 + (n^2a)V – (n^3ab) = 0
Differentiate V with respect to T
3PV^2 dV/dT- (nRT) 2V dV/dT – (V^2) nR – (nbP)2V + (n^2a) dV/dT – 0 = 0
3PV^2 dV/dT- 2nRTVdV/dT – nR (V^2) - 2(nbP)V + (n^2a) dV/dT = 0
3PV^2 dV/dT- 2nRTVdV/dT + (n^2a) dV/dT = nRV^2 - 2nbPV
dV/dT (3PV^2 - 2nRTV + n^2a) = nRV^2 - 2nbPV
dV/dT = (nRV^2 - 2nbPV)/ (3PV^2 - 2nRTV + n^2a)
dV/dT = nV(RV - 2nP)/ (3PV^2 - 2nRTV + n^2a)

The second one dV/dT, with volume is kept constant
PV^3 - nRTV^2 - nbPV^2 + (n^2a)V – (n^3ab) = 0
Differentiate V with respect to P
P3V^2 dV/dP +V^3 - 2(nRT) V – (V^2) nb – (nbP)2VdV/dP + (n^2a) dV/dP – 0 = 0
P3V^2 dV/dP +– (nbP)2VdV/dP + (n^2a) dV/dP – 0 = 2nRTV + nbV^2 - V^3
dV/dP (3PV^2 – 2(nbP)V + an^2) = (2nRTV + nbV^2 – V^3)
dV/dP = (2nRTV + nbV^2 – V^3)/ (3PV^2 – 2nbPV + an^2)

Run out of space...

2012-09-26 12:17:01 補充：
The third one dP/dT, with volume is kept constant
P = nRT/(V – nb) - an^2/V^2
Differentiate P with respect to T
dP/dT = nR/(V – nb)

The fourth one dP/dV with temperature constant
P = nRT/(V – nb) - an^2/V^2
Differentiate P with respect to V
P = nRT/(V – nb) - an^2 V^(-2)
dP/dV = -nRT/(V – nb)

2012-09-26 12:23:17 補充：
They cut off the last part. Re-post it.

The fourth one dP/dV with temperature constant
P = nRT/(V – nb) - an^2/V^2
Differentiate P with respect to V
P = nRT/(V – nb) - an^2 V^(-2)
dP/dV = -nRT/(V – nb)^2 +2 an^2 V^(-3)
dP/dV = -nRT/(V – nb)^2 + 2 an^2 /V^3

2012-09-26 12:27:49 補充：
I don’t think Form 6 students in H.K. learn about multivariable calculus, let alone Laplace Transform and eigenvalue/eigenvector.

Implicit Differentiation is tedious. Please double check dV/dT and dV/dP.
I did the easy one. Let others do the rest of your questions. Good luck!

2012-09-26 12:38:18 補充：
Correction:
The second one dV/dP, with volume is kept constant

I advise you to ask these Qs separate so that it's not necessary for 1 person to ans all these.

p.s.
How come 1) a) i) and 1) a) ii) are the same?

我知道只是看題目就頭痛呀，所以我在這裡求救....
這是我在澳洲的大學第一年的數學問題，但我認為澳洲的數學程度是比香港慢所以是中學水平?中6,7?

請問這些幾年級問題?
牽涉到如此複雜的問題，甚至是積分。
只是看題目已經頭暈目眩了。