Expert Answer:Problems about thermodynamics – equations

  

Solved by verified expert:need help for the following problems. 1.17 (Virial and Van-der-Waal’s equation of state) – seperate file as hints is attached1.33 (3-step cyclic process)1.51 (Combustion of glucose)1.54 (Climbing Mount Ogden)2.25 (Random Walk)2.29 (The Entropy of 2 Einstein solids.)
6_problems_about_thermo.pdf

hints_for_problem_1.17.pdf

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Check the hints for 1.17 (attached) as a second file
For problem 1.51; the reference table is attracted at the end of this file
Do not solve problem 2.24, I need help for the solution of problem 2.25
Below is the section 1.17 mentioned in the problem 2.25 section (c)
Reference table for problem 1.51
Question 1.17
The Van-der-Waals equation of state is


an2
P + 2 (V − nb) = nRT
V
Pulling the factor of V out of the second set of parentheses,

 

an2
nb
P + 2 V 1−
= nRT
V
V
(1)
(2)
and divide both sides by (1 − nb/V )




an2
nb −1
P + 2 V = nRT 1 −
V
V
(3)
Then we use the binomial expansion given in the book on the term on the RHS (1 − nb/V )−1 . I’ll do
the first two terms:
(1 + x)p ≈ 1 + px + . . .
(4)
Looking at what we’ve got, x = −nb/V and p = -1, and so


nb
nb −1
≈1+
+ …
1−
V
V
(5)
Inserting this into equation (3):




an2
nb
P + 2 V = nRT 1 +
+ …
V
V


b
= nRT 1 +
+ …
(V /n)
(6)
Moving the second term on the left hand side over to the right hand side gives us:
an2
P V = nRT 1 +
+ … −
V


(a/RT )n
= nRT 1 +
+ . . . − nRT
V


(a/RT )
= nRT 1 +
+ . . . − nRT
(V /n)


(a/RT )
= nRT 1 +

+ …
(V /n)


1
= nRT 1 + (b − a/RT )
+ …
(V /n)


b
(V /n)
b
(V /n)
b
(V /n)
b
(V /n)
1
(7)
Comparing this to the Virial equation of state:


B(T )
C(T )
P V = nRT 1 +
+
+ …
(V /n) (V /n)2
(8)
we can see by comparing the two equations, and looking at the term proportion to
1
(V /n) ,
B(T ) = b − (a/RT )
To get C(T ) you need to include the next term in the binomial expansion (the
1
which will result in a term proportional to (V /n)
2.
2
that
(9)
1
2 p(p
− 1)x2 term),

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