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3rd year Thermodynamics-Entropy problems

  1. Oct 14, 2011 #1
    Hi there!

    Long time lurker and fan of PF. Two questions are giving me trouble from my Thermodynamics homework. A nudge in the right direction would be extremely appreciated!

    1. The problem statement, all variables and given/known data
    Question 1:
    One mole of a monatomic gas is prepared at room temperature and atmospheric pressure (STP).
    (a) What is its internal energy and what is the volume occupied?
    (b) The temperature is raised to 100°C and the pressure is incresed by a factor of ten. What is the new internal energy and the volume occupied?
    (c) What is the change in entropy going from (a) to (b)? Express your answer in terms of the universal gas constant R.
    (d) What is the change in chemical potential going from (a) to (b)?

    Question 2:
    The entropy of N moles of a real (not ideal) gas has been determined to be
    S = N s0 + N R ln [(v - b)(u + a/v)c]
    where v = V/N, u = U/N, and s0 is a constant.

    (a) Is this an acceptable Fundamental Relation? (Hint: check for extensivity, monotonicity in U, differentiability, and whether S vanishes at T = 0)
    (b) Obtain U=U(S,V,N)
    (c) Obtain the three equations of state from U
    (d) Obtain the pressure as a function of temperature and volume
    (e) What is μ/RT at T = 90K and atmospheric pressure for a = 0.138E-6 Pa m6, b = 32.6E-6 m3 and c = 5/2, appropriate for molecular oxygen? [μ is the chemical potential]

    2. Relevant equations
    μ/RT - μ0/RT0 = ln[(P/P0)(T0/T)5/2]


    3. The attempt at a solution
    Question 1:
    (a) just looking for a numeric value here, and the conditions suggest an ideal gas, so I thought:
    U = (3/2)NRT = (3/2)(1)(8.314)(298) = 3.7 kJ
    and from PV=NRT → V = 0.0367 m3
    (b) U = (3/2)NRT = (3/2)(1)(8.314)(373) = 4.65 kJ
    V = 0.00459 m3
    (c) Again, I figured we could assume it is an ideal gas and use:
    ΔS = R ln[(u/u0)3/2 (v/v0)]
    =-1.74R
    (d) I'm absolutely lost on this part. As far as I can tell, we can only calculate the change in chemical potential as a fraction of temperature (ie. μ/T - μ0/T0) giving us a linear relationship between possibilities (since we don't know μ or μ0)

    Question 2:
    (a) while the relation is monotonic in U, differentiable, and extensive, it does not go to zero at zero T. Therefore not an acceptable fundamental relation.
    (b) very confused... wasn't sure what to do with s0:
    U = (v-b) exp(S/cNR) - a/V
    (c) just PDs:
    T = [(V-Nb)/cNR] exp(S/cNR)
    μ = -b exp(S/cNR) + (bS/cNR) exp(S/cNR) - 2aN/V
    P = exp(S/cNR) + aN2/V2
    (d) P = TcR/(v-b) + a/v2 note that these are per mole quantities again, not sure how to eliminate N
    (e) no idea how to eliminate S to do this part

    Any advice on any part of this question will be greatly appreciated. I have and will read any literature I can find, but these questions are not becoming any clearer!
     
  2. jcsd
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