April 13, 2007
PHY 3513, Midterm III_Solutions
You are allowed use of a calculator and a formula sheet. Please make sure that the
formula sheet does not contain anything else. Please show all work.
1. (a) Show that the mean speed of a nitrogen molecule at 300K is 476m/s.
⎡ 8kT ⎤
Mean speed is v m = ⎢
⎥
⎣ πm ⎦
1/ 2
=
8RT
πμ
where µ is the molecular weight of the gas
molecule (here for N2, µ = 28 kg/kmole). Thus v m =
8 × 8314 × 300
=476.3m/s
3.1416 × 28
(b) What is the ratio of the probability of finding a nitrogen molecule with speed
476 m/s at 300K to the probability of finding an oxygen molecule with the same
speed at 600K?
Recall that the molecular weight for O2 is 32 kg/kmole. For N2, it is 28 kg/kmole
The probability distribution is given by
N (v )
⎡ m ⎤
= f (v ) = 4πv 2 ⎢
N
⎣ 2πkT ⎥⎦
3/ 2
⎡ mv 2 ⎤
exp ⎢−
⎥
⎣ 2kT ⎦
Assuming the same total number of particles, the ratio then is for the same speed but
different temperatures and masses. One can write:
⎡ v2 ⎛ m
f N ⎡ m ⎤ 3/ 2 ⎡T ⎤ 3/ 2
m ⎞⎤
exp ⎢− ⎜
|N −
|O ⎟⎥
=⎢ ⎥
⎥
⎢
fO ⎣ T ⎦ N ⎣ m ⎦O
kT ⎠⎦
⎣ 2 ⎝ kT
For convenient calculation, use m/k = μ/R so that μ/RT|N = 28/8314×300 = 1.123×10-5
and μ/RT|O = 6.415×10-6. The ratio is 1.34.
2. A container of volume V is divided by partitions into three parts containing 1
kilomole of helium gas, 2 kilomoles of neon gas and 3 kilomoles of argon gas.
The temperature of each gas is initially 300K and the pressure is 3 atm. The
partitions are removed and the gases diffuse into one another. Calculate the
change in the Gibbs function and in entropy of the system in the mixing process.
The change on mixing in the Gibbs free energy and the entropy is given by (with a total
of 6 kilomoles, x1=1/6, x2=1/3 and x3=1/2)
ΔG = nRT [x1 ln x1 + x2 ln x2 + x3 ln x3 ] = 6 × 8314 × 300 × −1.0114 = −1.5 × 10 7 J
ΔS = −ΔG / T = 5.045 × 10 4 J / K
3. (a) Show that for an open system,
V
⎛ ∂S ⎞
⎜ ⎟ =−
T
⎝ ∂P ⎠ H ,n
(b) For an ideal gas the entropy S (H,P,n) has the form
⎡ φ ( H , n) ⎤
S (H , P, n ) = nR ln ⎢
⎣ P ⎥⎦
Where ϕ is a function of enthalpy and the number of kilomoles? Derive the
equation of state.
For an open system du = Tds –Pdv +µdn. Since h = u + Pv, it follows that
dh =Tds + vdP + µdn
V
⎛ ∂S ⎞
For constant h and n, Tds = -vdP, i.e ⎜ ⎟ = − .
T
⎝ ∂P ⎠ H ,n
(b) Given S(P, T), the equation of state v (P, T) can be derived here. Thus differentiating
S with respect to P, we get PV=nRT.