CHM 3411 Physical Chemistry I Fall 1999
FINAL EXAM SOLUTIONS 8 December 1999
Work any 8. Identify the 2 not to be graded!
mT = (¶H/¶p)T = V - T(¶V/¶T)p which you may assume for the purposes of this problem.
Using the full expression for the Van der Waals gas, show that an excellent approximation to mT is given by
mT = nb - 2an/RT
or, for a single mole,
mT = b - (2a/RT)
(The approximations that reduce the final expression to this are that V/(V-nb) ~ 1 and 2an2 << nRTV, both of which are excellent. And remember the trick to set p = stuff so that partials with p fixed will set the partials of stuff to zero.)
(p + an2/V2) (V - nb) = nRT
p = nRT/(V-nb) - an2/V2 0 = (¶p/¶T)p = nR/(V-nb) - [nRT/(V-nb)2](¶V/¶T)p + [2an2/V3](¶V/¶T)p (¶V/¶T)p = { nR/(V-nb) } / { [nRT/(V-nb)2] - 2an2/V3 } T(¶V/¶T)p = { nRT/(V-nb) } / { [nRT/(V-nb)2] - 2an2/V3 } V = nb + (V-nb) so V - T(¶V/¶T)p = nb + (V-nb) - T(¶V/¶T)p = nb + { [(V-nb)nRT/(V-nb)2 - 2an2(V-nb)/V3] - nRT/(V-nb) } / { [nRT/(V-nb)2] - 2an2/V3 } = nb - 2an2(V-nb)/V3 / [ nRT/(V-nb)2 - 2an2/V3 ] = nb - 2an2(V-nb)3/V3 / [ nRT - 2an2(V-nb)2/V3 ] ~ nb - 2an2 / [ nRT - 2an2/V ] ~ nb - 2an2/nRT = nb - 2an/RT = mT |
DH ~ Dp ×
(¶H/¶p)T
= Dp × m T
= Dp × (b - 2a/RT)
DH ~ 99 atm × (0.06380 L - [2×5.562 atm L2 / {0.08206 atm L / mol K × 298 K}] DH ~ - 38.9 atm L (8.314 J/0.08206 atm L) = - 3.94 kJ |
k = L° c
L° = n+l+ + n-l- = 237.0 S cm2 / mol c = k / L° = 1.53×10-4 S cm-1 / 237.0 S cm2 mol-1 = 6.45×10-7 mol/mL c = 6.45×10-4 M = [BH+] = [OH-] [B] = 0.0312 - c = 0.0306 Kb = [BH+] [OH-] / [B] = (6.45×10-4)2 / 0.0306 = 1.36×10-5 |
Water mixing with water represents no entropy change, but moles of A mixing with moles of B
surely does. And we have 0.1 moles of A and 0.15 moles of B for a total of 0.25 moles of
solute. That's n. And the final mole fraction of A is XA = 0.1/0.25 = 0.4
meaning that XB = 0.6.
DmixS = - n R (XAlnXA
+ XBlnXB) =
- ¼ mol 8.314 J/mol K (0.4ln0.4 + 0.6ln0.6)
DmixH = 0.00 by definition of ideal mixture. |
log10(p) = 7.9741 - (1318.9 / T)
Obtain its normal boiling point temperature and DvapH there.
First thing first; we ought to convert to natural logarithm via ln(x) = 2.3026 log(x), so
Even without that, the normal boiling point is easy; p = 760 mm Hg. So
T = 3036.9 / [ 18.361 - ln(760) ] = 258.95 K = - 14.20°C Now oddly enough, we don't have to change to bars or atmospheres to proceed to the enthalpy of vaporization. That's because a conversion factor inside the logarithm just becomes an additive term, altering the magnitude of the 18.361 but not changing the slope with respect to 1/T, which, by the Clausius-Clapeyron equation, is DvapH.
DvapH/R = - d(lnp)/d(1/T) = 3036.9 |
(All values pertain to 37°C.)
DoverallG° = + 23.8 - 31.9 =
- 8.1 kJ/mol
K = e- DoverallG° / RT = e8100 J/mol / ( 8.314 J/mol K × 310 K ) = 23.2 > 1 |
XI | 0 | 0.059 | 0.521 | 0.924 | 1 |
PI(torr) | 0 | 12.9 | 30.5 | 42.2 | 44.0 | PB(torr) | 94.4 | 86.6 | 75.3 | 24.2 | 0 |
(a) Does this solution exhibit positive, negative, or no deviation from ideality?
Were the solution ideal, we'd expect p0.5 to be ½ way from 0 to p*, but 30.5 > 22.0 and 75.3 > 47.7 for XI~0.5, hence the deviations are quite positive. We wouldn't have expected otherwise from the mixture of an organic (no hydrogen bonding) and an alcohol (lots of hydrogen bonding). |
(b) Estimate KHenry for both substituents. Are they consistent with your answer to (a)?
psolute ~ KH Xsolute
KH ~ psolute / Xsolute KH(I) ~ 12.9 torr / 0.059 = 219 torr >> 44 torr (positive deviation) KH(B) ~ 24.2 torr / (1-0.924) = 318 torr >> 94.4 (positive deviation) |
(Of course [cis]+[trans] = A0, the original concentration of cis. So there's really only one variable to solve the rate equation for, [trans]=x, as a function of time.)
Let x=[trans] and A0 be [cis]0, the initial cis
concentration, then
dx/dt = k1[cis] - k-1[trans] =
k1(A0-x) - k-1x = k1A0 -
(k1+k-1)x
x = A0 [ k1 / (k1+k-1) ] (1 - e- (k1+k-1) t ]
Although we know that
k1[cis]eq = k-1[trans]eq,
or rather K = [trans] / [cis] = k1/k-1,
we can actually arrive at that by setting t to infinity (where equilibrium has been
established). Then
So k-1 = k1 / K = 3.3×10-4 s-1 / 0.16 = 2.1× 10-3 s-1 and k1+k-1 = 2.39×10-3 s-1 That's all we need to establish t½ where x = ½A0 [k1 / (k1+k-1)]
t½ = - [1 / (k1+k-1)] ln { [ k1A0 -
½A0(k1+k-1)k1 /
(k1+k-1) ]
/ k1A0 }
|
- d[O3]/dt = k1 [O3]2 +
k2 [O] [O3]
d[O]/dt = k1 [O3]2 - k2 [O]ss [O3] ~ 0
- d[O3]/dt = k1 [O3]2 +
k2 (k1 /k2) [O3]2
= 2 k1 [O3]2
|
In case you don't remember it, an auto battery reacts Pb and PbO2 to produce PbSO4 from each in sulfuric acid solution. For our purposes, we'll let the acid activity be 1, but in real batteries it's more concentrated.
Species | DfG° (kJ/mol) | DfH° (kJ/mol) | S° (J/mol K) |
Pb | 0 | 0 | 64.81 |
PbO2(s) | -217.33 | -277.4 | 68.6 |
PbSO4(s) | -811.24 | -981.39 | 147 |
H+(aq) | 0 | 0 | 0 |
SO42-(aq) | -744.53 | -909.27 | 20.1 |
H2O(aq) | -237.13 | -285.83 | 69.91 |
- nFE° = DG°
E° = - DG° / nF DG° = DH° - TDS° dE°/dT = - (dDG°/dT) / nF = + DS° / nF
dE°/dT = DS° / nF = 260 J/mol K / [2 × 96,450 C/mol] = + 1.35 mV/K
DE° ~ (dE°/dT) DT
= 1.35 mV/K (- 50 K) = - 0.067 V per cell
|