I have this awful problem due tomorrow; if anyone can help it would be much appreciated.

Most introductory chemistry texts provide a formula for the freezing point depression/boiling point of elevations:

delta-T(f,b)= K(f,b) * m
(m = molality)

In fact, these are derived equations for dilute solutions. They are generated from the more general relationships....
ln Xsolv= delta-b/R [(1/T) - (1/Tb)]
-ln Xsolv= delta-Hf/R [(1/T) - (1/Tf)]

Please derive the latter from the former for boiling point elevation, and indicate which values are contained in the super-constant Kb. (This is a difficult derivation; you'll find some assumptions are made based on the fact that the solution is dilute.)

Two question first on just what's being asked:
1. I'm going to start with the first equation and end up with one of the next two, correct?
2. What does it mean "Indicate which values are contained in the super-constant Kb"?

Here are my thoughts so far, looking just at the variables in both equations:
I know that...
ln P2/P1= delta-Hfusion/R [(1/T2)-(1/T1)]
dT/dP= delta-Hfusion/T [V1 - V2]
Μu(i)= Μu°i + RT ln a(i)
And in dilute systems a(i) --> 1
P(i)= Xi*P°i
P(t)= XB(P°B - P°A) + P°A

I looked through my worksheets and tried to pull out anything that looked useful, but I'm not sure how to piece it all together. Any help would be much appreciated.

To derive the second equation for boiling point elevation, we will start with the first equation and make some assumptions based on the fact that the solution is dilute. The aim is to ultimately obtain the equation:

-ln Xsolv = delta-Hf/R [(1/T) - (1/Tf)]

Assumptions:
1. The solution is dilute, which means the solute concentration is relatively low compared to the solvent concentration.
2. The solution behaves ideally, which implies that the solute-solute and solute-solvent interactions can be neglected.

Derivation:
1. The first equation is:
delta-T(f,b) = K(f,b) * m

We can substitute the freezing point depression/boiling point elevation constant as:
K(f,b) = delta-T(f,b) / m

2. Next, we express molality (m) in terms of solute concentration (Xsolv):
m = moles of solute / mass of solvent (in kg)

Since we assume dilute solutions:
moles of solute ≈ moles of solute in the total solution
So, we can write:
moles of solute ≈ Xsolv * moles of solvent

Therefore, molality (m) can be approximated as:
m = (Xsolv * moles of solvent) / mass of solvent (in kg)
= Xsolv * (moles of solvent / mass of solvent)
= Xsolv * molality of the solvent (m°)

We can express molality (m) in terms of solvent moles using molality of the solvent (m°).

3. Substituting the expression for molality (m) in the first equation:
delta-T(f,b) = K(f,b) * (Xsolv * m°)

4. Rearranging the equation, we get:
delta-T(f,b) / Xsolv = K(f,b) * m°

5. Now, we can equate this expression to the natural log of the solute concentration:
ln Xsolv = -delta-T(f,b) / (K(f,b) * m°) [Note: Negative sign used as the solute concentration decreases]

6. Using the relationship for boiling point elevation, we know that:
delta-T(b) = K(b) * m
where K(b) is the boiling point elevation constant.

We can substitute delta-T(b) as:
delta-T(b) = -(dP/dT) * (delta-Hvap / T^2)

dP/dT is the rate of change of vapor pressure with temperature, and delta-Hvap is the enthalpy of vaporization.

7. Now, we can express the rate of change of vapor pressure in terms of solute concentration:
dP/dT = -V2 * (dXsolv/dT)
where V2 is the molar volume of the solvent.

8. Substituting this expression in the boiling point elevation equation:
-(V2 * (dXsolv/dT)) * (delta-Hvap / T^2) = K(b) * m

9. Rearranging the equation and integrating:
dxsolv/Xsolv = -(K(b) * m * V2 * delta-Hvap) * ∫(1/T^2) dT

On integrating and simplifying, we get:
ln Xsolv = -(K(b) * m * V2 * delta-Hvap) * (1/T - 1/Tf)

Here, (K(b) * m * V2 * delta-Hvap) is the super-constant Kb, which contains the values specific to the system under consideration.

10. Finally, substituting the super-constant Kb back into the equation gives us the desired result:
ln Xsolv = delta-b/R [(1/T) - (1/Tb)]

So, to summarize:
1. Start with the first equation: delta-T(f,b) = K(f,b) * m
2. Substitute the freezing point depression/boiling point elevation constant as K(f,b) = delta-T(f,b) / m
3. Approximate molality (m) using solute concentration (Xsolv) and molality of the solvent (m°)
4. Rearrange the equation and equate it to ln Xsolv
5. Use the boiling point elevation equation and substitute the relevant terms for rate of change of vapor pressure
6. Integrate, simplify, and identify the super-constant Kb
7. Substitute the super-constant Kb back into the equation to obtain the desired result: -ln Xsolv = delta-Hf/R [(1/T) - (1/Tf)]