Solve Ethanol ΔHvap as a Function of Temp at Constant Pressure

[NepToon]

Homework Statement

The Clausius-Clapyeron equation ln(p2/p1) = -ΔH_vap/R (1/t2 - 1/t1) is derived assuming that ΔH_vap is independent of temperature.
Find ΔH_vap as a function of temperature for ethanol at constant external pressure.

Homework Equations

these are not relevant equations but the equations that i tried solving using-
dU = TdS − pdV

The Attempt at a Solution

i tried starting with the equation for dh and integrating it. i also tried to derive using the clapyeron equation dp=(ΔH/tΔv) dt but i keep getting stuck. the other half of the question asks to derive the clausius clapyeron using the ΔH of this question so I am skeptic about the way I have approached this question.

 

[nate808]

To find ΔHvap as a function of temperature, we can use the following steps:

1. Start with the definition of enthalpy: dH = TdS + Vdp

2. Use the Clapyeron equation to write dp = (ΔHvap/TΔv)dt

3. Substitute this expression for dp into the definition of enthalpy, and rearrange the terms to get:
dH = TdS + (ΔHvap/TΔv)dtV

4. Use the definition of entropy, dS = dq/T, to rewrite dS in terms of heat (dq) and temperature (T).

5. Substitute this expression for dS into the equation for dH, and rearrange the terms to get:
dH = dq + (ΔHvap/TΔv)dtV

6. Since the process is at constant external pressure, dq = dHvap. Substitute this into the equation to get:
dH = dHvap + (ΔHvap/TΔv)dtV

7. Integrate both sides of the equation with respect to temperature (t), from t1 to t2:
∫dH = ∫dHvap + (ΔHvap/TΔv)∫dtV

8. Simplify the integral on the right side to get:
H2 - H1 = ΔHvap + (ΔHvap/TΔv)(V2 - V1)

9. Use the ideal gas law, pV = nRT, to rewrite V in terms of pressure (p) and temperature (T).

10. Substitute this expression for V into the equation and rearrange the terms to get:
H2 - H1 = ΔHvap + (ΔHvap/TΔv)(nR/p)(T2 - T1)

11. Finally, use the definition of enthalpy change (ΔH = H2 - H1) to get:
ΔHvap = ΔH - (ΔHvap/TΔv)(nR/p)(T2 - T1)

This is the desired equation for ΔHvap as a function of temperature for ethanol at constant external pressure.