**Introduction.**

**Objective: **Understanding the
equilibrium between liquid and vapor phases is very important for
such practical applications as distillation and vapor pressure
calculation. The purpose of this lab is:

- to learn the relation between enthalpy of vaporization, D
*H*_{vap}, and temperature variation in vapor pressure for different pure liquids and its relation to liquids' structure. - to learn about relation between vapor pressure above mixture of two liquids

The heat of vaporization of a liquid is a useful thermodynamic quantity because it allows calculation of the vapor pressure of a liquid at any temperature.

In this experiment the vapor pressure of four pure liquids
(water, methanol (or ethanol), benzene and CCl_{4}) will
be measured as a function of temperature and their heats of
vaporization, D*H*_{vap},
will be calculated.

**Vapor Pressure of Pure Liquids.**

Two phases (a and b) in equilibrium at constant pressure and temperature have the same Gibbs free energy:

*
G*_{a} = *G*_{b}

Recalling that *dG* = *VdP* - *SdT*
yields:

* V*_{a }*dP* - *S*_{a
}*dT* = *V*_{b
}*dP* - *S*_{b }*dT*

or *dP*/*dT* = (*V*_{b} -
*V*_{a})/(*S*_{b }- *S*_{a}*)* = DS/DV

For a phase transition occurring at constant temperature and
pressure, the definition of entropy *dS* = *dq*/*T*
implies that D*S*_{transition} = D*H*_{transition}/*T*.
Thus, the variations of pressure and temperature along the
phase coexistence line are linked via the Clapeyron equation:

* * (1)

Again, *P* and *T* here are the pressure
and the temperature, respectively; *S*_{x}
and *V*_{x} are the molar entropies and
molar volumes for corresponding phases and D*S*
and D*V *are their changes; D*H* is the change in molar enthalpy,
*i.e. *it is the molar enthalpy of vaporization, D*H*_{vap}, for a
liquid-vapor transition.

Assuming that the molar volume of liquid can be neglected (explain why) and vapor behaves as an ideal gas, the general Clapeyron equation can be rewritten as the Clasius- Clapeyron equation:

* *d
(ln *P*)/d(1/*T*) = - D*H*_{vap}/*RZ
*(2)

Here the compressibility factor:

* **Z*
= *PV*_{gas}/R*T*,
(3)

was introduced, which for the vapor is very close to 1. Then
after integrating the Clasius- Clapeyron equation (2), for
temperature independent D*H*_{vap}
and *Z* = 1, one gets:

ln (*P*/*P*_{o})
= - D*H*_{vap}/*R
*(1/T - 1/T*_{o}*) *(4)* *

This dependence predicts linear slope for ln *P* when
plotted as a function of inverse absolute temperature, 1/*T*.
Some assumptions we've made are not strictly correct:

a) Nonideality of the vapor makes *Z* to differ from
unity and become temperature dependent (explain how), which
should introduce some curvature to the dependence (4).

b) Since enthalpy of both liquid and vapor changes with
temperature, the difference between them, *i.e.* the
enthalpy of vaporization, D*H*_{vap},
is also temperature dependent. After all, above critical
temperature it should zero. Temperature dependence of enthalpy
can be estimated based on heat capacity, *C*_{p},
which, for simplicity can be taken as a constant. Then one can
write:

D*H*_{vap}
(T_{2}) = D*H*_{vap}
(T_{1}) + D*C*_{p}
(T_{2} - T_{1})
(5)

where D*C*_{p}
= *C*_{g} - *C*_{l}
(6)

c) Neglecting molar volume of liquid causes less than 1% error. I suggest you to evaluate it yourself.

It turns out that the effects a) and b) are each quite
significant; however, they very nearly cancel each other out^{1}
and Clausius-Clapeyron plots tend to be quite linear.

**Vapor Pressure of Solutions.**

The vapor pressure of an *ideal solution* of a
nonvolatile solute in a volatile liquid should be given by
Raoult's law:

*
P*_{l }=* x*_{l}*
P*_{l}^{*}
(7)

where *x*_{l} is the mole fraction of
the volatile liquid and *P*_{l}^{*}
is the vapor pressure of the pure volatile liquid. Thus, for an
ideal solution:

a) vapor pressure should follow Raoult'slaw, Eq(7),

b) D*H*_{vap }for
the solution should be the same as D*H*_{vap
}for the volatile solvent.

You will study mixtures between ethanol (or methanol) and glycerol (in proportions 1:1, 2:1 and 3:1) on the same setup and see if the Raoult's law is valid for these solutions.

__Your own project.__

You can come up with your own project within the scope of the setup you use.

**References:**

1. Waldenstrom *et al.*, *Journal of Chemical Education*
**59**, 30 (1982) for a discussion of these effects.

2. GNS (experiment 13) pp.199-207.

*Last updated 1/9/05.*