Question

The Clausius-Clapeyron equation and the Antoine equation. The vapor pressure of a liquid can be written in the empirical form known as the Antoine equation, ln (p / p ◦ ) = A1 − (A2 /(T + A3)) where A1, A2, and A3 are empirical constants determined from measurements, and T is the temperature in Kelvin.

(a) Starting with this equation, derive an equation for the enthalpy of vaporization as a function of temperature.

(b) For ethanol (l), A1 = 23.58, A2 = 3.67 × 103 , A3 = −46.702, and p ◦ = 1 Pa. Given that the standard pressure is 1 bar = 1.01325 × 105 Pa, what is the standard boiling temperature?

(c) Calculate ∆h at 298 K and at the standard boiling temperature.

Answer 1

answer
(a). The vaporization curves of most liquids have similar shape. The vapour pressure steadily increase as the temperature increases. A good approach is to find a mathematical model for the pressure increase as a function of temperature. Experiments showed that the pressure P, enthalpy of vaporization, DH_vap, and temperature T are related,

P = A exp (- DHvap / R T)

where R (= 8.3145 J mol^-1 K^-1) and A are the gas constant and unknown constant. This is known as the Clausius- Clapeyron equation. If P₁ and P₂ are the pressures at two temperatures T₁ and T₂, the equation has the form:

     P1    DHvap    1     1
ln (---) = ----  (--- - ---)
     P2R      T2T1

The Clausius-Clapeyron equation allows us to estimate the vapor pressure at another temperature, if the vapor pressure is known at some temperature, and if the enthalpy of vaporization is known.