The Antoine equation often is used to describe vapor pressur

The Antoine equation often is used to describe vapor pressures: lnP = A B /(T + C) . For temperatures near the normal boiling point:

a. Derive an expression that gives the behavior of Hvap as a function of temperature which is thermodynamically consistent with the Antoine equation.

Solution

The Antoine equation is a mathematical expression of the relation between the vapour pressure and the temperturer of pure substances. The equation was first proposed by Ch. Antoine, a french researcher, in 1888. The basic form of the equation is:

lnP = A - B / ( T + C )

and it can be transformed into this temperature-explicit form:

T = B ( A - lnP ) - C

where: P is the absolute vapor pressure of a substance

T is the temperature of the substance

A, B and C are substance-specific coefficients (i.e., constants or parameters

log is typically either log₁₀ or log_e

A simpler form of the equation with only two coefficients is sometimes used:

lnP = A - ( B \\ T )

which can be transformed to:

T = B ( A - lnP )

The Antoine equation is a semi-empirical equation which expresses vapour pressure as a function of temperature. A new, rapid and highly accurate method for obtaining its three constants from experimental data is presented and applied to ethanol, water and 14 anaesthetic substances where p is the vapour pressure, T is temperature and A, B and C are component-specific constants.

Usually, the Antoine equation cannot be used to describe the entire saturated vapour pressure curve from the triple point to the critical point, because it is not flexible enough. Therefore, multiple parameter sets for a single component are commonly used. A low-pressure parameter set is used to describe the vapour pressure curve up to the normal boiling point and the second set of parameters is used for the range from the normal boiling point to the critical point.

The Antoine Equation is a vapor pressure equation and describes the relation between vapor pressure and temperature for pure water between 0°C and 373.946°C.

For T in Kelvins and P in kPa:

273.150 < T 373.150:     A = 16.5699, B = 3984.92, C = 39.724
373.150 < T 647.096:     A = 16.7285, B = 4169.84, C = 28.665

Antoine(T) = exp(A B / (T + C))

For temperatures above the critical temperature (373.946°C / 647.096K), where water is a superfluid, the increase in vapor pressure appears essentially linear† with the same slope (i.e. dP/dT 235.88 kPa/°C). This behavior appears consistent with the Ideal Gas Law and statements that supercritical water behaves like an ideal gas. Now if we define Tc = 647.096 K, we can write

  P(T Tc) = Antoine(T)
P(T > Tc) = 235.88 (T Tc) + Antoine(Tc)

Hence, the behaviour of delta hvap as a function of temperature which is termodynamically consistent with the the antoine equation is derived .