The conditions and assumptions for the Antoine equation?

[Ortanul]

Homework Statement

Derive the basic relationship that the Antoine equation represents. Most importantly, explain the underlying condition when the Antoine equation applies and the underlying assumptions for the Antoine equation to be valid.

Homework Equations

Clausius-Clapeyron Equation: dP_sat/dT=ΔH/TΔV
Antoine Equation: lnP^sat=A-B/(T+C)

The Attempt at a Solution

Assume ΔV=V_gas-V_liq≈Vgas
From the Clausius-Clapeyron Equation, dP_sat/dT=ΔH/(T*nRT/P)=ΔH/R * P/T²
Rearrange: dP_sat/P=ΔH/R * dT/T²
Perform the integration, lnP_sat=A-ΔH/RT=A-B/T, A, B are the constant

I think this should be the basic relationship of the Antoine Equation, even though C is not involved in the equation, as Antoine Equation is an empirical relationship.
However, I don't know what kind of conditions and assumptions I should make before using the Antoine Equation, and they are not explicitly stated in my textbook. Should I consider ΔV=V_gas-V_liq≈V_gas as one of the assumptions?
Any help will be appreciated!

 

[haruspex]

I doubt it is valid to replace ΔV with V, as you have. That may explain the non-appearance of C.

 

[Chestermiller]

You need to work backwards. Start with the Antoine equation, and take the derivative with respect to T. Then compare you result with what you get from the Clausius-Clapeyron equation. Then you will see how ΔH is related to the constants in the Antoine equation.

 

[Chestermiller]

I doubt it is valid to replace ΔV with V, as you have. That may explain the non-appearance of C.

A key approximation in the derivation of the Clausius-Clapeyron equation is to neglect the specific volume of the saturated liquid in comparison to the specific volume of the saturated vapor. So ΔV is taken as the specific volume of the saturated vapor V.

 

[Ortanul]

A key approximation in the derivation of the Clausius-Clapeyron equation is to neglect the specific volume of the saturated liquid in comparison to the specific volume of the saturated vapor. So ΔV is taken as the specific volume of the saturated vapor V.

You need to work backwards. Start with the Antoine equation, and take the derivative with respect to T. Then compare you result with what you get from the Clausius-Clapeyron equation. Then you will see how ΔH is related to the constants in the Antoine equation.

I'll try to derive the equation again later. Thank you for your help!
Apart from that, if Antoine Equation can be fully derived from Clausius-Clapeyron Equation, I wonder why it is a empirical equation as stated on my book. Besides, could you please tell me if there is any other necessary assumption for the Antoine Equation itself apart from the negligible volume of liquid?

 

[Chestermiller]

I'll try to derive the equation again later. Thank you for your help!
Apart from that, if Antoine Equation can be fully derived from Clausius-Clapeyron Equation, I wonder why it is a empirical equation as stated on my book. Besides, could you please tell me if there is any other necessary assumption for the Antoine Equation itself apart from the negligible volume of liquid?

The Antoine equation assumes a particular functional form for the effect of temperature T on the heat of vaporization ΔH. You can see what that functional form is by taking the derivative of lnP_sat with respect to T, and then setting that equal to the derivative of lnP_sat with respect to T from the Clausius-Clapeyron equation. This will give you the functional form they assume for ΔH vs T in the Antoine equation.

Chet