Why is Clapeyron equation so important? Context: I'm studying basic thermodynamics. My textbook has a chapter on the Clapeyron equation which, as a reminder, is given by the following formula:
\begin{equation}
\frac{dP}{dT} = \frac{\Delta H_{a,b}}{T\Delta V}
\end{equation}
(a,b being the different phases)
Question: The equation describes the tangents of the coexistence curves in $P,T$ diagrams, but why is it so important ?


*

*Does indeed encompass some fundamental thermodynamics knowledge in it (that I am obviously missing) and/or

*Is it useful at calculating thermodynamic properties of substances[*] and/or

*Has it been artificially emphasized in our course, so that it serves as a question in the exams?


Possible answer: I've found in my lecture notes that "Clapeyron equation is the most important experimental confirmation of the 2nd law of thermodynamics". How would you comment on that ?

[*] I watched an online MIT lecture on Clapeyron (that SE won't let me link) and it had a nice example with an hypothetical RDX explosive detector and the least number of molecules that should detect, based on the vapor pressure of RDX in room temperature.
 A: You only gave 3 options as possible answers, but there are others which emphasize the importance of the Clapyeron equation.  These all relate to industrial processes.  If you are designing or operating a piece of equipment that involves vapor and liquid phases present at elevated temperatures, you need to know the relationship between pressure and temperature in order to properly design the equipment so that it won't burst, and so that the proper amount of heat transfer occurs.  Another example is in humidification and drying operations.  A very important example is in distillation columns in which you are trying to select the number of distillation trays to achieve a desired separation.  If you are using Raoults law to represent the vapor liquid equilibrium behavior of the system (in doing material and energy balances), you need to know the equilibrium vapor pressure vs temperature for each material in the mixture, since the partial pressure of each substance in the vapor is equal to its equilibrium vapor pressure at the liquid temperature times the mole fraction in the liquid.  So, in short, there are extremely important industrial applications for the Clapeyron equation.  The world does not begin and end at the laboratory door of a university.
