Final answer:
The relationship between equations like 1.7 and 1.8, in general, lies in the application of Schwartz's theorem, where mixed partial second derivatives can be obtained from the total differential.
Step-by-step explanation:
In physical chemistry, the total differential (equation 1.7) represents the small change in a function resulting from changes in multiple variables. This concept is crucial for understanding how different variables influence a system. Now, Schwartz's theorem comes into play when dealing with mixed partial second derivatives, as mentioned in equation 1.8. It states that for continuous functions, the order of differentiation doesn't matter. Specifically, in the context of physical chemistry, if you have the total differential of a function, you can use Schwartz's theorem to find the mixed partial second derivatives. This is particularly useful for analyzing complex systems where multiple variables are at play.
This mathematical tool simplifies the analysis of systems involving changes in multiple variables. Understanding this relationship between total differentials and mixed partial second derivatives enhances your grasp of the intricacies of physical chemistry.