Question

How do you generalize $\Delta (PV)\,and\,d(PV)$ ?

Answer 1

The matter is divided basically into the three states of solid, liquid and gas. The properties and the existence of the matter can be expressed and explained in the form of mathematical expressions and the rules of mathematics help in the simplification of the properties and the nature of matter.

Complete step by step answer:
The elements in nature exist in one of the three states, solid, liquid or gas. The gases are the freest in their mobility while the solids are the most rigid. All these properties arise from the intermolecular forces of attraction that act between the molecules and atoms of the elements. All the reactions that take place are along with a specific ratio in which the elements react.
The changes and the properties of the matter can be expressed in the form of mathematical expressions. The expressions can be further processed as the rules of mathematics guide.
Upon working with the differential equations we can see that the expression of $d(PV)$ may be expressed in the form of the product rule as
$d(PV) = PdV + VdP$
Which is the rule which is used for the differentials.
But when it comes to the delta function, then things change and the rules of differential don’t work.
So, $\Delta PV \ne {P_1}\Delta V + {V_1}\Delta P$
The delta function cannot be expressed in the terms of the differentials and the delta here is representative of the change from one state to the other.
The actual definition of the delta function is as
$\Delta (PV) = {P_2}{V_2} - {P_1}{V_1}$
Here the delta represents the change in the value.

Note: S.T.P is the acronym for the standard temperature and pressure. It defines the state in which the temperature and pressure are defined and the volume taken by any gas is found to be 22.4 L.
1 mole of all the gases at S.T.P is found to acquire a volume of 22.4L.