We know the qualitative relationship between vapor pressure, $P_{\text{v}\text{a}\text{p}}$ and Temperature for a pure liquid. $($a$.$ What is it$?)$ The quantitative relationship is given by

$P_{\text{v}\text{a}\text{p}}=e^{-\frac{H_{\text{v}\text{a}\text{p}}}{RT}}$

There are two other forms of this relationship that are more often used quantitatively. The first one is

$ln{P}_{\text{v}\text{a}\text{p}}=-\frac{H_{\text{v}\text{a}\text{p}}}{R}(\frac{1}{T})+ln$

b$.$ What is the first step in the derivation of equation $2$ from equation $1?$

c$.$ Why do we want to use equation $2$ and in what circumstances do we use this equation?

The other form of this same equation is

$ln\frac{P_2}{P_1}=-\frac{H_{\text{v}\text{a}\text{p}}}{R}(\frac{1}{T_2}-\frac{1}{T_1})$

d$.$ When is it appropriate to use equation $3?$

e$.$ What assumption is made in order to use equations $2$ and $3?$