(a) Charle's Law: It states that if pressure (P) remains constant, the volume (V) of a fixed amount of gas (n) is directly proportional to its absolute temperature (T).

V \(\propto\) T (P Constant)

Or \(\frac{V}{T}\) = constant or \(\frac{V_1}{T_1}=\frac{V_2}{T_2}\)

Charles also found that for a given mass of gas, if pressure is kept constant, the volume increases linearly with temperature.

V = V_o (1 + \(\propto\)t), where \(\propto\)= \(\frac{1}{273}\) and,

V_o is the volume at 0ºC

Volume at temperature T is

V_T = V_o\(\big(1+\frac{t}{273}\big)\) = V_o\(\big(\frac{273+t}{273}\big)\)

Where t is temperature in Celsius

Or V_T = \(\frac{V_o}{273}\times T\)

Where, T = 273 + t, T is the temperature on Kelvin scale.

(b) PV = nRT

Or \(\frac{n}{V}=\frac{P}{RT}\)

Since n = \(\frac{m}{M}\)

\(\therefore\) \(\frac{m}{MV}\)= \(\frac{P}{RT}\)

\(\therefore\) \(\frac{d}{M}\) = \(\frac{P}{RT}\) \(\big(\therefore d=\frac{m}{V}=density\big)\)

or d = \(\frac{MP}{RT}\)

\(\therefore\) d \(\propto\) P