$(a)\;\large\frac{-R^2}{P^2}\qquad(b)\;\large\frac{R^2}{P^2}\qquad(c)\;+1\qquad(d)\;-1$

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- The behaviour of gases has been expressed in terms of various gas laws obtained experimentally i.e Boyle's law , Charles law , Pressure-temperature law , Dalton's law of partial pressure , Graham's law of diffusion and Avogadro's hypothesis . Later on Maxwell derived kinetic equation $PV = \large\frac{1}{3}mnu^2$ , theoretically by assuming the concept of molecules and their motion . The term u represents root mean square speed of molecule.

$PV = RT$ for 1 mole of gas

At constant T : $(\large\frac{\delta V}{\delta P})_T = -\large\frac{V}{P}$

At constant V : $(\large\frac{\delta P}{\delta T})_V = \large\frac{R}{V}$

At constant P : $(\large\frac{\delta V}{\delta T})_P = \large\frac{R}{P}$

$\therefore (\large\frac{\delta P}{\delta T})_V.(\large\frac{\delta V}{\delta T})_P . (\large\frac{\delta V}{\delta T})_T = \large\frac{-R^2V}{P^2V} = -\large\frac{R^2}{P^2}$

Hence answer is (A)

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