By Sunil Bhardwaj

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According to Kinetic Gas equation, $$ PV = \frac { 1 }{ 3 } { mnu }^{ 2 }= \frac { 2 }{ 3 } \frac { 1 }{ 2 } { Mu }^{ 2 } $$ Where \(m \times n = M\) is the total mass of gas. But \(\frac { 1 }{ 2 } { Mu }^{ 2 } =\) Kinetic Energy of the gas, $$ \therefore PV = \frac { 2 }{ 3 } KE \qquad ...(1) $$ From the postulates of Kinetic Theory of gases, $$ KE \propto Absolute \ temperature (T) $$ $$ KE = kT $$ where k is constant of proportionality. Putting these values in eq (1) $$ \therefore PV = \frac { 2 }{ 3 } kT \qquad ...(2) $$ as \(\frac { 2 }{ 3 }\) and k is constant we can say $$ PV \propto T \qquad ...(3) $$ This is Boyles Law.

From eq 2 $$ PV = \frac { 2 }{ 3 } kT $$ $$ \therefore \frac { V }{ T } =\frac { 2 }{ 3 } \frac { k }{ P } $$ Now \(\frac { 2 }{ 3 } \) is a constant quantity, k is also constant, therefore, if P is kept constant, $$ \therefore \frac { V }{ T } =Constant $$ This is Charles law.

Consider any two gases. According to kinetic gase quation, For 1st gas $$ { P }_{ 1 }{ V }_{ 1 }=\frac { 1 }{ 3 } { m }_{ 1 }{ n }_{ 1 }{ u_{ 1 } }^{ 2 } $$ For 2nd gas $$ { P }_{ 2 }{ V }_{ 2 }=\frac { 1 }{ 3 } { m }_{ 2 }{ n }_{ 2 }{ u_{ 2 } }^{ 2 } $$ where P=Pressure of the gas
V=Volume of the gas
m=Mass of each molecule of the gas
n=Total number of molecules of the gas
u=Root mean square velocity of the molecules of gas.
If conditions of pressure and volume are similar for both the gases, then, \({ P }_{ 1 }{ V }_{ 1 }={ P }_{ 2 }{ V }_{ 2 }\) $$ \therefore \frac { 1 }{ 3 } { m }_{ 1 }{ n }_{ 1 }{ u_{ 1 } }^{ 2 }=\frac { 1 }{ 3 } { m }_{ 2 }{ n }_{ 2 }{ u_{ 2 } }^{ 2 } $$ $$ \therefore \frac { 2 }{ 3 } { n }_{ 1 }\frac { 1 }{ 2 } { m }_{ 1 }{ u_{ 1 } }^{ 2 }=\frac { 2 }{ 3 } { n }_{ 2 }\frac { 1 }{ 2 } { m }_{ 2 }{ u_{ 2 } }^{ 2 } $$ $$ \therefore \frac { 2 }{ 3 } { n }_{ 1 }KE_{ 1 }=\frac { 2 }{ 3 } { n }_{ 2 }{ KE }_{ 2 } $$ $$ \therefore { n }_{ 1 }KE_{ 1 }={ n }_{ 2 }{ KE }_{ 2 } $$ Further, according to one of the postulates of the kinetic theory of gases, Average K.E. of a gas \(\propto\) ?absolute temperature. Since conditions of temperature are similar in both the cases, $$ KE_{ 1 }={ KE }_{ 2 } $$ $$ \therefore { n }_{ 1 }={ n }_{ 2 }\qquad ...(5) $$ This is Avogadros law.