Derive the gas laws from the kinetic gas equation.

Explanation

a) Deduction of. Boyle.s law:
Kinetic gas equation is `PV = 1/3 m n u^2`
` PV = 1/3 m n u^2 `
` =1/3 xx 2/2 m n u^2`
` = 2/3 xx 1/2 m n u^2`
` PV = 2/3 `[ kinetic energy (KE ) ] ` [ :. KE_( " .n . moles") =1/2 m n u^2 ]`
` PV = 2/3 KT [ :. KE =KT ` According to kinetic theory ]
` PV= 2/3 KT `
According to Boyle.s law T is constant
` therefore PV = 2/3 ` ( constant )
` therefore PV = ` constant
` (or)`
`P prop (1)/(V)`
Hence Boyle.s law proved from kinetic gas equation.
Deduction of Charle.s law:
Kinetic gas equation is ` PV = 1/3 m n u^2 `
`PV = 1/3 m n u^2`
` =2/3 xx 1/2 m n u^2`
` =2/3 (KE ) ` [ Kinetic energy (KE) `= 1/2 m n u^2`]
` PV = 2/3 KT ` [According to kinetic theory KE = KT]
` (V )/(T ) = 2/3 xx (K )/( P)`
According to Charle.s law .P. is constant.
` therefore (V )/(T )` constant (or) V` prop `T
Hence Charle.s law proved from kinetic gas equation.
c) Graham.s law : At constant temperature and pressure the rate of diffusion of a gas is inversely proportional to the square root of its dersity, ` r prop (1)/( sqrt(d))`
Deduction : Kinetic gas equation is
` PV = 1/3 m n u^2 = 1/3 M u^2`
` u^2 = (3 PV)/( M ) = (3P )/(d )`
` therefore u = sqrt((3 P)/(d ))`
At constant pressure u =` K. (1)/( sqrt(d))`
` K=` constant , or ` u prop (1)/( sqrt(d))`
` :. ` The rate of diffusion of gases depends upon the velocity of the gas molecules.
So ` r prop (1)/( sqrt(d))`
This is Graham.s law.
d) Dalton.s law of partial pressures : The total pressure exerted by a mixture of chemically non .reacting gases at given temperature and volume, is equal to the sum of the partial pressures of of the component gases.
Deduction :
Consider a gas present in vessel of volume =V
no. of molecules` = n_1`
mass of each molecule` =m_1`
RMS velocity ` =u_1`
According to kinetic gas equation the pressure of the gas. ` P_1 = 1/3 (m_1 n_1 u_(1)^(2))/( V )`
When this gas is replaced by another gas in the same vessel, ` P_2 =1/3 (m_2 n_2 u_(2)^(2))/(V )`
When these two gases are mixed in the same vessel
the total pressure of the mixture is ` P=1/3 (m_1 n_1 u_(1)^(2))/(V ) +1/3 (m_2 n_2 u_(2)^(2))/(V ) =p_1 +P_2`
` therefore P=P_1 +P_2`
This is Dalton.s law of partial pressures.