Define most probable speeds of gas molecule. Give their interrelationship.

Explanation

a) RMS speed :
The square root of mean of the squares of the speeds of all molecules of a gas is known as RMS speed `(u_(RMS))`
` u_(rms) = sqrt((( u_(1)^(2)) +U_(2)^(2) + " ….." +u_(n)^(2))/( n_1 +n_2 +……))`
` u_(rms) = sqrt((3RT)/(M ) = sqrt(( 3PV )/(M ) = sqrt((3p )/(d))`
b) Average speed :
The arithematic mean of speeds of gas molecules is known as average speed `(u_(av))`
` u_("average ") = (u_1 +u_2 + ..... +u_n)/( n_1 +n_2 +......)`
` u_(av)= sqrt((8RT)/(pi M )) = sqrt((8 PV )/( pi M )) = sqrt(( 8 P)/(pi d))`
c) Most probable speed:
The speed possessed by the maximum number of molecules of the gas is known as most probable speed `(u_(mp))`
`u_(mp) = sqrt((2RT)/( M ) = sqrt((2P )/(d))`
inter relationships : -
` to U_(mp ) : u_(av) : I_(rms) = sqrt((2RT)/(M )) : sqrt(( 8RT)/( pi M )) : sqrt((3 RT)/(M))`
`= 1 : 1.128 : 1.224 `
` to u_(av) = 0.9213 xx u_(rms)`
` to u_(mp) = 0.8166 xx u_(rms)`