Question

A container is filled with a sample of gas having n molecule with speed

Answer 1

Average speed is given as,

V₍avg₎ = a+2a+3a+......... / n

V₍avg₎ = a (1+2+3+........) / n

V₍avg₎ = a / n . n(n+1) / 2

V₍avg₎ = a (n+1) / 2

Now, root mean square speed is given as,

V₍rms₎ = √ (a²+4a²+9a³+.........) / n

V₍rms₎ = √ a² (1+4+9+............) / n

V₍rms₎ = a √ [n (n+1) (2n+1) / 6n]

V₍rms₎ = a √ [(n+1) (2n+1) / 6]

Now,

V₍avg₎ / V₍rms₎ = a(n+1)/2 . 1/a √ [6/(n+1) (2n+1)] 

V₍avg₎ / V₍rms₎ = √[6(n+1)² / 4(n+1)(2n+1)]

V₍avg₎ / V₍rms₎ = √[3(n+1) / 2(2n+1)]

This is the required answer.