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A container is filled with a sample of gas having n molecules with speed a,2a,3a.....na .The ratio of average speed to root mean square speed is

A container is filled with a sample of gas having n molecules with speed a,2a,3a.....na.The ratio of average speed to root mean square speed is

Grade:12

1 Answers

Saqib
11 Points
6 years ago
Average speed is given as,V₍avg₎ = a+2a+3a+......... / nV₍avg₎ = a (1+2+3+........) / nV₍avg₎ = a / n . n(n+1) / 2V₍avg₎ = a (n+1) / 2Now, root mean square speed is given as,V₍rms₎ = √ (a²+4a²+9a³+.........) / nV₍rms₎ = √ a² (1+4+9+............) / nV₍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)] @ thanks

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