What is the de-Broglie wavelength of nitrogen molecule in air at 300K ? Assume that the molecule is moving with the root mean square speed of molecules at this temperature. (Atomic mass of nitrogen `=14.0076U`) Plank's constant=`6.63xx10^(-34)Js` , Boltzmann constant `=1.38xx10^(-23)JK^(-1)`

Explanation

Given , T = 300 , K = `1.38 xx 10^(-23) J//k, m = 28 015 u = 28.0152 xxx 1.67 xx 10^(-27)` kg
` h = 6.63 xx10^(-34) js,` Mean KE of molecules `1/2 "mv"^(2) = 3/2` KT
` v = sqrt((3KT)/m)= sqrt((3xx 1.38 xx10^(-23)xx300)/(28.0152xx1.66 xx10^(-27)))`
` :. v = 516.78 ` m/s
de - Brogile wavelength , `lambda = h/(mv) = (6.63 xx 10^(-34))/(28.0152 xx 1.66 xx10^(-27) xx516.78)=2.75xx10^(11)` m ,
` :. lambda = 0.0275 xx10^(-9) m = 0.028 ` nm