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  • In a crude model of a rotating diatomic molecule of chlorine, the two chlorine atoms are <img alt="2\times 10^{-10}m" src="https://entrancecorner.oncodecogs.com/gif.latex?2%5Ctimes%2010%5E%7B-

In a crude model of a rotating diatomic molecule of chlorine, the two chlorine atoms are 2\times 10^{-10}m apart and rotate about their

center of mass with angular speed  w= 2\times 10^{12}rad/s. What is the rotational kinetic energy of one molecule of Cl2 which has a molar mass of 70g/mol?

Option: 1

9.07\times 10^{-21}g


Option: 2

2.07\times10^{-21}g


Option: 3

3.09\times10^{-22}g


Option: 4

none


Answers (1)

best_answer

<p>moment of inertia ,&nbsp; I = 2(mr<sup>2</sup>) = 2mr<sup>2</sup></p>

<p>Here ,&nbsp; <img alt="m=\frac{70\times10^{-3}}{2\times6.02\times10^{23}}=5.81\times10^{-26}kg" src="https://latex.codecogs.com/gif.latex?m%3D%5Cfrac%7B70%5Ctimes10%5E%7B-3%7D%7D%7B2%5Ctimes6.02%5Ctimes10%5E%7B23%7D%7D%3D5.81%5Ctimes10%5E%7B-26%7Dkg" /></p>

<p>and&nbsp;<img alt="r=\frac{2.0\times10^{-10}}{2}=1\times10^{-10}m\\ \therefore l=2(5.81\times10^{-26})(1\times10^{-10})^{2}=1.16\times10^{-45}kgm^{2}" src="https://latex.codecogs.com/gif.latex?r%3D%5Cfrac%7B2.0%5Ctimes10%5E%7B-10%7D%7D%7B2%7D%3D1%5Ctimes10%5E%7B-10%7Dm%5C%5C%20%5Ctherefore%20l%3D2%285.81%5Ctimes10%5E%7B-26%7D%29%281%5Ctimes10%5E%7B-10%7D%29%5E%7B2%7D%3D1.16%5Ctimes10%5E%7B-45%7Dkgm%5E%7B2%7D" /></p>

<p><img alt="K_{R}=\frac{1}{2}lw^{2}=\frac{1}{2}\times(1.16\times10^{-45})\times(2\times10^{12})^{2}=2.32\times10^{-21}g" src="https://latex.codecogs.com/gif.latex?K_%7BR%7D%3D%5Cfrac%7B1%7D%7B2%7Dlw%5E%7B2%7D%3D%5Cfrac%7B1%7D%7B2%7D%5Ctimes%281.16%5Ctimes10%5E%7B-45%7D%29%5Ctimes%282%5Ctimes10%5E%7B12%7D%29%5E%7B2%7D%3D2.32%5Ctimes10%5E%7B-21%7Dg" /></p>

<p>At T= 300K,rotational K.E. should be equal to&nbsp;<img alt="\frac{1}{2}kT" src="https://latex.codecogs.com/gif.latex?%5Cfrac%7B1%7D%7B2%7DkT" /></p>

<p><img alt="=\frac{1}{2}\times(1.38\times10^{-23})\times300=2.07\times10^{-21}g" src="https://latex.codecogs.com/gif.latex?%3D%5Cfrac%7B1%7D%7B2%7D%5Ctimes%281.38%5Ctimes10%5E%7B-23%7D%29%5Ctimes300%3D2.07%5Ctimes10%5E%7B-21%7Dg" /></p>

<p>&nbsp;</p>
 

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Pankaj

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