An atom of carbon has a radius of 67.0 pm and the average orbital speed of the electrons in it is about 1.3 x 10⁶ m/s.
calculate the least possible uncertainty in a measurement of the speed of an electron in an atom of carbon. write your answer as a percentage of the average speed, and round it to significant digits.

Answer :

The least possible uncertainty in an electron's velocity is,

The percentage of the average speed is, 33 %

Explanation :

According to the Heisenberg's uncertainty principle,

...........(1)

where,

= uncertainty in position

= uncertainty in momentum

h = Planck's constant

And as we know that the momentum is the product of mass and velocity of an object.

or,

     .......(2)

Equating 1 and 2, we get:

Given:

m = mass of electron =

h = Planck's constant =

radius of atom =    

= diameter of atom =

Now put all the given values in the above formula, we get:

The minimum uncertainty in an electron's velocity is,

Now we have to calculate the percentage of the average speed.

Percentage of average speed =

Uncertainty in speed =

Average speed =

Percentage of average speed =

Percentage of average speed = 33.2 % ≈ 33 %

Thus, the percentage of the average speed is, 33 %

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