Question

With what velocity should be a particle journey (expressed in, take the value of n to the nearest integer) towards the nucleus of a copper atom to ${10^{ - 13}}$ meter from the nucleus of the copper atom?

Answer 1

Hint: We know that Kinetic energy is defined as the energy possessed by the moving object. The kinetic energy of an object increases with the speed of an object.
The kinetic energy is given by,
$K.E = \dfrac{1}{2}m{v^2}$
Now, we see electric potential energy.
Electrical potential energy is the energy gained by an object when it is moved against the electric field. The potential energy is given by,
$U = K\dfrac{{{q_1}{q_2}}}{r}$
Where,
- K is the coulomb's constant.
- q is the charge.
- r is the distance between the charges.

Complete step by step answer: We know that the alpha particle approaches the copper nucleus due to repulsion it decelerates and its velocity becomes zero after that alpha particle accelerates. Now we find the distance from the nucleus, equate the kinetic energy of an alpha particle and the electrical potential energy of the particle by using the formula as,
$\dfrac{1}{2}m{v^2} = K\dfrac{{{q_1}{q_2}}}{r}\,\,\,\,\xrightarrow{{}}1$
We also know that,
The mass of an alpha particle$ = 4 \times 1.67 \times {10^{ - 27}}Kg$
The coulomb’s constant $ = 9 \times {10^9}N{m^2}/{C^2}$
The charge of an alpha particle $ = 2 \times 1.6 \times {10^{ - 19}}C$
The charge of copper $ = 29 \times 1.6 \times {10^{ - 19}}C$
The distance from the nucleus of the copper$ = {10^{ - 13}}\,m$
Substitute all the known value in equation 1 we get,
$\dfrac{1}{2}\left( {4 \times 1.67 \times {{10}^{ - 27}}\,kg} \right){v^2} = 9 \times {10^9}\,N{m^2}/{C^2}\,\dfrac{{\left( {3.2 \times {{10}^{ - 19}}\,C} \right)\left( {46.4 \times {{10}^{ - 19}}C} \right)}}{{{{10}^{ - 13}}m}}$
$
  {v^2} = 4000.9 \times {10^{10}} \\
  v = 63.25 \times {10^5}m/s \\
$
Therefore, the velocity of an alpha particle is $63.25 \times {10^5}m/s$.

Note: Let us discuss about the potential energy of a molecule as,
Potential energy:
The energy possessed by a molecule when it is in rest is defined as potential energy. The potential energy of a molecule is given by,
$P.E = mgh$
Where,
- m is the mass of an object.
- g is the gravitational constant.
- h is the height of an object.