Initial and final states of the electron in a hydrogen atom:
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The initial state of the electron in a hydrogen atom is in the M shell, while the final state is in the L shell.

Centripetal acceleration of the electron:
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The centripetal acceleration of an object moving in a circular path is given by:

a = v^2 / r

where "a" is the centripetal acceleration, "v" is the velocity of the object, and "r" is the radius of the circular path.

Since the electron is moving in a circular orbit around the nucleus, there must be a centripetal force acting on it, which is provided by the electrostatic attraction between the negatively charged electron and the positively charged nucleus.

The magnitude of the centripetal force is given by:

F = (k * q1 * q2) / r^2

where "F" is the force, "k" is the electrostatic constant, "q1" and "q2" are the charges of the interacting particles (in this case, the charge of the electron and the charge of the nucleus), and "r" is the distance between the particles.

Since the force is equal to the mass of the electron times its centripetal acceleration, we can rewrite the equation as:

(m * a) = (k * q1 * q2) / r^2

where "m" is the mass of the electron.

Using the equation for centripetal acceleration, we can rearrange the equation as:

a = (k * q1 * q2) / (m * r^2)

Ratio of initial to final centripetal acceleration:
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To find the ratio of the magnitudes of the initial to final centripetal acceleration of the electron, we need to compare the values of "a" for the initial and final states.

Let's denote the initial state as "a_i" and the final state as "a_f".

Using the equation for centripetal acceleration, we can write:

a_i = (k * q1 * q2) / (m * r_i^2)

a_f = (k * q1 * q2) / (m * r_f^2)

where "r_i" and "r_f" are the radii of the M and L shells, respectively.

Dividing the equation for the initial state by the equation for the final state, we get:

(a_i / a_f) = [(k * q1 * q2) / (m * r_i^2)] / [(k * q1 * q2) / (m * r_f^2)]

Simplifying the expression, we find:

(a_i / a_f) = (r_f^2 / r_i^2)

Since the electron is transitioning from the M shell to the L shell, the radius of the L shell is greater than the radius of the M shell.

Therefore, we can conclude that (r_f^2 / r_i^2) > 1.

Hence, the ratio of the magnitudes of the initial to final centripetal acceleration of the electron is greater than 1.

Among the given options, the only one that satisfies this condition is option D, which states that the ratio is 16:81.