Final answer:
According to Bohr's theory, an electron's orbital velocity decreases as its orbit radius increases. Comparing revolutions of electrons at radii R and 4R shows that the electron at 4R takes 16 times longer to complete one revolution. None of the provided answer choices match this calculation, indicating a potential error in the options.
Step-by-step explanation:
The planetary model of the atom, which is referenced in the question, allows us to consider electrons orbiting around the nucleus in a manner somewhat similar to planets orbiting a star. According to this model and Bohr's theory, the velocity of an electron in orbit is inversely proportional to the radius of its orbit. That is, an electron in a smaller orbit moves faster than one in a larger orbit.
In this scenario, we're comparing two electrons, with one orbiting at a radius R and the other at a radius 4R. Using the relation that the velocity of the electron is inversely proportional to the radius, the electron at radius R will move faster than the electron at radius 4R. Furthermore, since the circumference of their orbits are proportional to their respective radii, the time taken for one revolution will be proportional to the radius of the orbit if the speed remains constant. However, here, as the radius increases, the speed decreases. The electron orbiting at 4R has a path four times longer to complete, but it also moves slower due to the larger orbit radius.
Since the velocity of an electron orbiting in a Bohr model is inversely proportional to the orbit radius r, and the circumference C of an orbit is directly proportional to r (C=2πr), the time t to complete an orbit is the ratio of the circumference to the velocity t=C/v. Thus for an electron in an orbit four times the radius, its velocity is one-fourth, and its path is four times longer, which implies that the time to complete one revolution is 4 times 4, which is 16 times longer than that of the electron in the smaller orbit.
Therefore, the ratio of the time taken by them to complete one revolution is 1:16, as the electron in the larger orbit (4R) takes 16 times longer than the electron in the smaller orbit (R). Given the provided options A through D, none correctly match this ratio, suggesting that there might be an error with the given choices.