Ratio of the radius of the first excited state to the second excited state of hydrogen atom

Introduction
The hydrogen atom is the simplest atom, consisting of a proton and an electron. In its ground state, the electron is in the lowest energy level, or orbital, around the nucleus. When the electron absorbs energy, it can move to higher energy levels, or excited states. The radius of the orbit increases as the energy level increases.

Formula for the radius of an electron orbit
The radius of an electron orbit is given by the formula:

r = n2h2/4π2meke2

Where:
n = the principal quantum number
h = Planck's constant
me = the mass of the electron
ke = Coulomb's constant

Calculating the ratio of the radii of the first and second excited states
To calculate the ratio of the radii of the first and second excited states, we need to calculate the radius of each state using the above formula and then divide the two radii.

For the first excited state, n = 2.
For the second excited state, n = 3.

We can calculate the radii as follows:

r1 = 22h2/4π2meke2 = 0.529 × 10-10 m
r2 = 32h2/4π2meke2 = 1.18 × 10-10 m

The ratio of the radii is:

r2/r1 = 1.18 × 10-10 m / 0.529 × 10-10 m = 2.23

Therefore, the ratio of the radius of the first excited state to the second excited state of the hydrogen atom is 2.23.

Conclusion
The ratio of the radius of the first excited state to the second excited state of the hydrogen atom is 2.23. This ratio can be calculated using the formula for the radius of an electron orbit and the values of the principal quantum numbers for the two states.