Shortest Wavelength of the Balmer Series:
The Balmer series is a series of spectral lines in the hydrogen atom spectrum that corresponds to electron transitions from higher energy levels to the second energy level (n=2). The shortest wavelength in the Balmer series occurs when the electron transitions from the third energy level (n=3) to the second energy level (n=2). This transition is known as the Hα line.

Shortest Wavelength of the Lyman Series:
The Lyman series is a series of spectral lines in the hydrogen atom spectrum that corresponds to electron transitions from higher energy levels to the first energy level (n=1). The shortest wavelength in the Lyman series occurs when the electron transitions from the second energy level (n=2) to the first energy level (n=1). This transition is known as the Lyman-alpha line.

Ratio of Shortest Wavelengths:
To find the ratio of the shortest wavelengths of the Balmer series to the Lyman series, we need to compare the energy levels involved in the transitions.

The energy levels in the hydrogen atom are given by the formula:
E = -13.6 eV / n^2

For the Balmer series, the electron transitions from n=3 to n=2, giving an energy difference of:
ΔE = E2 - E3 = -13.6 eV / 2^2 - (-13.6 eV / 3^2) = -13.6 eV / 4 + 13.6 eV / 9 = -3.4 eV + 1.511 eV = -1.889 eV

The corresponding wavelength can be calculated using the formula:
λ = hc / ΔE

For the Lyman series, the electron transitions from n=2 to n=1, giving an energy difference of:
ΔE = E1 - E2 = -13.6 eV / 1^2 - (-13.6 eV / 2^2) = -13.6 eV + 3.4 eV = -10.2 eV

The corresponding wavelength can be calculated using the same formula:
λ = hc / ΔE

Calculating the Ratio:
To find the ratio, we can divide the wavelength of the Balmer series by the wavelength of the Lyman series:

Ratio = (λ_balmer) / (λ_lyman)
= (hc / ΔE_balmer) / (hc / ΔE_lyman)
= ΔE_lyman / ΔE_balmer
= -10.2 eV / -1.889 eV
= 5.41

Since we are asked for the ratio of the shortest wavelengths, we can assume that the ratio is rounded to the nearest whole number.

Therefore, the ratio of the shortest wavelength of the Balmer series to the shortest wavelength of the Lyman series is approximately 5:1.

For Lyman series