If m number of lines of Balmer series lie in paschen series range of hydrogen spectrum and n number of lines of paschen series lie in Brackett series range then find (m + n)

To tackle the problem of finding the sum of the number of lines in the Balmer and Paschen series that fall within the specified ranges of the hydrogen spectrum, we first need to understand the series and their respective ranges. The Balmer, Paschen, and Brackett series are all part of the hydrogen emission spectrum, each corresponding to electron transitions between different energy levels.

Understanding the Series

The hydrogen spectrum is categorized into several series based on the electron transitions:

• Balmer Series: Transitions that end at the second energy level (n=2). The wavelengths of these transitions fall in the visible range.
• Paschen Series: Transitions that end at the third energy level (n=3). The wavelengths of these transitions are in the infrared range.
• Brackett Series: Transitions that end at the fourth energy level (n=4). These wavelengths are also in the infrared range but at longer wavelengths than the Paschen series.

Finding m and n

Now, let's analyze the conditions given in the question:

• m number of lines of the Balmer series lie in the Paschen series range.
• n number of lines of the Paschen series lie in the Brackett series range.

To find the values of m and n, we can use the formula for the wavelengths of the lines in these series, which is derived from the Rydberg formula:

1/λ = R_H (1/n1² - 1/n2²)

Where:

• λ is the wavelength of the emitted light.
• R_H is the Rydberg constant (approximately 1.097 x 10^7 m^-1).
• n1 is the lower energy level, and n2 is the higher energy level.

Calculating m

For the Balmer series, the transitions are from n=3, 4, 5, ... to n=2. The wavelengths for these transitions can be calculated, and we need to check which of these wavelengths fall within the Paschen series range (n=3 to n=∞). The first few lines of the Balmer series are:

• n=3 to n=2 (H-alpha)
• n=4 to n=2 (H-beta)
• n=5 to n=2 (H-gamma)
• n=6 to n=2 (H-delta)

Since the Balmer series is in the visible range, we can conclude that all lines (m = 4) fall within the Paschen series range, which is in the infrared.

Calculating n

For the Paschen series, the transitions are from n=4, 5, 6, ... to n=3. The first few lines of the Paschen series are:

• n=4 to n=3 (P-alpha)
• n=5 to n=3 (P-beta)
• n=6 to n=3 (P-gamma)
• n=7 to n=3 (P-delta)

These lines are in the infrared range, and since the Brackett series starts at n=4, we find that all lines of the Paschen series (n = 4) fall within the Brackett series range.

Final Calculation

Now that we have determined:

• m = 4 (lines of the Balmer series in the Paschen range)
• n = 4 (lines of the Paschen series in the Brackett range)

We can find the total:

m + n = 4 + 4 = 8

Thus, the final answer is that the sum of m and n is 8.