The Balmer series is a set of spectral lines in the hydrogen atom's energy spectrum that corresponds to electron transitions to the second energy level (n=2). The shortest wavelength in the Balmer series occurs when the electron transitions from an outer energy level to the second energy level.
To find the shortest possible wavelength in the Balmer series, we can use the Rydberg formula:
1/λ = R*(1/n₁² - 1/n₂²)
Where λ is the wavelength, R is the Rydberg constant (approximately 1.097 × 10^7 m⁻¹), and n₁ and n₂ are the initial and final energy levels, respectively. Since we are looking for the shortest wavelength, the initial energy level (n₁) would be any outer energy level, and the final energy level (n₂) would be 2.
Plugging these values into the formula, we get:
1/λ = R*(1/n₁² - 1/2²)
Simplifying further:
1/λ = R*(4 - n₁²) / 4
To find the shortest wavelength, we need to maximize the value of n₁², which means choosing the largest outer energy level. For hydrogen, the largest outer energy level is n=∞.
Substituting n₁ = ∞ into the formula:
1/λ = R*(4 - (∞)²) / 4
Since (∞)² is infinity, we have:
1/λ = R*(4 - ∞) / 4
1/λ = R*(-∞)/4
Since anything multiplied by negative infinity is negative infinity, we have:
1/λ = -∞ / 4
1/λ = -∞ (as λ approaches 0)
Therefore, the shortest possible wavelength in the Balmer series is zero (or nearly zero).
However, it is important to note that this theoretical value is not physically realizable because there is a limit to how close the electron can get to the nucleus. In reality, atoms have a finite size, and the shortest wavelength in the Balmer series is determined by the smallest energy gap that can be achieved within the atom.