Final answer:
The shortest wavelengths of the Paschen and Pfund spectral series differ by a factor of about 9, with the Pfund series exhibiting the longer wavelength as determined by the Rydberg formula.
Step-by-step explanation:
The ratio of the shortest wavelengths of two spectral series of the hydrogen spectrum, specifically the Paschen and Pfund series, can be described using the Rydberg formula. The shortest wavelength in a series occurs for transitions from infinity to the first energy level within that series (for the Paschen, n₁=3 and for the Pfund, n₁=5).
Given by the formula λ = 1 / R (1 / n₁² - 1 / n₂²), where R is the Rydberg constant, n₁ is the level electrons fall to, and n₂ (→∞) is the level electrons come from. For the Paschen series, the shortest wavelength occurs when n₂ = ∞, denoted by λ₁ = 1 / R (1 / 3²). For the Pfund series, the shortest wavelength is represented by λ₂ = 1 / R (1 / 5²).
To find the ratio of these wavelengths (Paschen's shortest to Pfund's shortest), we divide λ₁ by λ₂: Ratio = λ₂ / λ₁ = (1 / 5²) / (1 / 3²) ≈ 9. This result implies that the Pfund series has a longer shortest wavelength compared to the Paschen series by a factor of about 9.