Question

The wavelengths of spectral lines depends to some extent on the mass of the nucleus. This occurs because the nucleus is not an infinitely heavy mass that remains stationary. In fact, in reality, both the nucleus and the electron orbit about their common center of mass. It can be shown that a system of this type is entirely equivalent to a single object with a mass μμ (called the Reduced Mass) that orbits about the location of the nucleus at a distance equal to the separation between the nucleus and the electron. In order to account for the motion of the nucleus in the Bohr Model, one must replace the electron mass m_e with the Reduced Mass μμ in the Balmer Formula and the Allowed Energies Formula. The Reduced Mass is given by μ = m_eM/(m_e + M), where M = mass of the nucleus. By replacing the electron mass m_e with the Reduced Mass μμin the Balmer Formula (inside the Rydberg Constant), calculate the correct value for the wavelength emitted when a Hydrogen atom makes a transition from the n = 5 to the n = 3 state (the first Balmer line). In order to replace the electron mass m_e with the Reduced Mass μ inside the Rydberg Constant, you should multiply R by μ/m_e(you can verify this by looking at the definitions of R and a₀). The mass of the hydrogen nucleus is just the mass of the proton (M = m_p= 1.007276 u, where u = atomic mass unit). The mass of the electron is m_e = 0.000548580 u. Keep 6 significant figures in your answer.

How does the result from above compare to the wavelength that is obtained without using the reduced mass? Treat the result with the reduced mass as the accepted value and find the Percent Error (in %) between these two calculations for the wavelength. Keep 3 significant figures in your answer.

Using the Reduced Mass, calculate the wavelength emitted by a Deuterium atom (H – 2) when it undergoes the same transition from the n = 5 to the n = 3 state. Deuterium is an isotope of Hydrogen with one neutron in the nucleus. The change in the wavelength for the heavier isotope is known as the Isotope Shift in spectroscopy. The mass of the deuterium nucleus is just the mass of the proton & neutron (M = m_D = 2.014102 u). Keep 6 significant figures in your answer.

How does the wavelength for Hydrogen compare to the wavelength for Deuterium? Is the Isotope Shift significant? Treat the Deuteriumresult as the accepted value and find the Percent Error (in %) between these two calculations for the wavelength. Keep 3 significant figures in your answer.

Answer 1