The vibration in a diatomic molecule can be considered analogous to the stretching and compression o — Analytical Chemistry Chemistry Question
Raman spectroscopy
The vibration in a diatomic molecule can be considered analogous to the stretching and compression of a spring, as shown in Figure I. The strength of this hypothetical spring is expressed by the force constant, k, which is large for strong bonds and small for weak bonds. Quantum mechanical analysis of the vibrational motion of diatomic molecules shows that the vibrational energy can be expressed in discrete values. The vibrational energy Ev is expressed by the following equation:
Here, v is the vibrational quantum number, which can be any integral value 0, 1, 2,…, and µ represents the reduced mass of the molecule (1/µ = 1/m1 + 1/m2 : m1 and m2 are the atomic masses).
When the molecule is irradiated with intense radiation such as laser light, light with energy different from that of the incident radiation is scattered; this optical phenomenon is called Raman scattering. In this optical process, the difference between the energy of the Raman scattering light and the incident laser light is the vibrational energy of the molecule, as shown in Figure II.
Obtain the ratio of the reduced masses of H2, N2, and O2:
µH2 : µN2 : µO2 = a : b : c
Model Answer
µ = m1m2/(m1+m2); therefore, µH2 : µN2 : µO2 = (1×1)/(1+1) : (14×14)/(14+14) : (16×16)/(16+16) = 1/2 : 7 : 8 = 1 : 14 : 16
Wavelength λ (nm) and frequency ν (s-1 = Hz) are used to characterize light (radiation). In spectroscopy, the wavenumber (cm-1), which corresponds to the number of waves per cm, is also employed frequently. Calculate the frequency and wavenumber of green light at 500 nm.
Frequency = c s-1
Wavenumber = d cm-1
Model Answer
ν = c / λ = 3.0 × 10^8 m s-1 / (500 × 10^-9 m) = 6.0 × 10^14 s-1
wavenumber = 1 / λ = 1 / (500 × 10^-7 cm) = 2.0 × 10^4 cm-1
The energy difference between v = 0 and v = 1 for H2 is 4160 cm-1. Obtain the wavelength of Raman scattering light when H2 is irradiated with laser light at 500 nm.
The wavelength of Raman scattering light = e nm
Model Answer
According to the energy-conservation principle, the wavenumber of the Raman scattering light should be 20000 – 4160 = 15840 cm-1. The corresponding wavelength is ~631 nm.
Assuming that the force constant for O2 is twice that for H2, estimate the energy difference between v = 0 and v = 1 for O2. Obtain the wavelength of Raman scattering light when O2 is irradiated with laser light at 500 nm.
The energy difference = f cm-1
The wavelength of Raman scattering light = g nm
Model Answer
The vibrational energy of the oxygen molecule is 1 / (2√2) of the vibrational energy of the hydrogen molecule, that is, ~1475 cm–1. According to energy-conservation principle, the wavenumber of Raman scattering light should be 20000 – 1470 = 18530 cm–1. The corresponding wavelength is ~540 nm.