Physical Chemistry Laboratory II, CHEM 3155.001

April 20, 2012

Introduction and Objective

The experimental objective of this lab was to collect an IR spectrum of gaseous HCl and from it the experimental rotational constant, B, and fundamental vibration frequency, v0, can be calculated(1). The concept of infrared spectroscopy deals with the infrared region of the electromagnetic spectrum. Molecules absorb at specific resonant frequencies that are characteristic of their structure. It will match the frequency of the bond or group that vibrates because molecules are constantly in motion, both intermolecular vibrational and molecular rotational motion. The different modes are only IR active if it is associated with a dipole. Also, infrared absorption or emission can only occur at allowed transition levels. The frequencies of electromagnetic radiation absorbed or emitted by the transition between two of these levels for a diatomic molecule fall within the range of the infrared wavelengths. This allows the transitions to be measured using the method of IR for diatomic molecules, such as HCl. Because only specific transition levels are allowed, it is concluded that the values are quantized and quantum mechanical results are related to molecular motion. To understand the information contained in the HCl spectrum, we must take into account the vibrational and rotational energy levels of the molecule. One way the energy contributions from the various sources within the molecule can be separated and treated as independent contributions, equation 1, is through the Born-Oppenheimer approximation. E=EVIB+EROT= Ev+EJ (1)

Now each of the separate energies can be modeled independently. The rotational energy of the molecule can be modeled as a rigid rotor, which treats molecules as fixed masses on a spinning bar. The exact expression of rotational energy levels for rigid rotors can be obtained by solving the Schrödinger equation, equation 2, where J is the rotational quantum number, I is the moment of inertia, and B is the rotational constant, equation 3. EJ= h28π2IJ+1=ħ22IJJ+1=BhcJJ+1 (2)

B=h8π2cI 3

The other form of the energy is vibrational energy, and it can be modeled by the harmonic oscillator, which treats molecules as balls on a spring. For a classical system of a classical harmonic oscillator, the potential function for the vibration of diatomic molecules is equation 4. V= kx22=kR-Re22 (4)

Using equation 4 in the one-dimensional Schrödinger equation gives the energy of a quantum-mechanical harmonic oscillator, equation 5, where v is the vibrational quantum number, k is the spring constant, μ is the reduced mass, and υ0 is the wavenumber, equation 6. Ev=v+12ħk/μ=v+12hυ0 (5)

υ0= 12πkμ (6)

Using equation 1, equation 2 and equation 5 can be combined into equation 7. Ev,J=v+12hυ0+BJJ+1hc (7)

The change in energy due to the change in transition levels can be measured using equation 8, where in the excited state v=1 and J=J+1 and in the ground state v=0 and J=J. ∆E=Eexcited-Eground (8)

Plugging equation 7 into equation 8 with the appropriate variables gives equation 9, which is the equation for the R branch, or left side, of the experimental spectrum, which reduces to equation 10. ∆E=1+12hv0+BhJ+1J+1+1-0+12hv0-BhJJ+1 (9)

∆E=hv0+Bh2J+2 (10)

Equation 10 can then be rearranged into equation 11, where v = E/hc and v0= v0/h. v=v0+2BJ+1 (11)

Equation 11 is a linear plot of v, the wavenumber versus J+1, where J is the rotational quantum number. The slope of the graph is 2B and the y-intercept is v0. Equation 8 can also be used where in the excited state v=1 and J=J-1 and in the ground state v=0 and J=J. Plugging these variables into equation 7 and equation 8 gives equation 12, which is the equation for the P branch, or right side, of the experimental...