Question

\(\mathrm{P} 18.23 \quad\) The force constant for a \(^{1} \mathrm{H}^{127}\) I molecule is \(314 \mathrm{N} \mathrm{m}^{-1}\) a. Calculate the zero point vibrational energy for this molecule for a harmonic potential. b. Calculate the light frequency needed to excite this molecule from the ground state to the first excited state.

Short answer

a. The zero-point vibrational energy for the given molecule with a harmonic potential is \( E_0 = 4.56 \times 10^{-20} \,\text{J} \). b. The light frequency needed to excite the molecule from the ground state to the first excited state is \( f = 2.18 \times 10^{14} \,\text{Hz} \).

Step-by-step solution

Calculate the reduced mass

We need to calculate the reduced mass (\(\mu\)) for the given molecule. First, we need the masses of the atoms in the molecule, which are: 1H: m1 = 1.007825 u 127I: m2 = 126.90447 u (1 u = 1.66054 × 10^-27 kg) Now, using the formula for the reduced mass: \( \mu = \frac{m_1m_2}{m_1 + m_2} \)

Calculate the frequency of vibration

Now that we have the reduced mass, we can calculate the frequency of vibration (\(\nu\)) using the given force constant (\(k\)) and the formula: \( \nu = \frac{1}{2\pi}\sqrt{\frac{k}{\mu}} \)

Calculate the zero-point vibrational energy

Once we have the frequency of vibration, we can calculate the zero-point vibrational energy (\(E_0\)) using the following formula: \( E_0 = \frac{1}{2}h\nu \) where \(h = 6.626 \times 10^{-34} \,\text{Js}\) is Planck's constant.

Calculate the energy difference and light frequency

We need to compute the energy difference between the ground state (\(E_0\)) and the first excited state (\(E_1\)): \( E_1 = (1 + \frac{1}{2})h\nu \) The energy difference between these two states is: \( E_{diff} = E_1 - E_0 = h\nu \) Finally, we can calculate the frequency of light needed to excite the molecule from the ground state to the first excited state using: \( f = \frac{E_{diff}}{h} \) This will give us the light frequency needed to excite the molecule.

Key concepts

Harmonic Oscillator

In the realm of physics, a harmonic oscillator refers to a system where the force acting on an object is directly proportional to the displacement from its equilibrium position but in the opposite direction. This relationship is often likened to a spring; the further it is stretched or compressed, the stronger the force to return it to the center.

When studying molecules in quantum mechanics, we can model their vibrational motions as harmonic oscillators. This approach simplifies how we predict and calculate molecular behaviors. Even though real molecular vibrations can be more complex, the harmonic oscillator model is a useful approximation for the vibrations of a diatomic molecule, like the hydrogen iodide (HI) molecule in our exercise.

Reduced Mass Calculation

The concept of reduced mass becomes significant when two bodies, such as atoms in a molecule, are orbiting each other or vibrating relative to their center of mass. In such cases, we can simplify the system by considering a single body with the reduced mass, \( \mu \), rather than dealing with both masses separately.

To calculate the reduced mass of a diatomic molecule, we use the formula \( \mu = \frac{m_1m_2}{m_1 + m_2} \) where \( m_1 \) and \( m_2 \) are the masses of the two atoms. The reduced mass quantifies how the mass is 'distributed' in the vibrational motion of the molecule. It is crucial for understanding vibrational dynamics, as we use this value to determine both the vibrational frequency and the energy levels of the molecule.

Vibrational Frequency

Vibrational frequency, denoted by \( u \), is the rate at which the atoms in a molecule vibrate around their equilibrium positions. In a quantum mechanical context, we can calculate this frequency using a formula derived from the classic spring equation, \( u = \frac{1}{2\pi}\sqrt{\frac{k}{\mu}} \), where \( k \) is the force constant (much like the spring constant), and \( \mu \) is the reduced mass of the molecule.

This frequency is crucial for understanding a molecule's vibrational energy levels. It tells us how often a molecule's atoms move back and forth and is a factor in determining the quantized energy states that are permissible within the quantum mechanical model.

Quantum Mechanical Model

In contrast with classical physics, the quantum mechanical model describes systems at the atomic and subatomic levels, where energy is quantized into discrete levels. According to quantum mechanics, the energy of a harmonic oscillator is given by the formula \( E_n = (n + \frac{1}{2})hu \) where \( n = 0, 1, 2, \dots \) denotes the quantized energy levels, or 'vibrational states', and \( h \) is Planck's constant.

The lowest energy state (\( E_0 \)) is known as the zero-point energy, and it is the energy the molecule has even when it is in its ground state, due to the Heisenberg uncertainty principle. No harmonic oscillator can have zero energy, highlighting a fundamental difference between classical and quantum systems. Understanding this model allows us to predict how molecules interact with radiation, including how they absorb specific frequencies of light to move between energy levels.