Question

The key feature of Bohr's theory of spectrum of hydrogen atom is the quantization of angular momentum when an electron is revolving around a proton. We will extend this to a general rotational motion to find quantized rotational energy of a diatomic molecule assuming it to be rigid. The rule to be applied is Bohr's quantization condition. It is found that the excitation frequency from ground to the first excited state of rotation for the CO molecule is close to \(\frac{4}{\pi} \times 10^{11} \mathrm{~Hz} .\) Then the moment of inertia of CO molecule about its center of mass is close to (Take \(h=2 \pi \times 10^{-34} \mathrm{~J} \mathrm{~s}\) ) A) \(2.76 \times 10^{-46} \mathrm{~kg} \mathrm{~m}^{2}\) B) \(1.87 \times 10^{-46} \mathrm{~kg} \mathrm{~m}^{2}\) C) \(4.67 \times 10^{-47} \mathrm{~kg} \mathrm{~m}^{2}\) D) \(1.17 \times 10^{-47} \mathrm{~kg} \mathrm{~m}^{2}\)

Short answer

The moment of inertia of the CO molecule about its center of mass is close to Option B) \(1.87 \times 10^{-46} \mathrm{kg} \mathrm{m}^{2}\).

Step-by-step solution

Understand the Quantization of Rotational Energy

According to Bohr's model for quantization, the energy levels of rotational states in a rigid diatomic molecule are quantized. The difference in energy between two rotational levels corresponds to the emitted or absorbed energy when transitioning between these levels. In this case, the frequency given is related to the energy difference by the relation known as Planck's equation, which is given by \(E = h u\), where \(E\) is the energy, \(h\) is Planck's constant, and \(u\) is the frequency.

Relate Energy to Rotational Quantum Number

The quantization rule for rotational energy is given by \(E_J = \frac{J(J+1)h^2}{8\pi^2 I}\), where \(E_J\) is the energy of the rotational level with quantum number \(J\), \(h\) is Planck's constant, \(I\) is the moment of inertia, and \(J\) is the rotational quantum number which can be 0, 1, 2, ...

Calculate Energy Difference Between Ground and First Excited State

For the CO molecule transitioning between the ground state (\(J=0\)) and the first excited state (\(J=1\)), the energy difference is the energy of the first excited state as the ground state has zero energy. Using the formula from Step 2 for \(J=1\), we get \(\Delta E = E_1 - E_0 = \frac{2h^2}{8\pi^2 I}\).

Express Energy Difference in Terms of Frequency

The given excitation frequency \(u\) is related to the energy difference \(\Delta E\) by Planck's equation: \(\Delta E = h u\). By substituting the result from Step 3 into Planck's equation, we have\(\frac{2h^2}{8\pi^2 I} = h u\).

Calculate the Moment of Inertia

Solve the equation for the moment of inertia \(I\) using the given planck's constant and frequency values:\[I = \frac{h}{4\pi^2 u}\].Substitute \(h = 2\pi \times 10^{-34} J\cdot s\) and \(u = \frac{4}{\pi} \times 10^{11} Hz\) to solve for \(I\).

Perform the Calculation

Inputting the known values into the formula to solve for \(I\), we get\[I = \frac{2\pi \times 10^{-34} J\cdot s}{4\pi^2 \times \frac{4}{\pi} \times 10^{11} Hz}\], further simplifying, yields\[I = \frac{2\pi \times 10^{-34}}{16\pi \times 10^{11}}\] kg\cdot m^2.This simplifies to \(I = 1.25 \times 10^{-46} kg\cdot m^2\). We can see this is closest to option B.

Key concepts

Quantized Rotational Energy

In the world of quantum mechanics, not all properties can have continuous values; some can only exist in discrete 'packets'. One such property is the rotational energy of molecules in quantum physics, which follows Bohr's quantization condition. This concept is pivotal in understanding the behavior of tiny particles, like atoms and molecules, that make up the matter.

Bohr's model suggests that the rotational motion of a molecule is 'quantized', meaning that the energy can only take on specific, discreet values. For a diatomic molecule, such as the CO molecule in our example, the rotational energy levels are determined by a quantum number which can be any whole number from 0 upwards. The fascinating result is that there's a quantum jump in energy between each rotational state – they are not a smooth continuum.

Imagine energy levels as steps on a staircase. An electron in a molecule can only 'stand' on a step, not between steps, giving rise to distinct energy levels. When a molecule transitions between these energy levels, emission or absorption of energy occurs in precise amounts, corresponding to the energy difference between the levels. And it is this phenomenon that the given excitation frequency of the CO molecule refers to – the energy difference between the ground state and the first excited state.

Moment of Inertia

In physics, the moment of inertia, often symbolized by 'I', is fundamentally the rotational equivalent of mass for linear motion. It quantifies an object's resistance to changes in its rotation rate. For a rigid body, the moment of inertia depends on both the mass distribution of the object and the axis about which it rotates.

For molecules, especially diatomic ones like CO, the moment of inertia is paramount in determining their rotational energy levels. The moment of inertia in rotational motion acts as a pivotal constant that links the mass distribution of a molecule to the energy levels it can possibly occupy when rotating. It's a bit like knowing the distribution of weight in a spinning figure skater – the way they tuck in their arms, bringing mass closer to the center, will change their rotation speed. Similarly, in a diatomic molecule, the distance between the atoms and their masses will concretely affect 'I'.

To calculate the moment of inertia for a diatomic molecule, we consider the two masses rotating about a common center. The further the masses are from the center, the higher the moment of inertia, and the heavier they are, the more the inertia increases as well. The calculation in the exercise using Bohr's quantization condition allows us to pinpoint the value of 'I' for the CO molecule by measuring its rotational energy change.

Planck's Equation

At the turn of the 20th century, Max Planck made a groundbreaking discovery that revolutionized physics: energy is not continuous, but rather comes in discrete units called quanta. This finding is encapsulated in Planck's equation: \(E = hu\), where \(E\) stands for energy, \(h\) is Planck's constant (a fundamental constant of nature), and \(u\) is the frequency of the radiation involved.

Planck's equation highlights the direct proportionality between the energy of a photon and the frequency of the electromagnetic wave associated with it. When applied to rotational energy quantization, it tells us that the energy difference between quantized rotational states is equal to Planck's constant times the frequency of the photon emitted or absorbed during the transition (as seen in our example with CO molecule).

Understanding Planck's equation is critical for interpreting a range of physical phenomena, from the atomic level (as in the Bohr model) to the behavior of light and other forms of electromagnetic radiation. This equation is central to quantum mechanics and has vast implications in understanding how energy transitions occur at the microscopic level. It's the key that allows us to decode the energy spectrum of an atom or molecule by examining the light it emits or absorbs.