Question

The carbon monoxide molecule CO has an effective spring constant of \(1860 \mathrm{~N} / \mathrm{m}\) and a bond length of \(0.113 \mathrm{nm}\). Determine four wavelengths of light that CO might absorb in vibration-rotation transitions.

Short answer

The four wavelengths of light that CO may absorb in vibration-rotation transitions can be obtained by computing the respective energy differences. Using Planck's constant and the speed of light, convert these energy differences into wavelengths.

Step-by-step solution

Calculation of the frequency

The frequency of vibration can be determined using the formula \( f = \frac{1}{2\pi} \sqrt{\frac{k}{\mu}} \) where k is the spring constant and \(\mu\) is the reduced mass of the CO molecule, given by \( \mu = \frac{m1*m2}{m1+m2} \). CO is made up of Carbon (12 amu) and Oxygen (16 amu), hence \(\mu = \frac{12*16}{12+16} = 6.857 amu\). Convert this to kg (1 amu = 1.66*10^-27 kg). Now, compute the frequency.

Calculating the moment of inertia and the rotational constant

Compute the moment of inertia I for the CO molecule using the bond length r and the reduced mass \(\mu\). The moment of inertia is given by \(I = \mu * r^2\). The rotational constant B can then be calculated as \(B=h/8π^2Ic\).

Calculation of the Energy levels

For the vibrational and rotational energy levels, use the quantization formula: \(E(v, J) = (v + 1/2)hf + BJ(J+1)\). Calculate the energies for the different vibrational (v) and rotational (J) states.

Determining wavelengths of light absorbed

The energy difference ΔE between energy levels corresponds to the energy of the light absorbed. This energy can be converted to wavelength by using the equation \(E = \frac{hc}{λ}\). Isolate λ to determine the four wavelengths of light the CO molecule can absorb: \(λ = \frac{hc}{ΔE}\).

Key concepts

Spring Constant

The spring constant is a key concept in understanding molecular vibrations. In simple terms, it tells us how stiff or strong a bond is. It is represented by the symbol \(k\) and measured in units of Newtons per meter (N/m). The spring constant is like a measure of the "springiness" of a bond, similar to how a stiff spring differs from a loose spring.

When it comes to molecules, such as carbon monoxide (CO), the atoms are connected by bonds that can stretch and compress like tiny springs. The spring constant helps us quantify this behavior. In the case of CO, the spring constant is given as \(1860 \, ext{N/m}\). This means the CO bond is fairly strong and resistant to stretching.

• The larger the spring constant, the stronger the bond and the higher the frequency of vibration.
• This is crucial in vibration-rotation transitions, where molecules absorb specific energies that match these natural frequencies.

Reduced Mass

Reduced mass is an essential concept when studying diatomic molecules like CO. It helps in simplifying calculations involving two-atom systems. The reduced mass of a system is denoted by \( \mu \), and it is a combination of the individual masses of two atoms forming a molecule. This is given by the formula: \[ \mu = \frac{m_1 m_2}{m_1 + m_2} \] where \( m_1 \) and \( m_2 \) are the masses of the two atoms.

In the carbon monoxide molecule, where carbon is 12 atomic mass units (amu) and oxygen is 16 amu, the reduced mass becomes critical in calculating the frequency of vibration. After finding \( \mu = 6.857 \text{ amu} \), we convert it to kilograms (since 1 amu = \(1.66 \times 10^{-27} \text{ kg}\)).

• Reduced mass is used to make the problem of a two-body system simpler by converting it into an equivalent one-body problem.
• It's pivotal in determining the frequency of vibration using the spring constant.

Moment of Inertia

Moment of inertia is a crucial concept in understanding the rotational aspects of molecules. It is akin to a rotational analogue of mass for linear motion and is critical for calculating rotational energy levels. For a diatomic molecule like CO, the moment of inertia \(I\) can be found using the formula: \[ I = \, \mu r^2 \] where \( \mu \) is the reduced mass and \( r \) is the bond length. The bond length is essentially the fixed distance between the two atoms, which is given in nanometers--in the case of CO, it's \( 0.113 \text{ nm} \).

This parameter directly contributes to determining the rotational constant \(B\), which is then used to calculate the energy levels. The rotational constant \(B\) is found using the formula: \( B = \frac{h}{8 \pi^2 I c} \), where \(h\) is Planck's constant and \(c\) is the speed of light.

• The moment of inertia informs us about the molecule's resistance to change in its rotational state.
• A larger moment of inertia means the molecule rotates more slowly.

Wavelength Calculation

Wavelength calculation is a fundamental step in understanding what specific light wavelengths a molecule can absorb due to vibration-rotation transitions. This involves conversion from energy differences to wavelengths.

Typically, after calculating the energy difference \( \Delta E \) between excited and ground states using vibrational and rotational energy formulas, we can find the corresponding wavelength \( \lambda \). This is achieved through the equation: \[ \lambda = \frac{hc}{\Delta E} \] where \( h \) is Planck's constant and \( c \) is the speed of light.
The formula highlights the inverse relationship between energy and wavelength. Higher energy results in a shorter wavelength and vice versa.

• This calculation allows us to predict which wavelengths or colors of light a molecule will absorb.
• It’s crucial for applications like spectroscopy, where we use light absorption to study molecular structure.