Question

The force constant for the \(^{35} \mathrm{Cl}_{2}\) molecule is \(323 \mathrm{N} \mathrm{m}^{-1}\). Calculate the vibrational zero point energy of this molecule. If this amount of energy were converted to translational energy, how fast would the molecule be moving? Compare this speed to the root mean square speed from the kinetic gas theory, \(|\mathbf{v}|_{r m s}=\sqrt{3 k_{B} T / m}\) for \(T=300 .\) K.

Short answer

The vibrational zero point energy for the \(^{35}\mathrm{Cl}_2\) molecule is calculated as \(E = \frac{1}{2}h \cdot \frac{1}{2\pi}\sqrt{\frac{K}{\mu}}\). Using given values, we find the vibrational energy, convert it to translational energy, and calculate the speed using \(v = \sqrt{\frac{2E}{m}}\). Finally, we compare this speed to the root mean square speed \(|\mathbf{v}|_{rms}= \sqrt{\frac{3k_{B}T}{m}}\) for \(T = 300K\).

Step-by-step solution

Calculate the Vibrational Zero Point Energy

Using the formula for the energy of a harmonic oscillator \(E=\frac{1}{2}hf\), where \(h\) is Planck's constant and f is the frequency of the oscillator. The frequency is calculated using the formula \(f = \frac{1}{2\pi}\sqrt{\frac{K}{\mu}}\), where \(K\) is the force constant and \(\mu\) is the reduced mass. For the \(^{35}\mathrm{Cl}_2\) molecule (mass 35u), the reduced mass \(\mu = \frac{35}{2}\). Thus, the vibrational zero point energy is: \[E = \frac{1}{2}h \cdot \frac{1}{2\pi}\sqrt{\frac{K}{\mu}}\]

Insert given values

Given: \(h = 6.63 \times 10^{-34} Js\), \(K = 323 \mathrm{Nm}^{-1}\), \(\mu = \frac{35/2}{6.02 \times 10^{23}} kg\) (converted from atomic mass to kg). Plugging in these values, we calculate the vibrational zero point energy.

Energy Conversion to Speed

To convert this energy to translational kinetic energy, we use the formula \(E = \frac{1}{2}mv^2\), where \(v\) is the speed of the molecule and \(m\) is its mass. Rearranging the equation to solve for speed, we get: \(v = \sqrt{\frac{2E}{m}}\)

Compare to Root Mean Square Speed

Given the formula for root mean square speed \(|\mathbf{v}|_{rms}= \sqrt{\frac{3k_{B}T}{m}}\), where \(k_B = 1.38 \times 10^{-23} JK^{-1}\) is the Boltzmann constant, \(T = 300K\) is the temperature, and \(m\) is the mass of the molecule. Thus, we calculate the root mean square speed and compare it to the speed calculated in the previous step.

Conclusion

Through the above steps, we calculated the vibrational zero point energy of a molecule, converted this energy to translational energy to determine the speed, and compared this speed to the root mean square speed predicted by the kinetic gas theory.

Key concepts

Harmonic Oscillator

In the realm of molecular vibrations, the harmonic oscillator model provides a foundational approach to understanding how molecules behave. It represents molecules like springs that can stretch and compress in a regular, periodic fashion.
The concept of a harmonic oscillator in chemistry relates to the well-known physics principle involving spring-like motion. The model assumes a force that pulls the molecules back toward their equilibrium positions when they are displaced, just like a spring would. This force is the key to why molecular vibrations occur in such a periodic fashion.
Key characteristics include:

• A restoring force proportional to the displacement from equilibrium.
• Energy quantization where the molecule can only possess certain discrete energy levels.
• Vibrations described by a sinusoidal wave function.

Understanding harmonic oscillators offers valuable insights into molecular vibrational energies, helping predict various behaviors of molecules like absorption of energy and how they interact with light.

Reduced Mass

When dealing with systems involving two interacting masses, such as two atoms in a diatomic molecule, we often utilize the concept of reduced mass. Reduced mass simplifies the two-body problem into a single-body problem, making it much easier to handle with calculations.
Reduced mass \(\mu\) is mathematically represented as:
\( \mu = \frac{m_1 m_2}{m_1 + m_2} \)
where \(m_1\) and \(m_2\) are the masses of the two particles.
This value helps in characterizing the vibrational motion of the molecule as if it were a single particle with mass \(\mu\), especially in the context of the harmonic oscillator model.
By using the reduced mass in calculations, physicists and chemists can better understand the energy levels and the corresponding frequencies of molecular vibrations. This concept plays a vital role in determining how molecules absorb infrared light, providing insight into molecular structures.

Root Mean Square Speed

The root mean square speed is a statistical measure used to define the average speed of particles in a gas. This concept stems from the kinetic theory of gases, which explains that the kinetic energy of particles is directly related to temperature.
The formula for calculating the root mean square speed \( |\mathbf{v}|_{rms} \) is:
\( |\mathbf{v}|_{rms} = \sqrt{\frac{3k_{B}T}{m}} \)
where \( k_{B} \) is the Boltzmann constant, \( T \) is the temperature in Kelvin, and \( m \) is the mass of the particle.
This speed provides insight into the behavior of gas particles and helps to predict how they will respond under various conditions such as changes in temperature. It is essential when comparing calculated velocities, such as those converted from vibrational energy, to their theoretical average speeds.
For a given temperature, the root mean square speed offers a valuable metric to quantify the movement of particles in a molecular gas.

Zero Point Energy

Zero point energy refers to the minimal vibrational energy that molecules possess even at absolute zero temperature. It is a fundamental concept of quantum mechanics.
This energy arises because, contrary to classical physics expectations, molecules can never be completely at rest. Instead, they continue to exhibit small vibrations due to the inherent uncertainty in quantum mechanics.
In a harmonic oscillator, the zero point energy is given by:
\( E_0 = \frac{1}{2}hu \)
where \( h \) is Planck's constant and \( u \) is the frequency of the oscillator.
Understanding zero point energy has significant implications, particularly in predicting molecular behavior at low temperatures. It influences various properties and behaviors, like reaction rates and molecular stability, making it a cornerstone of molecular quantum theory.

Kinetic Gas Theory

Kinetic gas theory provides an essential framework for understanding the behavior of gases at the molecular level. This theory requires several assumptions which include that gas particles are in constant random motion and their mutual interactions can be considered as elastic collisions.
Through kinetic gas theory, one can explain and predict macroscopic properties of gases, such as pressure and temperature, based on their molecular activity.
Some core principles include:

• Gases consist of a large number of particles that occupy negligible space compared to the overall volume of gas.
• The average kinetic energy of gas molecules is proportional to the absolute temperature.
• Collisions between gas molecules (and with the container walls) result in pressure.

This theory lays the groundwork for calculating the root mean square speed of gas molecules, crucial in comparing experimental and theoretical speeds of gas molecules, as illustrated when translating molecular vibrational energies into such metrics.