Question

The apparatus shown here can be used to measure atomic and molecular speeds. Suppose that a beam of metal atoms is directed at a rotating cylinder in a vacuum. A small opening in the cylinder allows the atoms to strike a target area. Because the cylinder is rotating, atoms traveling at different speeds will strike the target at different positions. In time, a layer of the metal will deposit on the target area, and the variation in its thickness is found to correspond to Maxwell's speed distribution. In one experiment it is found that at \(850^{\circ} \mathrm{C}\) some bismuth (Bi) atoms struck the target at a point \(2.80 \mathrm{~cm}\) from the spot directly opposite the slit. The diameter of the cylinder is \(15.0 \mathrm{~cm},\) and it is rotating at 130 revolutions per second. (a) Calculate the speed (in \(\mathrm{m} / \mathrm{s}\) ) at which the target is moving. (Hint: The circumference of a circle is given by \(2 \pi r\), where \(r\) is the radius.) (b) Calculate the time (in seconds) it takes for the target to travel \(2.80 \mathrm{~cm} .\) (c) Determine the speed of the \(\mathrm{Bi}\) atoms. Compare your result in part (c) with the \(u_{\mathrm{rms}}\) of \(\mathrm{Bi}\) at \(850^{\circ} \mathrm{C}\). Comment on the difference.

Short answer

(a) 61.23 m/s, (b) 0.0004577 s, (c) 61.17 m/s, different from \( u_{\text{rms}} \) (354.6 m/s).

Step-by-step solution

Calculate Target Speed

To find the speed at which the target is moving, we start by calculating the circumference of the cylinder. The radius \( r \) is half of the diameter, so \( r = \frac{15.0 \text{ cm}}{2} = 7.5 \text{ cm} = 0.075 \text{ m} \). The circumference \( C \) is \( 2 \pi r = 2 \pi \times 0.075 = 0.471 \text{ m} \). The cylinder rotates at 130 revolutions per second, so its speed is \( C \times 130 = 0.471 \times 130 = 61.23 \text{ m/s} \).

Calculate Time for Target to Travel

Next, we calculate the time taken for the target to travel \( 2.80 \text{ cm} \), which is \( 0.028 \text{ m} \). Using the speed from Step 1, \( v = 61.23 \text{ m/s} \), the time \( t \) is \( \frac{0.028}{61.23} \approx 0.0004577 \text{ s} \).

Calculate Speed of Bi Atoms

The speed of the Bi atoms can be calculated by considering the position they travel to in the given time. Since the atoms hit a point \( 2.80 \text{ cm} \) away during the time calculated, their speed is \( \frac{0.028}{0.0004577} \approx 61.17 \text{ m/s} \).

Calculate RMS Speed of Bi at 850°C

The root-mean-square speed \( u_{\text{rms}} \) of bismuth atoms is given by the formula \( u_{\text{rms}} = \sqrt{\frac{3kT}{m}} \). For \( T = 850^{\circ} \text{C} = 1123 \text{ K} \), and assuming the molar mass of Bi is \( 208.98 \text{ g/mol} \), converting to kg gives \( m = 208.98 \times 10^{-3} / 6.022 \times 10^{23} \text{ kg} \). Use \( k = 1.38 \times 10^{-23} \text{ J/K} \). Thus, \( u_{\text{rms}} = \sqrt{\frac{3 \times 1.38 \times 10^{-23} \times 1123}{208.98 \times 10^{-3}/6.022 \times 10^{23}}} \approx 354.6 \text{ m/s} \).

Compare Speeds

The calculated speed of Bi atoms (\(61.17 \text{ m/s}\)) is much slower than the \( u_{\text{rms}} \) value (\(354.6 \text{ m/s}\)). This discrepancy is expected due to experimental conditions and measurement accuracy. Many atoms might not be going straight through the slit or at peak speed.

Key concepts

Atomic and Molecular Speeds

When discussing the behavior of atoms and molecules, the concept of speed is pivotal. Atoms and molecules are perpetually in motion, and in gases, they move in all directions with a variety of speeds. The distribution of these speeds at a given temperature is described by Maxwell's speed distribution. This distribution predicts that:

• Some atoms or molecules will move very slowly.
• Some will travel fast.
• Most will have a speed around a certain average value.

In the exercise example, bismuth (Bi) atoms' speeds were gauged using a rotating cylinder, showcasing the application of Maxwell’s distribution in a real-world scenario.
Understanding the range and distribution of atomic and molecular speeds helps physicists and chemists predict how gases will behave under various temperatures and pressures.

Root-Mean-Square Speed

The root-mean-square speed (\( u_{\text{rms}} \) ) provides a statistical measure of the average speed of particles in a gas, derived from kinetic theory. It is not a simple average but a calculation that accounts for the square of velocities:

• \[ u_{\text{rms}} = \sqrt{\frac{3kT}{m}} \]
• \( k \) is Boltzmann’s constant, linking temperature to energy, and \( T \) is the absolute temperature in Kelvin.
• \( m \) is the mass of one particle.

Based on the step-by-step solution, the calculated \( u_{\text{rms}} \) for bismuth at 850°C (1,123 K) was approximately 354.6 m/s. This indicates that on average, bismuth atoms are moving at this speed.
However, the problem highlights a notable exception: the calculated speed of the atoms impacting the target was much lower. Such differences can be explained by factors like experimental setup and conditions that affect the atoms' trajectories.

Circumference Calculation

The exercise involves calculating the circumference of a rotating cylinder, a key step towards determining the rotational speed of the target. The formula for the circumference \( C \) of a circle is given by:

• \[ C = 2 \pi r \]
• Where \( r \) is the radius of the cylinder.

In this problem, the cylinder had a diameter of 15.0 cm, giving a radius of 7.5 cm (or 0.075 m).
Thus, using this radius:

• The circumference was calculated as 0.471 m.

This value enabled the calculation of the cylinder's rotational speed (in m/s), essential for further analysis in the experiment, like determining how long it takes the target to cover a certain distance.

Thermal Motion of Atoms

The thermal motion of atoms refers to the constant movement of atoms and molecules within materials, becoming more vigorous as temperatures rise. This thermal activity is central to understanding how temperature influences the speed of particles:

• At higher temperatures, atoms and molecules have more kinetic energy.
• This energy influences their speed, often resulting in broader distributions in speed according to Maxwell's distribution.

In the given exercise, bismuth atoms at 850°C were analyzed. Higher temperatures provide even greater energy, imparting greater speeds to the atoms on average.
By comparing real speed measurements to theoretical values like \( u_{\text{rms}} \), researchers can glean insights into the nature of atomic interactions and the effects of external conditions on atomic velocities.