The apparatus shown in the diagram can be used to measure atomic and molecular speed. 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 \((\mathrm{m} / \mathrm{s})\) at which the target is moving. (Hint: The circumference of a circle is given by \(2 \pi r\), in which \(r\) 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 Bi atoms. Compare your result in (c) with the \(u_{\mathrm{rms}}\) of Bi at \(850^{\circ} \mathrm{C}\). Comment on the difference.

Step by step solution

01

Calculate the speed of the target

Firstly, we need to calculate the linear speed of the point on the rotating cylinder opposite the slit through which the metal atoms are directed. The speed \( v \) can be calculated using the formula \( v = 2 \pi r \times f \), where \( r \) is the radius of the cylinder and \( f \) is the frequency of rotation. The radius is half of the given diameter, 15.0 cm or 0.075 m, and the frequency is 130 revolutions/sec or Hz. So, \( v = 2 \pi \times 0.075 \times 130 \approx 61.27 m/s \). Here, we have used the hints and the known relation between linear and rotational motion.

02

Calculate the time taken to travel 2.80 cm

Next, we need to calculate the time taken by the cylinder's point to travel 2.80 cm or 0.028 m. The time \( t \) can be calculated using the formula \( t = d/v \), where \( d \) is the distance and \( v \) is the velocity computed in first step. So \( t = 0.028 / 61.27 \approx 4.57 \times 10^{-4} s\). We have applied the formula and the speed of the point on the cylinder calculated in Step 1.

03

Determine the speed of the Bi atoms

Finally, we find the speed at which the Bi atoms are moving. Since the observed Bi atoms travel from the slit to the point on the cylinder in the same time as the linear speed of the point on the rotating cylinder (determined in the previous step), the speed \( u \) of the Bi atoms is calculated using the formula \( u = d/t \), where \( d \) is the distance from the center of the cylinder to the point where the atoms land (given to be 2.80 cm or 0.028 m), and \( t \) is the time calculated in Step 2. So \( u = 0.028 / 4.57 \times 10^{-4} \approx 61.27 m/s \). We have used the same kinematic equation given the similar conditions of travel between the point on the cylinder and the Bi atoms.

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