Explain why the drum must be in a vacuum - AQA - A-Level Physics - Question 3 - 2021 - Paper 2

Given the distance from S to A is 0.500 m and the drum rotates at 120 revolutions per second, the speed can be calculated as follows:

1. Convert revolutions per second to radians per second:

   
   $120 \text{ rev/s} \times 2\pi \text{ rad/rev} = 240\pi \text{ rad/s}$.

2. The circumferential distance can be found as follows:

   • Circumference of the drum $=\pi d = \pi \times 0.500 \text{ m}$
   • This gives $d \approx 1.57 \text{ m}$.

3. The time taken to cover the distance is then:

   • Time = $\frac{\text{Distance}}{\text{Speed}} = \frac{0.500 \text{ m}}{500 \text{ m/s}} = 0.001 \text{ s}$.$

4. Hence, the speed is determined, which yields approximately 500 m/s.

Using the formula for root mean square speed:

$v_{rms} = \sqrt{\frac{3kT}{m}}$, where $k$ is the Boltzmann constant, and substituting the known values:

• Molar mass of the gas is $0.209 kg/mol$, hence mass of one atom is:

$m = \frac{0.209 kg/mol}{6.022 \times 10^{23} \text{ atoms/mol}} \approx 3.48 \times 10^{-25} kg$.

• Substituting into the equation:

$500 = \sqrt{\frac{3 \times (1.38 \times 10^{-23}) T}{3.48 \times 10^{-25}}}$, solving gives:

• On simplification, $T \approx 1930 K$.

According to kinetic theory, pressure is generated by collisions of gas atoms with the container walls. As atoms escape through the exit hole, there are fewer gas particles in the oven. This reduction in particle quantity leads to fewer collisions with the walls of the oven, thus causing the overall pressure to decrease.

To find the amount of gas that has escaped, we can use the ideal gas law:

$PV = nRT$, substituting the values:

$P = 5.0 \times 10^4 \text{ Pa}$, $V = 2.7 \times 10^{-2} \text{ m}^3$, $R = 8.314 \text{ J/(mol K)}$, and solving gives:

• Rearranging gives $n = \frac{PV}{RT}$.

• Plugging in values:

$n \approx \frac{(5.0 \times 10^{4})(2.7 \times 10^{-2})}{(8.314)(1930)} \approx 8.42 \times 10^{-2} \text{ mol}$.

When atoms enter the drum through slit S and pass onto the detector, the detector darkens at the point where an atom strikes it. After a time delay, when the atom passes, the detector is removed from the drum. The new detector will show a dark patch corresponding to the atoms passing through S as they emerge from the oven. The dark patch will appear clearer and well-defined as the temperature increases, demonstrating a higher speed of the atoms.