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Calculating and comparing speeds
Look at the values of the speed of sound in a gas and the speed of its molecules. You will find that they are comparable in size, with the speed of the molecules always a bit greater, and you can think about why this should be true.
Comparing speeds of sound
Here is a table of measured values of the speed of sound in three gases:
GasMolar mass / gSpeed of sound at 273 K / m s1Speed of sound at 300 K / m s1Helium4972.51019Nitrogen28337.0355.5Carbon dioxide44257.4269.8
1. Which gas has the highest speed of sound, at either temperature?
2. Which gas has the least massive molecules?
3. Which gas has the lowest speed of sound, at either temperature?
4. Which gas has the most massive molecules?
5. At which temperature is the speed of sound the higher?
6. At which temperature are the molecules moving faster?
Comparing speeds of molecules
The next questions may suggest to you a reason for the pattern you have seen.
7. If there are N molecules in an ideal gas at temperature T, pressure P, volume V then
EMBED Equation.3,
where the molecules have mass m and mean square speedEMBED Equation.3, and k is
the Boltzmann constant.
Show that the mean square speed is given by
EMBED Equation.3
8. Calculate the mass of a helium atom, given that 4 g (= 0.004 kg) of helium contains
N = 6.02 1023 atoms.
9. Calculate the square root of the mean square speed for helium atoms, at 300 K, given that the Boltzmann constant k = 1.38 10 23 JK 1.
10. The masses of nitrogen molecules and helium atoms are in the ratio 28 / 4. What should be the ratio of their mean square speeds at any given temperature?
11. Using the answer to question 9, predict the square root of the mean square speed (the rms speed) for nitrogen molecules at 300 K.
12. Repeat question 9 for carbon dioxide molecules.
Comparing speeds of sound and speeds of molecules
A sound wave in a gas consists of a moving wave of compressions and expansions of the gas. A compressed region must compress the gas next to it for the wave to move forward. The molecules in the compressed region must move into, or knock others into, the region next to them. The wave cant have arrived before the molecules do. So the speed of the wave cannot be larger than the speed of the molecules; the two speeds may be comparable.
13. Copy the table of speeds of sound and add to it the values of speeds of molecules calculated for helium, nitrogen and carbon dioxide. How do the two sets of speeds compare?
Effect of temperature
14. If the temperature of a gas falls from 300 K to 273 K, by what factor do you expect the root mean square speed of its molecules to change?
15. Do the speeds of sound shown in the table follow a similar pattern?
Practical advice
These questions do not give a carefully argued reason why the speed of sound in a gas cannot exceed the speed of the molecules. But they show that there is a pattern, by taking molecules of different masses, and comparing two temperatures. Students see that despite large variations in the speeds in the different gases, the two speeds remain comparable. The questions give plenty of practice in calculation, manipulation of equations and reasoning about ratios.
The first questions (questions 19) are quick and simple, at the level of simple practice, and are suited to students of all abilities. The remaining questions take the level up to that at or a bit beyond the post-16 level examination.
You may have shown the rapid expansion of bromine into a vacuum, as evidence of the speed of molecules. Looked at a different way, this is a pressure shock wave (i.e. sound) travelling into the vacuum.
It may be helpful to have at the back of your mind the actual relation between the rms speed of molecules and the speed of sound. The speed of sound is
EMBED Equation.3
and since
EMBED Equation.3
then
EMBED Equation.3
Thus the ratio of the rms speed to the speed of sound is
EMBED Equation.3
If for example g = 1.4, the speed of molecules is approximately 46% greater than the speed of sound.
Alternative approaches
A more qualitative discussion could be better for less able candidates.
Social and human context
The theory of the speed of sound was worked out long before anyone had an idea of the speeds of molecules.
Answers and worked solutions
1. Helium, at 972.5 m s1 at 273 K and 1019 m s1 at 300 K.
2. Helium at 4 g mol1.
3. Carbon dioxide, at 257.4 m s1 at 273 K and 269.8 m s1 at 300 K.
4. Carbon dioxide, at 44 g mol1.
5. The speed of sound is larger in all cases at the higher temperature 300 K.
6. From the kinetic theory, the kinetic energy and so the speed of the molecules will be higher at the higher temperature, 300 K.
7.
EMBED Equation.3
Divide both sides by N giving:
EMBED Equation.3
Multiply both sides by 3 and rearrange, obtaining:
EMBED Equation.3
8.
EMBED Equation.3
9.
EMBED Equation.3
whence taking the square root the rms speed is 1370 m s1.
10. Since
EMBED Equation.3
then the ratio of the mean square speeds is 4 / 28.
11. The rms speed for nitrogen will be
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12. The factor is now (4 / 44) = 0.301 giving a speed of 0.301 1370 m s 1 = 413 m s 1.
13. The table is now:
GasMolar mass / gSpeed of sound at 273 K / m s 1Speed of sound at 300 K / m s 1rms speed of molecules at 300 K / m s 1Helium4972.510191370Nitrogen28337.0355.5517Carbon dioxide44257.4269.8413
14. The squares of the speeds are proportional to the temperature, so if the temperature falls from 300 K to 273 K the speeds fall in the ratio (273 / 300) = 0.954.
15. Yes. The ratios 972.5 / 1019, 337.0 / 355.5 and 257.4 / 269.8 are all in the ratio 0.954 approximately.
External reference
This activity is taken from Advancing Physics chapter 13, 100S
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