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Speed of sound and speed of molecules

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TAP 603- 4: Speed of sound and speed of molecules

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:

    Gas Molar mass Speed of sound at 273 K Speed of sound at 300 K 11 / g / m s/ m s

    Helium 4 972.5 1019

    Nitrogen 28 337.0 355.5

     Carbon dioxide 44 257.4 269.8

    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

    21, PV?NkT?Nmv3

    2where the molecules have mass m and mean square speedv, and k is

    the Boltzmann constant.

     Show that the mean square speed is given by

    3kT2.v?m

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 ? 10231 JK.

    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 can’t 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

    ?Pc? ?

    and since

    12 ??Pc3

    then

    3Pv? rms?

    Thus the ratio of the rms speed to the speed of sound is

    3 ?

    If for example ? = 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

    111. Helium, at 972.5 m s at 273 K and 1019 m s at 300 K.

    12. Helium at 4 g mol.

    113. Carbon dioxide, at 257.4 m s at 273 K and 269.8 m s at 300 K.

    14. Carbon dioxide, at 44 g mol.

    25. The speed of sound is larger in all cases at the higher temperature 300 K. 1Nkt?Nmv.3 6. From the kinetic theory, the kinetic energy and so the speed of the molecules will be Divide both sides by N giving: higher at the higher temperature, 300 K.

    7. 21kt?mv.3

     Multiply both sides by 3 and rearrange, obtaining:

    3kT2.v?m

    8.

    ?3?14?10kg mol?27m??6.64?10kg23?16.02?10mol

    9.

    ?23?13?(1.38?10J K)?300K262?2v??1.87?10ms?276.64?10 kg

    1whence taking the square root the rms speed is 1370 m s.

    10. Since

    3kT2?vm

    then the ratio of the mean square speeds is 4 / 28.

    11. The rms speed for nitrogen will be ? (4 / 28) = 0.378 of the rms speed for helium,

    11giving a speed of 0.378 ? 1370 m s = 517 m s.

    1112. The factor is now ? (4 / 44) = 0.301 giving a speed of 0.301 ? 1370 m s = 413 m s.

    13. The table is now:

    Gas Molar mass / Speed of sound Speed of sound rms speed of 11g at 273 K / m s at 300 K / m s molecules at 300 1K / m s

    Helium 4 972.5 1019 1370 Nitrogen 28 337.0 355.5 517 Carbon dioxide 44 257.4 269.8 413

    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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