Smoke particles in the air typically have masses of the...

Smoke particles in the air typically have masses of the order of 10${^-}{^1}{^6}$ kg. The Brownian motion (rapid, irregular movement) of these particles, resulting from collisions with air molecules, can be observed with a microscope. (a) Find the rootmean-square speed of Brownian motion for a particle with a mass of 3.00 $\times$ 10${^-}{^1}{^6}$ kg in air at 300 K. (b) Would the root-meansquare speed be different if the particle were in hydrogen gas at the same temperature? Explain.

Transcript

00:01 So this question is only asking about finding the root mean squared velocity.

00:05 So we have the mass of this molecule, so three times 10 to the negative 10 kilograms.

00:15 And then we have the temperature of 300 kelvin.

00:19 And the formula for the root mean squared, when we're talking about one molecule, it's going to be three times the boltzman constant times the temperature all over the mass.

00:31 Of that molecule.

00:33 And a case of b, boltsman constant is going to be equal to 1 .31 times 10 to the negative 23rd.

00:44 And then for if you were to, if you're, if you want to look into the dimensional analysis, it's going to be meter squared times kilograms times second to negative times seconds to the negative second and then times times times.

01:03 Kelvin to the negative first.

01:05 So this would be the unit for boltzmann constant.

01:08 This would if you if you're interested in finding the dimensional analysis that could help you.

01:13 And so at this point we can just solve directly and say that this is going to be equal to three times 1 .31 381 i apologize.

01:26 This is this is actually 1 .381 times 10 to the negative 23rd.

01:53 Times the temperature in kelvin, 300, and then divided by that mass of 3 times 10 to the negative 10th kilograms.

02:03 This is going to be equal to 6 .434 times 10 to the negative 3rd meters per second.

02:14 So that would be our answer for a...

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