Photons of light scatter off molecules, and the distance you can see through a gas is proportional to the mean free path of photons through the gas. Photons are not gas molecules, so the mean free path of a photon is not given by Equation 20.3, but its dependence on the number density of the gas and on the molecular radius is the same. Suppose you are in a smoggy city and can barely see buildings 500 m away.

a. How far would you be able to see if all the molecules around you suddenly doubled in volume?

Question

Photons of light scatter off molecules, and the distance you can see through a gas is proportional to the mean free path of photons through the gas. Photons are not gas molecules, so the mean free path of a photon is not given by Equation 20.3, but its dependence on the number density of the gas and on the molecular radius is the same. Suppose you are in a smoggy city and can barely see buildings 500 m away.

a. How far would you be able to see if all the molecules around you suddenly doubled in volume?

Accepted Answer

Hi everyone. In this practice problem, we are being asked to calculate the farthest distance that we will be able to see through a forest. Imagine that you are in a dense forest where you can barely see trees that are 50 m away due to the fok. If the number of fok particles was to decrease by a factor of three, we are being asked to call the farthest distance that we will be able to see through this forest. Consider that Faulk particles are similar to gas molecules with respect to their dependence on the number density of the gas and molecular radius. The options given for the distance are a 50 m b, 100 and four m C 60 m and D 40 m. So we will consider the Faulk particles as spherical masses where a number of Faulk particles decreases by a factor of three, then the volume will be a third time off the actual initial volume. The volume of the sphere is proportional to are to the power of three. So volume is proportional to our cube. Therefore, the radius decreases by a factor of one over cubic root of three and number density remains constant. So the radius radius will decrease by a factor of one divided by S A cubic group of three. And the number density number density will remain constant. The mean free path will depend only on R. So let the new mean free path be Lambda prime and the new radius be R prime. So you wanna consider lambda prime divided by lambda will be equal to R squared divided by R prime squared. So it's an inversely proportional. So in this case, R squared will remain R squared and R prime squared, as I have described previously, the radius will decrease by a factor of one divided by this cubic root of three. So I'm gonna um substitute this R prime squared with one divided by cubic root of three of R. And remember that we still want to put that in the square and then calculating this, we can actually cross out the R squared here and then one divided by cubic root of three can actually be calculated into an actual um decimal value. So that lambda prime divided by lambda will then be equal to two point oh eight. So then Lambda prime rearranging this lambda and put it onto the other side will equal to two point oh eight multiplied by a Lambda or two point oh eight times lambda. So the lambda is initially fif given the problem statement to be 50 m away or 50 m. So we want to substitute that value into this equation that we have here. So two point oh eight multiplied by 50 m and there will be the new mean free path or the new Lambda prime, which will come up to a value of 100 and four m. So therefore the farthest distance that we will be able to see through the forest when the fok particles was to decrease by a factor of three will be 100 and four m or Lambda prime the new lambda, the new Lambda, the new mean free pat will be 100 and four m. So that'll be it for this practice problem. If you guys still have any sort of confusion, please feel free to check out our other lesson videos on similar topics available on our website. But other than that, that will be it for this one. Thank you.

Verified video answer for a similar problem:

Key Concepts

Mean Free Path

Number Density

Scattering of Photons