Oxygen (O2) has a molar mass of 32.0 gmol. What is (e) Suppose a...

Question

Oxygen (O2) has a molar mass of 32.0 g>mol. What is (e) Suppose an oxygen molecule traveling at this speed bounces back and forth between opposite sides of a cubical vessel 0.10 m on a side. What is the average force the molecule exerts on one of the walls of the container? (Assume that the molecule's velocity is perpendicular to the two sides that it strikes.)

Accepted Answer

Hey everyone today, we're being told that we have a 0.15 m long cube filled with this nitrogen gas that has an RMS velocity of 524 m per second. Imagining that the molecule is repeatedly moving between the left and right faces of the cube. We're being asked to determine the average force exerted on one of the cube's faces. We're taking the motion of the perpendicular or the motion from left to right to be perpendicular to said left and right faces. So because we're dealing with left to right and nothing else. This is a one dimension problem. We're only dealing with one dimension and we can go ahead and use our impulse formula. Her impulse formula which states that impulse and yours is equal to force into time, which is equal to the change in P. The change in momentum which is equal to mass times the change in velocity. So we're going to keep this here and we're going to go through a few steps that we're going to use to solve this problem. The very first, the first step is finding what is the RMS velocity? Well, it's already given for us in this question, thankfully The root mean squared velocity is simply 524 m/s. So we have that. Now the second step here is to determine delta P. The change of momentum by using the X access to be left and right orientation in the positive direction will be to the right. So using the formula that we used earlier, we know that the X access because we're dealing with one dimension, the change of momentum in the X direction will be the mass of the particle, multiplied by the change of velocity which is the final velocity in the X direction, minus the initial velocity in the X direction. Now we know that velocity magnitude is conserved when molecules collide with the wall, which means that the absolute value of the of the initial, sorry of the initial velocity will be the same as the absolute value of the final velocity. And we're taking the absolute value because the only thing that changes is the direction or the sign of the value, which is why the absolute value makes sense. So we can say that the collision on the left wall would be the initial velocity will be negative and the um collision on the right wall would be positive. So substituting that back in. So using this definition, we can also assume that this will be equal to the rMS velocity as the velocity is or the magnitude of the velocity is not changing only the direction is. So let's go ahead and use that in our equation. So we can say that mass go to the final velocity minus the initial velocity which is simply um mass into the final velocity. Remember we determined the sign of the initial velocity to to be negative because we'll say that's on the left face of the cube because we're doing this on an X coordinate basis in a one dimensional axis or one dimensional plane. My bad. So it will be the final velocity minus negative. The final velocity, I'm sorry, negative. The initial velocity which is simply um is equal to the initial velocity plus the final or the initial final velocity plus the initial velocity. Using this identity. Yet again, we can also say that this is simply equal to two M. V, not X, which is simply to M V. Rms because X and V R M s are the same. So this is our formula. Now we can go on to step three. We need to determine the force from the impulse formula mentioned earlier and we can use this part of the formula. So let's do this in blue. So step three is determined f. So we can say that F. And we're doing this in the X direction is equal to the change in momentum in the X direction over T. Because again F. T. Is equal to the change of momentum. So equating for F gives us delta P over T. Now time is the interval for the molecule to collide with the left wall and then the next time it collides with the left wall. So the entire time it takes to bounce off the left wall, go to the right side and come back to the left wall. So the distance that's being traveled is twice the length of the cube. So we can say, we can say that time is equal to the distance divided by the RMS velocity or that it is two times the length of the cube divided by the RMS velocity. Again, this is because it's the uh we know that time is equal to distance by velocity. And the distance here is two times the length of the cubits, hitting the left wall, bouncing off the right wall and then coming back to this initial starting point. So substituting that into our formula, we get F. The force is equal to two Mv RMS V R M s multiplied by one over T. Simplifying this or rather expanding it. I should say it's too M V R M S. Multiplied by V. R. M s over two L. Let's write this little neater L. And simplifying this, we get Are two cancels out. Can we get mm multiplied by V. R. S squared over. L. Is equal to F. No. Here's where things get a little tricky. Let me scroll down so we have a little more space. We're told in the question That we have a molar mass for nitrogen, right? We have a molar mass for nitrogen, it is 28.01 g per mole. So let's write that down here. Were given that we have a molar mass. Leaving leaving this formula, we just derived, We'll come back to it in just a second. But the molar mass of nitrogen is equal to 28. g per mole. No, uh molar mass uh can be converted to mass simply using avocados number. So molar mass, it's equal to avocados number multiplied by the mass, which means therefore mass is simply equal to molar mass divided by avocados number. And we can substitute this back into our formula. Now by saying that force is simply equal to the Moeller mess, Multiplied by VRMS two, divided by I've got his number multiplied by L the length of the cube. However, we can't just substitute in our values just yet. We need to convert 28. The molar mass into kilograms for S. I units 28.1 g per mole can be converted to kilograms with a simple conversion factor. By remembering that one kg is equal to 10 to the 3rd g. So simplifying this, we get that this is 28.01 times 10 to the negative three kilograms per mole. Substituting this back into our equation with the rest of the information that we know. We know that L is equal to 0.15 m. It's the length of the Cube. V R M s is simply 524 m/s squared. And we're dealing with newton's force in newtons and remember That one Newton is nothing but one kg meter square. So equating everything that we have because the F. The force is equal to 28.01 times 10 to the kg per mole, multiplied by V. RMS square which is 524 m per second squared, divided by 6.22 times 10 to the negative or sorry, 10 to the 23rd molecules per mole. Multiplied by 0. m so our molds will cancel out and we will be left with a final answer of 8.5 two times 10 to the negative newtons. Looking back at her answer choices, We can see that the answer that we're looking for the average force exerted on one of the cube's faces is none other than answer choice. D 8.52 times 10 to the negative 20th Newtons. I hope this helps. And I look forward to seeing you all in the next one.

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