We have two equal-size boxes, A and B. Each box contains gas t...

Question

We have two equal-size boxes, A and B. Each box contains gas that behaves as an ideal gas. We insert a thermometer into each box and find that the gas in box A is at $50$ °C while the gas in box B is at $10$ °C. This is all we know about the gas in the boxes. Which of the following statements must be true? Which could be true? Explain your reasoning.

(a) The pressure in A is higher than in B.

(b) There are more molecules in A than in B.

(c) A and B do not contain the same type of gas.

(d) The molecules in A have more average kinetic energy per molecule than those in B.

(e) The molecules in A are moving faster than those in B.

Accepted Answer

Hello, fellow physicists today, we're gonna solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. A class is presented with two identical containers labeled D and E in terms of size and shape, each container is filled with an ideal gas. The temperature of sample D is found to be 15 °C while sample E has a temperature of 35 °C. What conclusions can be made from this given information? So that's our end goal. So ultimately, we're trying to figure out what conditions that are sorry based on these conditions, what conclusions we can make based off of the information that's provided to us. So we're trying to figure out based on the information that's provided to us what conclusions can be made. So with that in mind, we're also given some multiple choice answers and they're all different conclusions that can be made or possibly be made from the information above. But only one of these multiple choice answers is true. Some of these conclusions are false. So let's read them off to see what our final answer might be. So A is the sample contains different gasses. B is mean kinetic energy per molecule. KD is less than KEC is number of molecules. ND is less than NE D is pressure PD is less than pe and finally, E is speed of molecules motion VD less than VE. OK. So with that in mind to start solving for this problem, we need to first off, we need to recall and use the equations for the average kinetic energy per molecule. We also need to recall the equation for the R MS velocity which is the root mean square velocity of the gas molecules. And we also need to remember recall, I use the ideal Gasso equation. So let's call equation one, the kinetic, the average kinetic energy equation. So it's denoted as Kav. So average kinetic energy is equal to three halves multiplied by Boltzmann's constant multiplied by the temperature T. OK. And then our second equation will be the R MS velocity. The root means square velocity, which we're gonna denote as VR MS. And that is equal to the square root of three multiplied by the universal gas constant multiplied by the temperature all divided by the molar mass. That's what the capital M denotes is molar mass. And then finally, equation three is the ideal gas law equation. So it's pressure multiplied by the volume is equal to the number of moles in the sample multiplied by the universal gas constant multiplied by the temperature. So let's quickly note that identical containers in terms of size, which is what we're told in the prom itself. And because they're identical containers in terms of size, that will mean that the volume is the same. So I'm just gonna say volume equals same as a little note to ourselves. We also need to note that we're provided the temperature values in the prom itself and that all of our equations we listed above. So all of our equations that we listed above will include the use of temperature as you've probably noticed. So therefore, from all of the information that is provided, we can conclude that drum roll, we can conclude that VD is equal to VE. So that means that the volume of container D is equal to the volume of container E, that's what that means. So VD equals VE, so that's one of the important statements that we need to conclude. And then so the volume, once again, the volume of D is equal to the volume of E. We also can state that the temperature of container D is less than the temperature of container E. So now we need to investigate all of our multiple choice statements to see which answer is true. So let's start investigating multiple choice answer letter A. So let us substitute N equals lowercase M divided by capital M where M or lowercase M is the mass of the sample. And capital M, like we said before is the molar mass into the Ideal gas Law equation. So pressure multiplied by the volume is equal to the mass M divided by the molar mass multiplied by the universal gas constant multiplied by the temperature. So note that if the mass of each sample is different, it will lead to a different results. So therefore, we cannot tell if the gasses are the same or different. So I'm gonna say that once again. So we need to note that if the mass of each sample is different, it will lead to a different result. So that means, so their masses are different, then that means the results will be different. So also m can be different if the gasses are different. And due to that fact, we cannot tell the gasses are the same or different. So this means that this answer is false. OK, moving right along. So let's look at multiple choice answer. B now, so note that when we investigate equation one, we will see that the average kinetic energy per molecule will depend on the temperature value only. So thus, the kinetic or the average kinetic energy is directly proportional to the temperature value. So when the average kinetic energy increases, the temperature will also increase. And if the average kinetic energy decreases, the temperature will also decrease. So therefore, this answer is true. So that means that this is our final answer. So true. But even though we found the correct answer, let's look at our other multiple choice answers really quick to see why this one is most definitely the correct answer. So let's look at multiple choice answers C now, so we need to note that the number of moles molecules and Avogadro's number are related by N equals capital N divided by capital N A and P. So pressure multiplied by the volume is equal to capital N divided by capital N A multiplied by the universal gas constant multiplied by T where capital N is the number of moles and then, and then divided by Avogadro's number. So the number of molecules I should say not moles, moles is lowercase N. So capital N is the number of molecules divided by Ava Garos number. And we do not know the pressure since the pressure value is unknown to us. Therefore, N capital N can have any value. And we need to note that P and N so capital P, the pressure and capital N enable each other to have any value. So this answer is false. So now we can look at multiple choice answer letter D. So using the ideal gas law PV equals N RT can have any. So, so, so we're using the ideal gas law, so PV equals N RT N which is lowercase N. In this case, like we've been discussing is the number of moles and the number of moles is unknown. So P the pressure can have any value. So this answer has to be false and finally investigating multiple choice answer letter E. So using the equation two, which is the R MS velocity. So the R MS velocity depends on both the temperature and the molar mass where the molar mass is an unknown value and V RM ss. So the V can be any value. Thus V can be so either V either velocity or the R MS velocity for either sample, either container can be greater. So this answer has to be false. Awesome. So with that all in mind, looking at our multiple choice answers, the correct answer has to be the letter B which means that the mean kinetic energy per molecule KD is less than ke Thank you so much for watching. Hopefully that helped and I can't wait to see you in the next video. Bye.

Key Concepts

Ideal Gas Law

Kinetic Molecular Theory

Molecular Speed Distribution

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