A 100 cm³ box contains helium at a pressure of 2.0 atm and a temperature of 100℃. It is placed in thermal contact with a 200 cm³ box containing argon at a pressure of 4.0 atm and a temperature of 400℃. What is the final thermal energy of each gas?

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Hi, everyone in this practice problem, we will have to calculate the thermal energy of each sample of mono atomic gas. We'll have a sample of mono atomic gas, Y with a volume of 800 millimeter cube with the pressure of three ATM and temperature of 100 and 50 degrees Celsius. A different sample contains neon at volume of 750 millimeter cube pressure of five ATM and temperature of 250 degrees Celsius. The two samples are allowed to interact thermally through a boundary. We're being asked to calculate the thermal energy of each sample when the thermal equilibrium is attained. The actions given are A for the energy for the Y 1. Jules for neon, 0.47 Joles for the uh B for the Y 0.43 Joles and for neon 1.3 Joles C for the Y 0.86 Jules and for neon 0.86 jules, D for the Y 0.4 Joles and for neon 1.3 Joles. And lastly, E for the Y 1.3 Joles and for neon 0.4 Jos So let's start with actually um identifying that both gas samples are going to be mono atomic. The equilibrium condition is going to be given by E sample one or E S one average will then be equals two E two average or also equals to just E S and average for our ant sample and equals to the E total average as well. So this will then be equals to E Y F divided by N Y or essentially the average is just essentially the total E, the total energy divided by N and then E Y F divided by N Y will be equals to E neon F divided by a neon equals to total energy E total divided by N total. And also just equals two E Y plus E neon divided by and Y plus and neon just like. So we do not know what E Y is in neon S N Y and N neon. So we have to first calculate that in order for us to be able to solve this problem. So let's actually start with calculating um the number of particles or N. So the number of particles N will be equals to PV divided by K B multiplied by T moles or N will be equals to this number of particles and divided by an A or the avocado number. So we have to use uh we have to first calculate the number of particles to then calculate our moles or N and our thermal energy or our E can be calculated by using the formula for three divided by two, multiplied by N multiplied by K B multiplied by T for mono atomic gasses. At the same time, we can actually utilize the ideal gas law to obtain the same um answer where we'll have the N to be equals to PV divided by RT and E or the energy to be equals to three divided by two multiplied by N RT. So I'm gonna just use this blue collar to indicate that we are not going to utilize this inside of our calculation today. So um let's actually start calculating first, we have the volume given in millimeter cube. So I'm gonna first convert that into meter cube. So we have the volume to then be equal to um let's start with the Y. So 800 millimeter cube and converting that into meter cube, we will have to multiply this by one m divided by 1000 millimeter and then cube that and then our volume will then be equals to 800 times 10 to the power of negative nine m cube. And next, let's actually calculate our uh number of particles or N for Y. So our N will be equals to PV divided by K BT. The P is going to be the pressure for Y or N Y here and P is the pressure for Y which is three A PM and converting that into pascals. We have to multiply this where 1.13 times 10 to the power of five pascals for one A PM. And then multiply that with the volume which is 800 times 10 to the power of negative nine m cube divided with K B which is the Boltzmann constant, which is 1.38 times 10 to the power of negative 23 just divided by Kelvin. And then multiply that with our temperature, which is for the Y gas is 150 degrees Celsius, we have to convert that into Kelvin. So 100 and 50 plus 273 Kelvin. So let's do the calculation for this and why or the number of particles. So and Y will then be equals to 4.165 times 10 to the power of 19 molecules. Awesome. So that's the N Y or the number of particles for gas sample Y. And then moving on, we want to calculate the moles or number of moles or N Y where N Y equals to N Y or capital N Y divided by the alpha number or N A, the N Y capital N Y is 4.165 times 10 to the power of 19 molecules. And the avogadro number is just a constant which is 6.0 to 2 times 10 to the power of molecules. For more this will get uh give us the number of moles for N Y or the moles which is 6.916 times to the power of negative five mole. So now that we have the number of particles for N Y and the number of moles, we uh the last thing that we need for Y is just the energy where the energy will be calculated by multiplying three divided by two E Y will be equals to three divided by two and Y multiplied by K B multiplied by T. This is E Y I or the initial of uh the initial energy for gas sample. Y. So substituting out of this value, you will then get E Y I to then be equals to three divided by two multiplied by N Y which is 4.165 times 10 to the power of 19 molecules multiply it that way. But with K B which is a constant 1.38 times 10 to the power of negative 23 divided by Kelvin and then multiplying it with the temperature which is going to be 1 50 plus 73 Kelvin. So I'm just gonna put that down here and then doing all this calculation, we will get our E Y I value to then be equals to 0. Jules. Next we want to move on to neon and do pretty much exactly the same um calculation where and neon will then be equals to PV divided by K BT. And the pressure for neon is given to be five ATM. In the problem statement, the volume is 1000 750 millimeter cube and the temperature is degrees Celsius. So let's uh utilize off that information. So now the N for neon will then be equals to five AM multiplied by, we have to convert that into pascal. So multiplied by 1.13 times 10 to the power of five pascals for every one ATM. And then multiply that with the volume which is 1000 750 times 10 to the power of negative nine m cube and then divide that by the Boltzmann constant 1.38 times 10 to the power of negative 23 Joles divided by Kelvin multiply that with the temperature which is 250 C uh degrees Celsius converting that into Kelvin, we have to plus 273 Kelvin. So calculating all of this, we will get our neon or the number of particle for our neon to then be equals to 1.2281 times 10 to the power of 20 molecules. Next, we have already found the number of particles for neon. We can move on to calculate the number of moles. So an neon or little Aon, right, like this will be equals to the capital N of neon divided by the N A or the avogadro number. So uh capital N neon is 1.2281 times 10 to the power of 20 molecules. And the avogadro number is a constant which is going to be uh 6.22 times 10 to the power of molecules for every one male. So calculating this, we will get the most of neon to then be equals to 2.3935 times 10 to the power of negative four more. Lastly, we want to calculate the um energy for neon initially. So E neon, I will then be equals to three divided by two multiplied by the capital and neon multiplied by K B multiplied by T. So this will then give us three divided by two multiplied by um 1.2281 times 10 to the power of molecules multiply that again with K B or the Boltzmann constant, which is 1.38 times 10 to the power of native 23 Joel per Kelvin multiply that with the temperature which is 2 plus 2 73 Kelvin, this will give us the E neon I value of 1.3296 Jos. So now let's actually tie everything together. So the equation that we were uh looking at initially that we derived is E Y F divided by N Y equals to Eon F divided by a neon equals to E total divided by N total, which can be calculated by taking E Y plus E neon and dividing all of that with N Y plus and neon. So, um what we want to use the E, the final equation that we are going to use for E Y F and for E neon F are pretty much the same where E Y F is going to be N Y multiplied by the final term that we have, which is um E Y plus E neon of that divided by N Y plus and neon. And for Eon, F, we'll be using a neon multiplied by the same term which is E Y plus E neon divided by N Y plus and neon. So let's start substituting all of the values that we have obtained or calculated previously. So let's start with E Y F E Y F will then be equals to N Y which is 6.916 stems down to the power of negative five mole multiply this with open print ce Y plus E neon, which is going to be 0.3647 jules plus Eon which is 1. jules and then divide all of that with um N Y which is 6.916 times 10 to the power of negative five moles plus a neon which is going to be two point oh +3935 times to the power of negative four moles calculating this, we will get our E Y F value to then be equals to 0.43 S. Now moving on to neon F, our eon F will be equals to and neon which is going to be 2.3935 times 10 to the power of negative four mole. And multiply that with the last term that we have here for E Y F as well, which is open parenthesis, 0.3647 Joles four E Y plus 1.3296 Jules for E neon close parenthesis divide all of that with N Y which is 6.916 times 10 to the power of negative five mole plus and neon 2.3935 times 10 to the power of negative four mole. And in calculating this, we'll get our E neon value to be equals to 1.3 jules. So E Y final or E Y F will be equals to 0. Joles and F will be equals to 1.3 jules. So in this case, those two answer will correspond to option B in our answer choices. So option B will be the answer to this particular practice problem. So that will be it for this video. 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 answer B with E Y F equals to 0. jules and E neon F equals to 1.3 jules will be the answer to this particular practice problem and that will be it for this video. Thank you.

A 100 cm³ box contains helium at a pressure of 2.0 atm and a temperature of 100℃. It is placed in thermal contact with a 200 cm³ box containing argon at a pressure of 4.0 atm and a temperature of 400℃. What is the final thermal energy of each gas?

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Key Concepts

Thermal Energy

Ideal Gas Law

Heat Transfer