A monatomic gas follows the process 1→2→3 shown in FIGURE EX19...

Question

A monatomic gas follows the process 1→2→3 shown in FIGURE EX19.26. How much heat is needed for (a) process 1→2 and (b) process 2→3?

Accepted Answer

Hey, everyone in this problem, we're asked to consider a system containing helium gas that undergoes the process A to B to C, which is described in the figure we're given and we wanna find the heat needed or, or removed A for process A to be and process B to C. OK. So looking at our diagram, we have the pressure and atmospheres on the Y axis, the volume in centimeters cubed on the X axis A starts at a pressure of six, a volume of 500 and to go from A to B, we're going straight downwards, the volume is gonna remain constant and the pressure is gonna drop to two. And then from B to C, we're going straight across and so straight horizontally, the pressure remains the same, but the volume is gonna increase to 1500. And so we have these two processes and we have four answer choices here. OK? We have that the heat removed is either gonna be negative 304 negative 305 or 403 joules. And that the heat in process two is going to be 507 joules or 705. And we're gonna come back to these answer choices when we're done working through this pro Now let's start with part one and part one is gonna be this process from A to B. OK. So from A to B, we have a constant volume. And if we have a constant volume, they recall that this is an Isoc Cork process, an Isoc Cork process has that the quantity of heat OK? From A to B is just equal to the change in internal energy because the work is zero, there's no work done on the surrounding area. And so this is gonna be calculated as N multiplied by CV multiplied by delta T. Yeah, we have this equation but there are some things we need to find. First. Number one, we don't know the number of moles. And so we're gonna need to calculate it. And number two, in order to calculate delta T, we need to know the temperature at point B and we don't have that. So we're gonna need to calculate the temperature at point B. Let's go back up and do both of these things starting with finding the number of moles. And yeah, now let's pay attention to point A. OK. And at point A, we know that the pressure is six atmospheres, 46 multiplied by 1.01325 multiplied by 10 to the exponent five pascals. We know that the volume is 500 centimeters, cut 45, multiplied by 10 to the exponent negative 4 m skewed. And we know the temperature as well. And we know the temperature because the process from C to A is isothermal. That means that the temperature doesn't change, it's constant at both of those points. And we're told in the diagram that that's 150 degrees and we're gonna take that 150 °C. We're gonna add 273 to get to our standard unit of Calvin. And that's gonna be 423 Calvin. Now, we know PV and T, so let's recall our ideal gas law. We're dealing with helium, we can treat that as an ideal gas. And so we're gonna write that P multiplied by V is equal to N multiplied by R multiplied by T. And is there a number of moles? That's what we're looking for. Our, is the ideal gas constant which we know. OK. And then all of those other variables we've just defined. So to find N, we're gonna take P A multiplied by VF and divided by R T A. We can now substitute in our values and it is gonna be equal to six multiplied by 1.01325 multiplied by 10 to the exponent five past scales multiplied by five, multiplied by 10 to the exponent negative 4 m cubed all divided by 8.3145 jewels per mole Calvin multiplied by 423 Calvin. When we work all of this out on our calculator, we get that N is going to be equal to about 0.086 43. And we're gonna put a red box around this. So we don't lose track of it because we're going to need it. And the other thing we needed to calculate was CV. And our temperature TB. So CV, we're gonna start there and CV. And because we're dealing with helium, which is a mono atomic gas, it is just going to be equal to three halves multiplied by R. Now, since we're dealing with CV, let's also find CV or CP. Sorry. OK. So CV is the heat capacity at constant volume CP at constant pressure. So we're gonna find CP as well because we're gonna need it for the second process. I recall that it's just CV plus R or in this case, five HS are right now, the temperature of TB. That was the other thing we needed to find in order to do our calculation from A to B. Let's go ahead and do that as well. We are going to find the temperature at point B and we're gonna do it in a very similar way to finding the number of moles. And we're gonna use the ideal gas law, but this time at point B instead of at point A. OK. So at point B, the pressure is going to be two atmospheres or two multiplied by 1.01325 multiplied by 10 to the exponent by costs the volume at point B is going to be equal to 500 centimeters. Cute. There are five multiplied by 10 to the exponent negative 4 m ski. And we wanna find the temperature using the exact same equation PV equals N RT. We're gonna rearrange for the temperature this time. Instead of the number of moles, the temperature at point B is going to be PB VB divided by an arm. This is equal to two multiplied by 1.01325 multiplied by 10 to the exponent five pa scales multiplied by five, multiplied by 10 to the exponent negative 4 m cubed all divided by the number of moles. And which we found to be 0.086 43, multiplied by the ideal gas pump, 8.3145 joules per mole cat. All righty when we work this out, OK, the unit of moles is gonna divide out half K meters cubed is equivalent to jewels. So that's gonna divide out. We're gonna be left with Calvin in the numerator, which is what we want for temperature. We get a temperature of 140.999 Calvin, right. We have our temperature, we have the number of moles we have CV. Let's get back to what we're trying to find, we're trying to find the heat needed or removed for process A to B. OK. So we're dealing with that, we have our equation, we can now substitute in our values Q from A to B is going to be equal to the number of moles, 0.086 or three moles multiplied by CV, three halves multiplied by R 8.3145 joules per mole Calvin multiplied by delta T. Now we have to be careful when we calculate delta T, we always do the final temperature minus the initial temperature. In this case, we're going from A to B. So the final temperature is the temperature at point B which is 140.999 Kelvin. And we're subtracting the temperature at a 423 Kelvin. OK? We're not always gonna have delta T being positive. OK? So resist that urge to put the bigger number first, always pay attention to your final temperature minus your initial temperature good. And when we work all of this out on our calculator, we get that Q from A to B is going to be equal to negative 303 point 978 jules. OK. So that heat is going to be removed. We have a negative value. So we have heat removed, we have about 304 jewels that's going to be removed. So that's part one. And now we're gonna move on to part two and part two is looking at the exact same thing. But for the process from B to C, from B to C, we have a constant pressure instead of a constant temperature. So let's take a look at how we're gonna work through that one. If we have a constant pressure, we can call this an isobaric process. And in an isobar process, the quantity of heat is gonna be calculated very similarly to what we just calculated for constant volume. And the only difference is that instead of having the heat capacity at constant volume, we're gonna have the heat capacity at constant pressure. So we have N DP multiplied by delta T. Now CP, we calculated when we did CV, we know the number of moles. We've already calculated that the temperature at point C A is going to be the same at point A. Remember the process from C to A is isothermal. So it has a constant temperature. So we already know that temperature at point C, we know the temperature at point B, we just calculated it. So in this case, we have everything that we need. Now we can get rates of substituting it in. So Q from B to C is going to be equal to 0.08643 mole multiplied by five halves, multiplied by 8.3145 joules per mole then multiplied by delta T. And in this case A delta T is going to be the opposite way around. We want the final temperature. So the temperature at point C which is 423 Kelvin minus the initial temperature, the temperature at point B which is 140.999 Kelvin. OK. And when we work all of this out, we're gonna get a quantity of heat from B to C about 506 0.63 jules. Now, this is a positive value. So this means we have at all right. And that's it for this question. We've done both part one and part two. Let's take a look at our answer. Choices. We're going round to the nearest jewel. Ok. For part one, we found that we had heat removed and it was negative 304 jewels. And for part two, we found that we had heat added and it was 507 jewels. And so the correct answer here is going to be option A. Thanks everyone for watching. I hope this video helped see you in the next one.

A monatomic gas follows the process 1→2→3 shown in FIGURE EX19.26. How much heat is needed for (a) process 1→2 and (b) process 2→3?

Key Concepts

First Law of Thermodynamics

Isothermal Process

Work Done by a Gas

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