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Physics

## Carnot cycle and Carnot engine

Introduction to the Carnot cycle and Carnot heat engine
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## Carnot cycle and Carnot engine

Discussion and questions for this video
I'm still pretty confused about the 2 adiabiatic processes in the Carnot cycle. If there is no exchange of heat during the move from B to C and from D to A, and the temperature is constant during that move, how is it that the temperature all of a sudden jumps from T2 back to T1? Please don't answer "because it is an adiabatic process", that doesn't help me...
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The temperature is not constant. In the adiabatic expansion, the work done by the gas is supplied by the energy in the gas, and the temperature then decreases.

In the equation U = Q - W (forgive the lack of delta's) adiabatic and isothermal are the edge cases. Isothermal means U=0, adiabatic means Q=0. In the first case, all work is at the expense of heat entering (or leaving) the system. I the second case, work is a "cash out" of internal energy, hence (for an ideal gas) temperature.
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Okay I understood that a Carnot cycle undergoes 4 states.
Isothermal expansion and compression , Adiabatic expansion and compression.
But why do we need a two adiabatic states? Work is being done during the isothermal expansion and compression so What's the point of isolating the system for adiabatic expansion and compression? Can't the system directly be taken to the cold reservoir after keeping it on the hot reservoir?
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Yes, one could take the system directly from the T1 reservoir to the T2 reservoir upon reaching the state B. But if you do that the system will not be in thermal equilibrium while it's making the adjustment to the sudden change in temperature. That would happen again if, after reaching state D isothermally, one moved the system directly back to the T1 reservoir. Those periods while the system is adjusting to the sudden change in temperature seriously decrease the efficiency of the cycle. That's not obvious at this stage of the discussion, but it is the case and Carnot was the first to argue that performing the cycle in his careful way maximized the efficiency of the cycle, i.e., maximized the net amount of work done by the system for a given net heat flow from the T! reservoir to the T2 reservoir. All real heat engines do have to deal with various instances of sudden temperature changes and departures from thermal equilibrium. To the extent that they do, they are less efficient than the Carnot cycle.
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What's the main purpose of the Carnot Cycle though??
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Sadi Carnot introduced the Carnot cycle in an analysis of the efficiency of heat engines in the early 19th century. He showed that efficiency was lost whenever heat engines deviated from being in thermal equilibrium and that any heat engine operating between a maximum temperature, T1, and a minimum temperature, T2, could not have greater efficiency than a Carnot cycle operating between the same temperatures. So in this sense the Carnot cycle is the theoretical ideal.
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Ok, like calgeko, I am confused why adiabatic expansion in carnot cycle decreases temp. if we apply the principle of PV=nRT, it obviouslt states that V varies directly with T. meaning V increase (expansion) means T increase. Pls. kindly help me with this. Thanks.
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In an adiabatic expansion, heat does not go in or out. But that does not mean that energy cannot go in or out. Energy can be transferred through the flow of heat, yes, but it can also be transferred out mechanically. That's what happens in adiabatic expansion. The gas does work to make the volume expand. In doing the work, the gas loses energy, and the result is lower temperature.

The problem with applying PV=nRT the way you are doing above is that you are implcitly assuming P is constant when you say that V has to vary directly with T. IF P is constant, you are correct. But in an adiabatic expansion, P is not constant. It goes down.

Think about the opposite of an adiabatic expansion: a compression. If I tell you I have an insulated piston, and I am going to compress the gas in it by pushing on the piston, your intuition will probably tell you that the pressure and volume are both going to go up, right? Try it with a bicycle pump; you will be able to feel the pump get hot as you try to pump up a tire that is under high pressure.

If it gets hot when you compress, it should get cooler when you decompress, right? So that's adiabatic expansion: lower pressure and lower temperature.

The first law of thermo confirms this. delta U = Q - W. Q is zero in adiabatic expansion. So delta U = - W. In expansion, W is a positive number, so delta U is a negative number. Delta U represents the change in internal energy. In an ideal gas, internal energy is all kinetic energy, so negative delta U means kinetic energy has decreased, which means temperature has decreased.
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From B to C , how come its follows a ln curve even though temperature isn't constant ?

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The curve from A to B is an isotherm, meaning it's temperature is contstant.
The curve from B to C is adiabatic, which is a different curve. (you can see that is doesn't follow the isotherm.)
That it is a curve is because the system can not jump from one volume to another.

Hope this helped.
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What equation does the adiabatic curve follow (specifically)?
I know that PV = nRT.
I'm guessing that the temperature is a function of pressure AND volume T(P,V), but what is the relationship?
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P(V)^n=Constant where n is called the adiabatic exponent IF the process is ADIABATIC For an adiabatic process
n=cp/cv. where cp is the specific heat at constant pressure and cv is the specific heat at constant volume....... Notice that when n=0 ,u get a const pressure line (horizontal). ,when n =1 u get a isothermal curve (rectangular hyperbola) and when n= infinity u get a constant volume curve (vertical line on PV graph.)... Now for adiabatic process ,lets say for system of air, the adiabatic exponent equals 1.40 whixh is >1 and hence the adiabatic curve is MORE sloping down than isothermal curve for air.
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Doesn't an adiabatic process violate the 1st law of thermodynamics ?? If the system is isolated and cant exchange heat with its surroundings and if the KE is used up in doing work , how does the system regain that energy ?? Is it converted to Potential Energy ?
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Energy that was used up from adiabatic process was partly regained in the isothermal process where exchange of energy is possible
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for the D to A process, it's adiabatic and u add pebbles to reach state A, but how do u know it will reach state A exactly instead of somewhere else. i mean how do you make sure it will be the same volume after u add the pebbles
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One has to choose the state, D, very carefully. In other words, the isothermal compression from state C must proceed only until the system reaches the intersection of the isothermal curve for temperature, T2, and the adiabatic curve coming down from state A. That intersection defines the state, D.
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At 15:25 when we start adding pebbles to the system to move from state D to state A was it an adiabatic process?
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The short answer is yes.
B to C and D to A are both adiabatic processes.
A to B and C to D are both isothermic processes.
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I couldn't exactly get what really happened in the case (not named, in the video) just after the 'C' case. When we have a gas at temp T2 put on a reservoir at T2, why and how will the heat be transferred to the reservoir(as the arrow depicts, at 12:59), when both are at same temp.??
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This hypothetical reservoir is infinitely large and can take in any amount of heat without raising its temperature. When pebbles are added after the C point and the pressure increases, there is a tendency for the temperature (hence the total kinetic energy) to also rise. The reservoir allows the system to bleed off this excess heat into the cold reservoir, or sink, and thus allows the system to retain the temperature of T2 from C to D. The system is therefore isothermic between C and D.
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In the adiabatic process from B to C, temperature and pressure are reducing. Wouldn´t the volume be increasing because of the piston moving up (as a result of "pebbles" being removed off of it) and the gas expanding?
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Sal's drawing is a little misleading. From B to C, volume is being held constant by definition so the drawing should be closer to a straight line (assuming it will increase or decrease in a real example). As temperature decrease so will the pressure.
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how come the temperature is not constant in a adiabatic process as dq = ncdt
and for a adiabatic process dq = 0 which means dt must be constant
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Because there's also work being done, one way or another. While heat doesn't flow in or out during an adiabatic process, work either becomes heat or internal heat becomes work (depending on direction of process).

When expanding, some of the internal heat is becoming work as it expands, pushing something out of the way for it's new larger, footprint. When compressing, something outside is doing work on the system to get it in a smaller volume. Without the ability for heat to flow elsewhere, these both result in an internal temperature change.
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why can not the whole heat be converted into work...?? if we think of isothermal process then from first law of thermodynamics, dQ =dW.... or Q=W....!here the whole heat is conveted into work....why so...?
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The best you can do in theory is for Q to be equal to W. In the real world, at least a little bit of the Q is going to leak away rather than be converted into W, because of friction and similar effects.
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I have a question and im sorry if this is slightly off topic but i am just curious. In biology class I just learned about cellular respiration . can i say that the generation of ATP through ATP synthase and the proton gradient across the mitochrondrial inner membrane is like the carnot engine? i didnt understand this video entirely but if the carnot engine is similar to the way ATP is made, it would simplify a lot for me. any help is much appreciated!
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Thermodynamics certainly comes into play in biological processes. In the case of cellular respiration, the laws of thermodynamics plays a huge factor! The electrochemical potential gradient that is created by the ETC is restored by allowing those protons back through, while ATP synthase harnesses the potential energy and turns it into chemical potential energy by creating ATP from ADP and Pi. The second law says that over time, chemical potentials tend to equilibrate in a closed system.

Carnot considered a heat engine in his famous theorems. The cell simply uses a different form of energy (electrochemical potential energy) to get work (the creation of ATP via ATP synthase) out of the system.
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Suppose 100 moles of a monatomic ideal gas undergo an isothermal expesion from 1.00m3 to 4.00m3 .How much work is done by by the gas during this expansion ie. W--------kj
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For an isothermal process, PV=constant as in PV=nRT, 'nRT' is constant when temperature is constant. And we know that workdone is integration of P.dV, so we write it as:
dW(i.e. small change in workdone) = P.dV
we have two equations, isothermal equation; PV=constant and ideal gas equation; PV=nRT.
Now in P.dV, we can write P as (constant/V), and this constant equals nRT as told earlier.
Now, dW= (nRT/V).dV
Integrating on both sides, W= nRT(lnV), where ln means natural logarithm.
now when gas goes from volume V1 to V2, W=nRT(lnV2 - lnV1)
or, W= nRT[ln(V2/V1)]. Remeber this result. Now you can put the given values and obtain the answer!
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During each cycle a reversible engine absorbs 2600j of heat from a high temperature reservoir and performs 2300j work, what is the efficiency of the engine ie. E=?
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2300/2600 *100% = 88%
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I understand what happened in the Carnot cycle, and I understand that work done by the system is the are of that Carnot loop in the PV diagram. My only question is, if gas did work, where did that energy go? Since we ended up back at point A with the same mass on piston, height of piston, pressure of gas, and volume of gas, there was no change in the sum of gravitational potential energy of the rocks and internal energy of the gas.
How could the gas have magically done work without any change energy?
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what is exactly carnot engine?
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How are the Rankine Cycle and the Carnot Cycle related? And is the Carnot Cycle related to the Otto Cycle? or am I just confusing all the cycles?
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If this is the wrong place to ask this, I apologize, but what is a _state variable_?
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ok so the engine gets energy from heat provided does work and thus releases heat while doing work. so if my engine doesnt give out heat while doing work then its 100% efficient. but thats not possible. am i right?
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Yes you are right thats not possible beacuase of friction due, to friction a large of heat is radiated out!
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Is the net work done in cycle is zero ? please tell me that.....?
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yes. This is due to couple. couple is nothing but a pair of equal and opposite forces acting on an object. When we peddle the couple acts and therefore the net force is zero. work is the product of force and displacement. Since force is zero work done will also be zero.
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In my book (atkins) it says that the integral of (dq_reversible/T) = zero for a carnot cycle. The passage is used to explain to us (students) that entropy is a state function. The book makes good sense. Why however is Q total for the carnot cycle not equal to that q_reversible that we were originally intending to find? Is it because the isothermal steps are not reversible?
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this may sound a stupid question but how is A to B an isothermal process when the work is done by the system?
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Why does the temperature increase or decrease when the volume changes? Wouldn't the pressure simply change as well to compensate and leave the temperature constant?
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PV = nRT. When volume changes, pressure AND temperature can both change.
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At 7:00, why is there a change in temperature when there should be no change in temperature in an isothermic process?
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I don't see a change in temperature. Is it possible you confused the labeling of the P-V diagram? The T1 written on the curve states that the temperature of the system is constantly T1 throughout the process from A to B. What changes is the volume of the system which increases, and the pressure of the system which decreases.
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I am confused as to how Q1 - Q2 = W. I mean the system got heat from the first reservoir and then gave heat to the second reservoir and converted some of that heat to work, then how does the system return to it's first state to complete the cycle? Where does that energy come from if the system gave some of that energy as work?
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I do not understand how Q can change when the temperature is constant, as in the first step of the carnot-cycle. Because Q = mcdT and m is not changing and c is a constant, or have I got i wrong? Is c the factor that makes Q change?
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The specific heat formula, Q = mcdT, in which m is the mass of the system and c is the specific heat, applies most straightforwardly to solids and liquids which do not expand very much, compared to gasses, when they are heated. For a gas there is a specific heat for heating at a constant volume, cv, and a specific heat for heating at constant pressure, cp. The latter, cp, is always the larger specific heat since the heat absorbed both changes the temperature and does some work as the volume expands at constant pressure. For ideal gasses the difference is, cp - cv = R, the universal gas constant (if the mass is measured in moles). Now an isothermal process can be thought of as equivalent to a sequence of very tiny volume changes at constant pressure followed by very tiny pressure changes at constant volume, each pair returning to the chosen fixed temperature from which it started. Depending on the direction one is moving along the isothermal, one of the changes in a pair will absorb heat into the system while the other change in the pair will release heat from the system. But since the corresponding specific heats have different values the net effect will be either to absorb heat along the isothermal (increasing volume direction) or to release heat along the isothermal (decreasing volume direction). In this way heat absorption or release at constant temperature is seen to be compatible with the specific heat formula.
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so if we move from left to right work is done by the system?
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Yes. The volume of the cylinder was increased by the particles inside the cylinder, so the system DID work.
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Between C and D, shouldn't Q2 be an arrow pointed away from the isotherm to indicate that heat was transferred into the reservoir from the system?
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I thought this was isothermal not adiabatic
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According to graph given in the vid:
i) A to B and C to D are isothermal processes.
ii) B to C and D to A are adiabatic processes.
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I am wondering since change in internal energy U = 0 if T=const. Is that also true if there is a phase change because then we would not need to have a temperature change but we would still have a change in latent energy which i thought was the internal energy.
Thank you.
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ah true thank you!!
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At @ 15:50 Sal draws the final adiabatic curve from D to A. Is this curve exactly the same shape, length etc as the adiabatic curve from B to C? Thanks!
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No, it is not exactly the same shape, length, etc. But it is the same kind of curve and that kind of curve is of the form,

PV^g = constant,

different constants for different adiabatics and where the exponent, g, on the volume, is the ratio of the specific heat at constant pressure, cp, to the specific heat at constant volume, cv, i.e., g = cp/cv. Since cp > cv, g > 1 and adiabatic curves are steeper than isothermal curves in a P V diagram.
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I got a question!! Usually this is done with a cylinder with a piston.
Suppose there is a some pump of volume 50.
But there is no piston fit into the pump.
Let's say there is a volume of 40 occupied with molecules
When the volume occupied by the molecules changes to 50, and the pressure remains the same then is there any work done even if there is no piston.?
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If there is no piston, then there is no pressure, and no way to define the volume.
No pressure means no work.
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At 12:32 Sal says" its kind of a cold reservoir"
I was hoping to please clarify:This is only because it keeps the temperature of the piston low right? Not because it is colder than the piston
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At 17:50 sal says " the system transfers a smaller amount of heat back to the ambient/other reservoir"
How do we know that it is a smaller amount of heat?
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To make ice, a freezer that is a reverse Carnot engine extracts 42kJ as heat as -15 degree celcius during each cycle, with coefficient of performance 5.7. The room temperature is 30.3 degree celcius. How much (a) energy per cycle is delivered as heat to the room and (b) work per cycle is required to run the freezer?

All I do is just using the formula Kc=QL/(QH-QL) for part (a) and K=QL/W for part (b)? Is that a correct method to do it? Anyone can help me with this?
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Sal, is putting on or off the pebbles to depict compression and expansion of vol respectively..But, what happens actually when we have a flexible piston and a container that is firstly put on a source(at T1), then on an insulating stand, further on a sink and again back on insulating stand?
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One last thing to get confirmed is that in a carnot cycle two adiabatic processes and two isothermic processes take place. Is that right ?
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If an completely isolated system(That doesn't exchange temperature with the surroundings), does work, then it does less work than it should (or less than it would when it would exchange the temperature with surroundings), because when the temperature decreases the, pressure and volume of the final state also decreases. Is that correct?
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why is adiabiatic curve steeper than the isothermic curve??
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Because temperature changes during adiabatic processes, so that means you have to move from one isotherm to another, and the only way to do that is to be steeper.
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What is the effect of the Work W done by the cycle? In what form is it wasted if there is no friction?

Basically why is Q1 > Q2?
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how are isothermal and adiabatic processes related to each other? during the state of thermal equilibrium(dT=0),there is no flow of heat too(dQ=0)...
explain....!
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The way I think about the relationship is this: you can draw a series of isotherm curves that each represent the PV combinations at a certain temperature. If you want to stay at the same temperature, you travel along an isotherm. If you want to move from one temperature to another, you have to move from one isotherm curve to another. There are three ways you can do that:
isobaric, where the pressure does not change and maximum work gets done
isochoric, where the volume does not change, and no work gets done
adiabatic, where no heat comes or goes, so the internal energy change (ie the temperature change) determines the work

In a way, an isothermal process is the opposite of an adiabatic one. In an isothermal process, you are allowing heat to come and go so that you are at the same temperature as your surroundings. You have to do an isothermal change slowly and with easy flow of energy into or out of the system. Adiabatic is the opposite. You are not letting any heat come in or go out. If your system gets very far from the temperature of the surroundings, still no heat can flow in or out to restore thermal equilibrium. The way you do that is to insulate your system as much as possible. You also do an adiabatic change quickly, sort of like you are trying to get away with it before the heat can leak in or out.
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If that (V,p)-co-ordinate diagram or pressure as a function of volume diagram is given with arrows, what can one deduce about it without further knowledge.

Basically what does a mere (V,p) with arrows diagram tell us?
Is it just the work done to/by the system or are we able to deduce also the isothermic/adiabatic processes between the points??
I guess we need the T:s as well to be able to say that?
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I don't understand why the net amount of heat is >0. I would think that Q1 and Q2 cancel eachother out completely.
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Think about it this way. As Sal mentioned, the net internal energy as the system is always the same at point A (assuming no energy loss). However, we know the system performed work (Since work performed is the integral of P(V), and the area under the curve on the path from A to C and the different path from C to A are not equal). Since Delta U = Q - W, U = 0, and we know W is positive, Q has to be positive. Therefore, the net heat added to the system is positive.
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I wonder why the videos after Thermodynamics 5 aren't in Physics playlist on youtube? Before using the actual website I used to watch video on youtube directly. I think that would be much more conveniently if they were added there as well. But still many thanks for the lessons! :)
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@10:12 Khan says that his pressure would go down at T2 but his volume would go down also.
I suppose that this is wrong and only a spoken mistake, because the move to the right o in the positive direction on the V axes can only mean more volume and lower pressure. Lower volume does not normally go hand in hand with lower pressure. So, although Khan probably meant more volume and lower pressure he said lower volume and lover pressure. Please confirm;)
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i noticed the same. Yes, I'm sure it was a small error
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we learned that in an adiabetic process pressure, volume and temperature go down - should' nt then the corresponding curve fall vertically (no increase in volume)?
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In an adiabatic process, no heat goes in or out. That means you move from one isotherm to another as pressure or volume changes. There's no reason you have to move straight up or down.

A process with no change in volume is called isochoric or isovolumetric. No work gets done because the volume does not change.
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I was not clear enough im my question yersterday. If in an
adiabetic process the volume decreases then the curve in the PV diagramm should not bent to the right but to the left or why not?
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The things that determine the lines in the diagram are Pressure and Volume. If the line would bend to the left, this would indicate that if the volume becomes smaller the pressure becomes smaller as well, which is not the case. (think of the pistons Sal so often uses to explain features in thermodynamics). So in the case of a cilinder with a piston, if the curve would bend the to left this would mean that you would DECREASE the volume and with this DECREASE the amount of bumps all the gas particles would deliver to the walls of the cilinder(pressure). This leads to a contradiction.

I'm not very good at English but I hope this helped.
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why are we taking change in internal energy _*ZERO*_ ???
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1)its given that in insothermal process Q=W In what way does the gas do the work?
2)why is is it only PV diagram why not PT diagram
3)why cant v keep the cylinder first on the cold reservoir and then keep it on the hot reservoir?
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The gas expands. The expansion does work.
You can draw a PT diagram, too. It will convey the same information. PV is sort of easier to think about.
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2 Comments
How can B go to C in an adiabatic process when B originally came from A in a isothermic progress, does this mean B was just 'pulled out' of the giant reservoir keeping it isothermic.
If this is done wouldn't this mean that work(the moving from B) would be done on system B additional to the work that comes from removing the rocks.
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How is work of adding or removing the pebbles accounted for? Without this work, nothing gets done.
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In physics Work is not the normal work. Its practically something humans will never find useful in future ('sigh").....I mean how much work did Peter do in lifting the weight (100kg) and in keeping it back on the stand ? Hes doin lots of work but since displacement is 0 , WORK IS 0. Imagine the poor thing's plight!
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3 Comments
Why does the volume continue to expand during the "adiabatic expansion" if no additional heat is added? And likewise, why does the volume continue to fall during the "adiabatic compression" after the heat sink is removed?
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The volume expands until the interior pressure falls to match the exterior pressure.
On the return cycle, the volume contracts due to outside pressure until the interior pressure rises to meet it.
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Why are the adiabatic parts of the PV diagram of the Carnot Cycle shaped like a rectangular parabola as well? .. They do not occur at a constant temperature so they need not be shaped that way right?
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how does he know (19:34) that there is more heat going in the systm than heat going out?
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Since work=heat, and from step C to A (that the system is receiving heat) the work is less than from step A to C.
And this is because when the temperature of a system is at lower state you need less force to do work to the system.
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Why is it that in most cases the PV diagram is a rectangular hyperbola ?? Like a downward curve ?? Could it also be like an upward curve , or a diagonal straight line maybe ?? Thanks !!! :)
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it cnt be an upward curve , ur decreasing the volume it will have to go down
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I'm still confused. What is a Carnot cycle, then?
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The Carnot Cycle has been mathematically proven to be the most efficient method for converting thermal energy into work and vise versa.
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20:52 I cannot understand why the point B to C can happen because of adiabatic situation. Their temperature changed T1 to T2, I guess. What happened between B to C?
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Remember that physically we allowed the gas to expand by decreasing the restraining force (pebbles). If you think about the gas laws PV/T = constant and since we did not keep T constant, the net result is V increase, P decrease, T decrease.
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How can cooling and heating things ever be adiabatic??
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System changes in the real world can't be perfectly adiabatic, but if you insulate something very well, you can isolate the system enough so that you can analyze it as adiabatic.
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I understand that for isothermal -w=nRT(lnV2-lnV1)

can I say that for adiabatic:

since Q=0 then DU=-w
since U=3/2 PV then DU=(3/2P2V2)-(3/2P1V1)

so: -w=(3/2P2V2)-(3/2P1V1)
so I can calculate the work done by the system in a adiabatic expansion knowing initial and final pressures and volumes? Does that correspond also to the area under the adiabatic curve in the PV graph?

Thanks,

G.
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Yes you can. watch the previous video again. :)
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from a to b system has a conducting wall because reservoirs transfer heat to the cylinder.but from b to c it is an adiabatic process meaning that the cylinder has a diathermic wall(no heat can enter or exit)how is this possible?
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How carnot engine works ?
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I have a formula for Work done per cycle of the Carnot cycle:
W = η R TH ln (V3/V4)
Where η is efficiency & TH is Temp hot.
I'm not sure how this was derived; any ideas? Thanks
Nic
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Can anyone please explain to me what an Adiabetic system is?
Am i correct that if it is Adiabetic the temperature in the system will change? (heat will be added or removed from a "reservoir"?) And if its a Isotherm the tempertature will not change? And that a system cannot be Adiabetic and Isothermic at the same time?
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Ok I have heard the term adiabatic before. What is adiabatic pressure or compression? I work at a ASU Air separating unit - cryogenics. I understood it was similar to lets say water hammer. But you got me think more on the macro and micro level of that system. Lets Use 99% oxygen at 160degree F traveling town a 5 inch pipe with a gauge pressure of 56psi. How would you draw up and explain the adiabatic pressure if that system flow was interrupted by a sudden stop of flow - a valve closing quickly
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Why does the Temperature goes down, in the process of B-C??
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hey i ve two questions regarding carnot cycle
1) when we reached to state D we will again remove the reservoir , again adiabatic process will occur ? what i meant to say is that , again the WHOLE process will work?

2) why we use carnot engine ? what are the uses of carnot engine?

i hope you understand the questions which i have asked you .
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Carnot Engine has no particular use....except in theory. It's just a module to show that maybe, when things become more advanced and efficient to the
limit that no heat is given out, ALL of the heat can be converted to useful work.
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• a question in Tips & Thanks
• an answer that should be its own question
about the site
• a question about Khan Academy
• a post about badges
• a technical problem with the site
• a request for features
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