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### Course: Thermal physics (Essentials) - Class 11th>Unit 3

Lesson 7: Efficiency of a Carnot engine

# Proof: Volume ratios in a Carnot cycle

Proof of the volume ratios in a Carnot cycle. Created by Sal Khan.

## Want to join the conversation?

• At , Sal integrated the entire equation. The right hand side of equation is zero. The integration of zero must be a constant as derivative of a constant is zero. But, instead after integrating the equation was written still equal to zero.
Can anyone explain? It will be very helpful. Thanks.
• As Andrew said, it's a constant, but it's the same constant at the beginning and end of the integral, so it subtracts to zero.
(1 vote)
• What is significance of using only adiabatic processes in the relation ignoring isothermal processes occurring in the the carnot's cycle?
• This proof is for a Carnot cycle, which has two adiabatic segments and two isothermal segments. He used the adiabatic processes to do the math, but since V is a state variable, the V ratio of the "four corners" holds true for the CYCLE regardless of how you derive it. It may be possible to derive it using the isothermal segments, but I have not tried. It would have to yield the same ratio, obviously, since it's still the same overall cycle. My guess is that the math is more complicated with the isothermals, or else Sal would have proved it with the isothermals. Not sure. You could try that proof yourself and see if it is easier.
• At , how is (T2/T1)^3/2 the same as (T1/T2)^3/2
• You should probably watch this again — Sal never says those are equal!

It may be helpful to you to write out the manipulations he is showing on paper and think about what is happening at each step ...
• Is there a reason behind the laws of Thermodynamics being named as Zeroth, First and Second rather than conventional 1st 2nd and 3rd laws?
• This happened because, instead of the three laws of motion, which were written in one instance by Isaac Newton, the first and second law of thermodynamics wer written before, and the zero law, which we see as more fundamental, was formalized after.
• At 2.35 Sal wrote delta U + P delta V = 0 which is true for adiabatic expansion only, not for adiabatic compression. Using this equation he obtained another expression at which should again be true only for adiabatic expansion. Then at how could it be applied for process D to A which is adiabatic compression ?
• Why is it true for expansion but not compression? The only difference between the two is the sign of delta v.
• at why is it (Tf/Ts)^2/3(Vf/Vs)=1????
why is it 1???
• To start with, ln [(Tf/Ts)^3/2(Vf/Vs)]=0. Now, ln (natural log) is logarithm base e (a constant), so log e [(Tf/Ts)^3/2(Vf/Vs)] = 0. From this, and by the property of logarithm, e to the power of 0 (e^0) = [(Tf/Ts)^3/2(Vf/Vs)]. Property of logarithm says whatever number to the 0th power equals 1. So since (e^0) = [(Tf/Ts)^3/2(Vf/Vs)], this whole [(Tf/Ts)^3/2(Vf/Vs)] expression equals 1.
• what is an initial volume and how do you find it on a graph?
• Depending on what your axis are it is usually when x=(0) assuming y is volume
• Could someone explain to me why at Sal is able to divide the 'work done' term by nRT, specifically by T? If it were an isothermal process then I would understand how since T isn't varying we can just cancel it but in an adiabatic process such as this, T is a variable so it is not erroneous to divide through by it in this way ? (He has written pressure as a function of volume AND temperature NOT just volume, as would be the case when dealing with work done in an isothermal process).
• I believe he can divide throughout by a constant or variable, as long as he does it equally on both sides of the equation (which he does)
There is no physical significance to what he is doing, it is simply a mathematical way of rearranging the equation to make the integeration possible (seperating V and T terms)

Make sense OK??
• Why are integrals common in thermodynamics? At one point Sal says, "Let's do these integrals. This tends to show up a lot in thermodynamics, these antiderivatives."
• The integral give the area under the adiabatic curve
(1 vote)
• I know delta means difference between two specific states. But, what is only V and T (i mean variables without delta) standing for? A very specific state between two states or any state between them?