Simple entropy calculation

In summary, the author is discussing the Carnot cycle and how the efficiency is related to the temperatures of various processes. He provides a sketch of the cycle and explains that the initial volume is not at the isothermal expansion. He also provides a reminder that the ratio of maximum to minimum volume is the volume at the tip of the adiabatic expansion.
  • #1
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Homework Statement



A Carnot engine operates on 1 Kg of CH4, which we shall consider an ideal gas. Take [tex]\gamma[/tex]=1.35. The ratio of max to min volume is 4 and the cycle efficency is 25%. Find the entropy increase during the isothermal expansion.
[Classical and Statistical Thermodynamics, Ashley H. Carter, pg.104]

Homework Equations



dU = TdS-PdV

The Attempt at a Solution



Ideal gas, isothermal expansion,[tex]\Rightarrow[/tex] dU = 0

[tex]\Rightarrow[/tex] dS=[tex]\frac{P}{T}[/tex]dV=nR[tex]\frac{dV}{V}[/tex]

[tex]\Rightarrow[/tex] [tex]\Delta[/tex]S=R(62.5)ln(4)=720J/K


The correct answer is 294J/K

I know my problem has to be with the number of moles, probably accounting for the efficiency. working backwards I see there should be 25.5 moles of methane for n.
How do I account for the efficiency and get this answer?
 
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  • #2
In a Carnot cycle, the efficiency is connected to the temperatures of various processes. Does this help? Have you sketched out the cycle?

(Also, 1 kg methane is not 25.5 moles.)
 
  • #3
Mapes said:
In a Carnot cycle, the efficiency is connected to the temperatures of various processes. Does this help? Have you sketched out the cycle?

[tex]\eta=1-\frac{T_{1}}{T_{2}}[/tex]

Is that what you mean? How does that help for an isotherm? I'm not sure what information to use regarding the Carnot cycle itself considering the question only asks about the isothermal expansion.

For just the expansion with the ratio of volumes given my answer should be correct, there is some relevant information I am not seeing.


(Also, 1 kg methane is not 25.5 moles.)

I know, its 62.5 moles. I got 25.5 from solving
[tex] nRln(\frac{V2}{V1})=294J/K[/tex]
for n.
 
  • #4
Sketch the cycle; are the volumes at the beginning and end of the isothermal process the maximum and minimum volumes in the cycle?
 
  • #5
Thank you Mapes.

Of course those are not the max/min volumes. I was able to obtain the correct volume ratio buy utilizing [tex]PV^{\gamma}=constant[/tex] for an an adiabat,[tex] \eta=1-\frac{T_1}{t_2}[/tex], and the given information.
 
  • #6
I'm doing this question right now, and I am wondering how to get 294 J/K as an answer. I don't understand why the final volume doesn't relate to the max volume, and how to use PV^gamma to get the correct ratio.

Thanks.
 
  • #7
Seem you guys forget that the initial volumn is not at the isothermal expansion, you have to trace it back to the start of adiabatic expansion then the isothermal expansion. (draw the diagram and you will see)

Ratio Max/Min volume is the volume at the tip of adiabatic expansion.

You'll need to use both PVg = PV/T = constant

Good brush up exercise though.
 

1. What is entropy?

Entropy is a measure of the disorder or randomness in a system. It is a fundamental concept in thermodynamics and is commonly used to describe the amount of energy in a system that is unavailable for work.

2. How is entropy calculated?

The simple entropy calculation formula is: S = k ln W, where S is the entropy, k is the Boltzmann constant (1.38 x 10^-23 J/K), and W is the number of microstates or possible arrangements of a system. This formula can be applied to various systems, such as gases, solids, and mixtures.

3. Can entropy be negative?

In a closed system, entropy can never decrease, so it cannot be negative. However, in an open system where energy can be exchanged with the surroundings, the entropy of the system can decrease while the total entropy of the universe increases.

4. What factors affect entropy?

Entropy is affected by the number of particles or molecules in a system, temperature, and the degree of disorder or randomness. An increase in any of these factors will lead to an increase in entropy.

5. How is entropy related to the second law of thermodynamics?

The second law of thermodynamics states that the total entropy of a closed system will always tend to increase over time. This means that systems naturally tend towards a state of greater disorder or randomness, which is reflected in the increase of entropy.

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