Imaginary Ideal Gas Cycle Proof

In summary, the conversation discusses the calculation of thermal efficiency for an imaginary ideal-gas cycle using constant heat capacities. The equation for thermal efficiency is η = 1 - γ[((V1/V2)-1)/((P3/P2)-1)]. The conversation also explores the use of the ideal gas law, PV=RT, in solving for the temperatures in the cycle and concludes with the simplified equation for thermal efficiency.
  • #1
jrklx250s
15
0

Homework Statement


Given an imaginary ideal-gas cycle. Assuming constant heat capacities, show that the thermal efficiency is

η = 1 - γ[((V1/V2)-1)/((P3/P2)-1)]

Since i can't show you the cycle we are shown that

l Qh l = which is absolute value of the heat at high temperature = Cv(T3-T2)
l QL l = which is absolute value of the heat at low temperature = Cp(T1-T2)
Cp/Cv = γ

The Attempt at a Solution



Ok so subing in these equations for thermal efficiency
which is

η = 1 - l QL l / l Qh l

we get...

η = 1 - γ(T1 - T2)/(T3 - T2)

η = 1 - γ((T1/T3) - 1)

This imaginary cycle only has a power stroke and we are assuming that its adiabatic...from this we concluded that

T1V1^(γ-1) = T3V2^(γ-1)
T1P2^((1-γ)/y)=T3P3^((1-γ)/y)

divide each equation we get

V1^(γ-1)/P2^((1-γ)/y) = V2^(γ-1)/P3^((1-γ)/y)

Now I am not sure how to rearrange from here to make T1/T3 = (V1/V2)/(P3/P2)

Any suggestions would be greatly appreciative.
Thanks!
 
Last edited:
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  • #2
Since no one has replied I'm assuming some are confused as to what I'm talking about so here is the ideal gas cycle that I need to calculate the thermal efficiency from.
Here is the link to the picture of the cycle
http://imageshack.us/photo/my-images/411/img1048u.jpg/
 
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  • #3
Hi jrklx250s! :smile:

Did you already try to apply the ideal gas law PV=RT?
 
  • #4
Hi Serena,

Yes I believe so when i calculated the adiabatic processes for the power stroke... which i concluded that they were

T1V1^(γ-1) = T3V2^(γ-1)
T1P2^((1-γ)/y)=T3P3^((1-γ)/y)

And Since I need to make T1/T3 = (V1/V2)/(P3/P2)

This means that T1 = (V1*P2)

and T3 = (V2*P3)

not sure how to conclude these from the two equations above. And I know its a simple alegbraic rearrangement that I am missing here.
 
  • #5
Careful. Let's start with (V1/V2)/(P3/P2).
With some fraction manipulations this is equal to (P2*V1) / (P3*V2).

Looking at the diagram you posted I can see that P1=P2 and that V1=V3.
Furthermore you have that for instance P1*V1 = R*T1.

Perhaps you can use that?
 
  • #6
Haha wow...thank you serena I was making this so much more complicated than it was.

Yea of course you can just conclude that since
P1=P2
V2=V3

so therefore...
P1V1=nRT1
P3V3=nRT3

P2V1=nRT1
P3V2=nRT3

solving for both T's

T1=P2V1/nR
T3=P3V2/nR

sub this in my previous equation and we get...
η= 1 - ((V1/V2)-1)/((P3/P2)-1)

Thank you.
 

What is an imaginary ideal gas cycle?

An imaginary ideal gas cycle is a theoretical model used in thermodynamics to study the behavior of gases. It is an idealized version of a real gas cycle that makes certain assumptions to simplify the analysis.

What are the assumptions made in an imaginary ideal gas cycle?

The main assumptions made in an imaginary ideal gas cycle are that the gas particles are point masses with no volume, there is no intermolecular attraction or repulsion between the particles, and the gas undergoes reversible processes. These assumptions allow for easier calculations and analysis of the gas cycle.

What is the purpose of an imaginary ideal gas cycle proof?

The purpose of an imaginary ideal gas cycle proof is to show that the theoretical model accurately represents the behavior of real gases under certain conditions. This helps scientists understand and predict the behavior of gases in various situations.

How is an imaginary ideal gas cycle different from a real gas cycle?

An imaginary ideal gas cycle is different from a real gas cycle in that it assumes ideal conditions that do not exist in the real world. Real gases have volume and experience intermolecular forces, while ideal gases do not. Additionally, real gas cycles can experience irreversible processes, while ideal gas cycles are assumed to be reversible.

What are some applications of an imaginary ideal gas cycle?

An imaginary ideal gas cycle is commonly used in thermodynamics and engineering to study and design various systems involving gases, such as heat engines and refrigeration systems. It is also used in the development of equations and laws, such as the ideal gas law, which have a wide range of applications in science and engineering.

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