Is PV^gamma Constant in Adiabatic Processes?

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In summary, the homework equation for proving PV^gamma = constant is: ΔEint = WnCvdT = PdVnRT / V dV.
  • #1
stratz
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Homework Statement


Proving PV^gamma = constant In adiabatic expansion. Q = 0

Homework Equations


N/A

The Attempt at a Solution



ΔEint = W

nCvdT = PdV = nRT / V dV

∫Cv/T dT = ∫R/V dV

Cv ln(T2/T1) = R ln(V2/V1)

ln(T2/T1) = (R/Cv) ln(V2/V1)

T2/T1 = (V2/V1)R⋅gamma / Cp

P2V2 / P1V1 = (V2/V1)R⋅gamma / Cp

I did some extra rearranging after this point, but it appears that the equation in this form cannot get PV^gamma = constant or for P1V1^gamma = P2V2^gamma. Is there a mistake somewhere?

Thanks.
 
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  • #2
stratz said:
ΔEint = W

nCvdT = PdV = nRT / V dV
Watch the signs. Does W stand for the work done by the gas or the work done on the gas?
 
  • #3
TSny said:
Watch the signs. Does W stand for the work done by the gas or the work done on the gas?
Well it is the differentiated form of work, but I guess that doesn't matter since the rate of change of work done on the gas is negative right?

I'll try adding a negative sign to that and see if it works. Thanks
 
  • #4
To see if dT and dV have the same sign, think about what happens to the temperature during an adiabatic expansion.
 
  • #5
Okay, so temperature will decrease as energy is lost in the form of work on the environment due to expansion of volume right?
 
  • #6
stratz said:
Okay, so temperature will decrease as energy is lost in the form of work on the environment due to expansion of volume right?
Yes.

Some people write the first law for an infinitesimal reversible step as dE = dQ - dWby where dWby is the work done by the gas: dWby = +PdV.

Others write it as dE = dQ + dWon where dWon is the work done on the gas: dWon = - PdV

Either way, dE = dQ - PdV.
 
  • #7
Alright, I got this to work correctly, I had to use the Cv + R identity which canceled out the Cp in the denominator. However this gave me P1V1^gamma = P2V2^gamma. Would it be possible to rearrange this somehow to make just one side equal to a constant?
 
  • #8
State 2 can be thought of as any state during the adiabatic process. So for any state (P, V) during the process, you have P V γ = P1V1γ.
 
  • #9
TSny said:
State 2 can be thought of as any state during the adiabatic process. So for any state (P, V) during the process, you have P V γ = P1V1γ.

Yeah, it's just that I saw a proof somewhere where they got PVgamma = econstant which ended up being PVgamma = constant. Anyways, it's pretty much the same result.

Thanks for the help
 
  • #10
Instead of doing definite integrals from state 1 to state 2, you could do indefinite integrals. This will lead to P V γ = const where the const is related to the constants of integration of the indefinite integrals.
 
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Related to Is PV^gamma Constant in Adiabatic Processes?

What is the significance of proving PV^gamma = Constant?

Proving PV^gamma = Constant is important because it is a fundamental law in thermodynamics, known as the ideal gas law. It describes the relationship between the pressure (P), volume (V), and temperature (T) of an ideal gas and is essential in understanding the behavior of gases.

What does PV^gamma = Constant represent?

PV^gamma = Constant represents the relationship between the pressure and volume of an ideal gas at a constant temperature. It states that the product of the pressure and volume of an ideal gas raised to a constant value (gamma) remains constant.

How is PV^gamma = Constant derived?

PV^gamma = Constant is derived from the combination of Boyle's law, Charles's law, and Avogadro's law. Boyle's law states that at a constant temperature, the product of pressure and volume is constant. Charles's law states that at a constant pressure, the volume of an ideal gas is directly proportional to its temperature. Lastly, Avogadro's law states that at a constant pressure and temperature, the volume of an ideal gas is directly proportional to the number of moles of gas present. When these three laws are combined, they result in the ideal gas law (PV = nRT), which can be rearranged to PV^gamma = Constant.

What is the value of gamma in PV^gamma = Constant?

The value of gamma in PV^gamma = Constant is dependent on the type of gas being studied. For monatomic ideal gases, gamma is equal to 5/3. For diatomic ideal gases, gamma is equal to 7/5. For polyatomic ideal gases, gamma can vary between 1.3 and 1.6, depending on the complexity of the molecule.

How is PV^gamma = Constant used in practical applications?

PV^gamma = Constant is used in various practical applications, such as in the design and operation of gas turbines, internal combustion engines, and refrigeration systems. It is also used in the analysis of atmospheric and oceanic processes, as well as in the study of the behavior of stars and planets. Additionally, it is used in the development of new materials and technologies, such as in the production of solar panels and other renewable energy sources.

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