Value of gamma in adiabatic process

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SUMMARY

The discussion focuses on the calculation of the final temperature in an adiabatic process involving a mixture of helium and oxygen, known as heliox. The initial temperature is 298 K, and the process is characterized by the equation T1 * (V1 ^(γ-1)) = T2 * (((V1/2) ^(γ-1))). The values of specific heat capacities are provided: for helium, Cp = 5/2 R and Cv = 3/2 R, resulting in γ = 5/3; for oxygen, Cp = 7/2 R and Cv = 5/2 R, leading to γ = 7/5. The confusion arises regarding the calculation of γ for the gas mixture, with the consensus that the heat capacity of an ideal gas mixture is the molar average of the individual gases' heat capacities.

PREREQUISITES
  • Understanding of adiabatic processes in thermodynamics
  • Knowledge of specific heat capacities (Cp and Cv) for monoatomic and diatomic gases
  • Familiarity with the concept of γ (gamma) in thermodynamics
  • Basic principles of gas mixtures and their properties
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  • Research the calculation of γ for gas mixtures in thermodynamics
  • Study the principles of adiabatic processes and their applications
  • Learn about the heat capacities of various gas mixtures
  • Explore the implications of using heliox in diving and its thermodynamic properties
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Students and professionals in thermodynamics, chemical engineering, and diving physiology, particularly those interested in the properties of gas mixtures and adiabatic processes.

Pouyan
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In conjunction with diving, a mixture of helium and oxygen called heliox. It will form the background to the following model invoice. Suppose that 1 mole of He and 1 mole of O2 is compressed as fast at half the volume that no heat exchange with the environment have time to take place. Temperature at the start is 298 K. What will be the final temperature?!I know this is an adiabatic process in this case
T1 * (V1 ^(γ-1)) = T2 * (((V1/2) ^(γ-1)))
And T2 = T1 * (2 ^ (γ-1))

The attempt at a solution
I know that Helium is a mono atomic gas and value for Cp= 5/2 R and Cv= 3/2 R. That means γ = 5/3
Oxygen is a gas with two atoms and Cp= 7/2 R and Cv = 5/2 R. The value of γ = 7/5

I don't understand the value of γ when we have a mixture. I see a solution that says we have equal quantities of He and O2 when the Cp for heliox 3 R and gamma is 3/2.
Why ?! Should I take the average of the Cp of Helium and Oxygen?
 
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Can someone help me ?!
 
The heat capacity of an ideal gas mixture is equal to the molar average of the heat capacities of the individual gases. This is because the heat of mixing of ideal gases is zero.

Chet
 
The book claims the answer is that all the magnitudes are the same because "the gravitational force on the penguin is the same". I'm having trouble understanding this. I thought the buoyant force was equal to the weight of the fluid displaced. Weight depends on mass which depends on density. Therefore, due to the differing densities the buoyant force will be different in each case? Is this incorrect?

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