Help with Adiabatic Derivation

In summary, an adiabatic derivation is a mathematical process used to describe the behavior of a thermodynamic system that undergoes a reversible, adiabatic process where no heat is exchanged with the surroundings and the internal energy of the system remains constant. Its purpose is to analyze the changes in thermodynamic variables and gain a better understanding of how the system behaves under specific conditions. The key equations used include the first and second laws of thermodynamics. Adiabatic derivations have various real-world applications, such as in the study of atmospheric processes and the design of refrigeration systems. However, they have limitations, such as assuming a reversible process and not considering external factors, and can become more complex for systems with more than three variables
  • #1
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Using [tex]dU = dW[/tex],

[tex]\frac{f}{2}NkdT = -PdV[/tex]

I eventually came to

[tex]T^\frac{f}{2}V = \mbox{constant}[/tex]

I tried to then get it to the form in the book - [tex]PV^\frac{f+2}{f}=\mbox{constant}[/tex] using the formula [tex]PV = NkT[/tex]:

[tex]\left(\frac{PV}{Nk}\right)^\frac{f}{2}V=\mbox{constant}[/tex]

[tex]P^\frac{f}{2}V^\frac{f+2}{2}=\mbox{constant}[/tex]

How do I get it in the right form? Thanks
 
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  • #2
[tex]P^{\frac{f}{2}} V^{\frac{f+2}{2}}=const[/tex]
so i have:
[tex]\frac{f}{2}lnP+\frac{f+2}{2}lnV=constant[/tex]
then:
[tex]lnP+\frac{f+2}{f}lnV=constant[/tex]
 
  • #3
Thanks quydau35 :)
 

1. What is an adiabatic derivation?

An adiabatic derivation is a mathematical process used to describe the behavior of a thermodynamic system that undergoes a reversible, adiabatic process. This means that no heat is exchanged with the surroundings, and the internal energy of the system remains constant.

2. What is the purpose of an adiabatic derivation?

The purpose of an adiabatic derivation is to analyze the changes in thermodynamic variables such as temperature, pressure, and volume of a system as it undergoes an adiabatic process. This allows for a better understanding of how the system behaves under specific conditions.

3. What are the key equations used in an adiabatic derivation?

The key equations used in an adiabatic derivation include the first law of thermodynamics, which states that the change in internal energy of a system is equal to the heat added to the system minus the work done by the system. This is represented by the equation dU = Q - W. The second law of thermodynamics is also used, which states that the change in entropy of a system is equal to the heat added to the system divided by the temperature. This is represented by the equation dS = Q/T.

4. What are some real-world applications of adiabatic derivations?

Adiabatic derivations are used in a variety of real-world applications, including the study of atmospheric processes, the behavior of gases in engines, and the design of refrigeration and air conditioning systems. They are also used in the analysis of chemical reactions and the behavior of materials under extreme temperatures.

5. What are the limitations of an adiabatic derivation?

One limitation of an adiabatic derivation is that it assumes a reversible process, which may not always be the case in real-world scenarios. It also does not take into account external factors such as friction and turbulence, which can affect the behavior of a system. Additionally, the equations used in an adiabatic derivation may become more complex for systems with more than three variables, making it more difficult to analyze.

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