Understanding Adiabatic Expansion in Thermodynamics

In summary, during an adiabatic expansion, PV^gamma is a constant for an ideal gas according to the first law and the ideal gas law. The formula nRdT/gamma - 1 can also be derived using definite integration and the adiabatic condition. This formula can be used to calculate work done during an adiabatic process.
  • #1
vaishakh
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Can anyone here help me to derive that during an adiabatic expansion, PV^gamma?is a constant, as well as other expressions similar to the above?
I have found the following equation using definite integration and the basic formulae, nRdT/gamma - 1 = work done.
 
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  • #2
vaishakh,
You can prove [tex] P V^{\gamma}[/tex] is constant for an ideal gas in an adiabatic process from the first law ( [itex] dU = dQ - pdV , dQ=0) [/itex] and the ideal gas law [itex] (pV=nRT) [/itex].
How did you get your formula nRdT/gamma - 1 = work done.?
 
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  • #3
vaishakh said:
Can anyone here help me to derive that during an adiabatic expansion, PV^gamma?is a constant, as well as other expressions similar to the above?
Since heat flow (Q) is zero, use:

[tex]nC_VdT = dU = PdV[/tex] and

[tex]VdP + PdV = nRdT = n(C_P - C_V)dT[/tex]

This will give two expressions for ndT. Integrate both expressions.
I have found the following equation using definite integration and the basic formulae, nRdT/gamma - 1 = work done.
This follows from the adiabatic condition. One can express the work as:

[tex]W = \int_{V_i}^{V_f} PdV = \int_{V_i}^{V_f} \frac{PV^\gamma}{V^\gamma}dV = K\int_{V_i}^{V_f} \frac{dV}{V^\gamma}[/tex]
Work that out to get the expression for Work.

AM
 
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  • #4
Thanks for clearing the second part Andrew.
 

1. What is the definition of thermodynamics?

Thermodynamics is the study of energy and its transformation from one form to another, as well as the effects of this transformation on matter and the surrounding environment.

2. What are the three laws of thermodynamics?

The first law states that energy cannot be created or destroyed, only transferred or converted. The second law states that the total entropy of a closed system always increases over time. The third law states that as the temperature of a system approaches absolute zero, its entropy also approaches a constant value.

3. What is the difference between heat and temperature?

Heat is the transfer of energy from a hotter object to a colder object, while temperature is a measure of the average kinetic energy of particles in a substance.

4. What is an example of the first law of thermodynamics in action?

An example of the first law of thermodynamics is when a hot cup of coffee is left to cool down in a room. The coffee will transfer its heat energy to the surrounding air until it reaches the same temperature, demonstrating the conservation of energy.

5. How is thermodynamics applied in real-life situations?

Thermodynamics is applied in various real-life situations, such as in the design of engines and power plants, refrigeration systems, and chemical reactions. It is also used in understanding weather patterns and climate change, as well as in the production of food, medicines, and other products.

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