Reversible adiabatic expansion

In summary, the conversation is about a tutorial question on showing the work done during a reversible adiabatic expansion of an ideal gas, which is given by the equation W = (P1V1 – P2V2)/(1 ‐ γ). The conversation includes attempts at solving the problem, including using the first law and the ideal gas law. The final solution is provided by applying the ideal gas law and substituting it into the equation, resulting in a simplified and easier solution.
  • #1
tensus2000
6
0

Homework Statement



I'm in a rutt for a tutorial question:

The question is basically to show that the work done during a reversible adiabatic expansion of an ideal gas is

W = (P1V1 – P2V2)/(1 ‐ γ) ... Y is gamma

Homework Equations





The Attempt at a Solution



I've got so far as to get W= (P1V1^γ – P2V2^γ)
due to P1V1^γ being constant for a reversible adiabat
also that W= -PdV

But i haven't a clue how to get the 1-Y at the bottom, me thinks its intergrating for V to get this but my maths is very bad so i don't know how to do this.
 
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  • #2
  • #3
tensus2000 said:

Homework Statement



I'm in a rutt for a tutorial question:

The question is basically to show that the work done during a reversible adiabatic expansion of an ideal gas is

W = (P1V1 – P2V2)/(1 ‐ γ) ... Y is gamma

Homework Equations


The Attempt at a Solution



I've got so far as to get W= (P1V1^γ – P2V2^γ)
due to P1V1^γ being constant for a reversible adiabat
also that W= -PdV

But i haven't a clue how to get the 1-Y at the bottom, me thinks its intergrating for V to get this but my maths is very bad so i don't know how to do this.
Apply the first law: [itex]\Delta Q = \Delta U + W[/itex].

Since [itex]\Delta Q = 0[/itex] and [itex]\Delta U = nC_v\Delta T[/itex] where [itex]C_v = R/(\gamma - 1)[/itex] you should be able to work it out quickly. (Hint: apply the ideal gas law: PV=nRT).

AM
 
  • #4
still lost
 
  • #5
tensus2000 said:
still lost

[tex]\Delta Q = \Delta U + W = 0[/tex]

(1) [tex]\therefore W = - \Delta U[/tex]


Now:

(2) [tex]\Delta U = nC_v\Delta T[/tex] and

[tex]C_v = C_p/\gamma = (C_v+R)/\gamma[/tex] so:

(3) [tex]C_v = R/(\gamma-1)[/tex]

Substitute (3) into (2) and then just substitute PV for nRT

AM
 
  • #6
Cheers, i got it
The way you showed was much easier then the way i was going about it
 

Related to Reversible adiabatic expansion

1. What is reversible adiabatic expansion?

Reversible adiabatic expansion is a process in thermodynamics where a gas expands without gaining or losing heat, and the process is reversible. This means that the expansion of the gas can be reversed by compressing it back to its original state without any change in temperature.

2. How is reversible adiabatic expansion different from irreversible adiabatic expansion?

In reversible adiabatic expansion, the process is carried out slowly and is in thermal equilibrium throughout. This means that the system remains at a constant temperature throughout the process. In contrast, irreversible adiabatic expansion happens quickly and is not in thermal equilibrium, resulting in a change in temperature.

3. What is the equation for reversible adiabatic expansion?

The equation for reversible adiabatic expansion is PV^γ = constant, where P is the pressure of the gas, V is the volume, and γ is the ratio of specific heats of the gas. This equation is also known as the adiabatic equation of state.

4. What is an example of reversible adiabatic expansion?

A common example of reversible adiabatic expansion is the expansion of a gas inside a cylinder with a movable piston. If the piston is slowly pulled out, the gas inside expands without gaining or losing heat, and the process is reversible. This is often used in heat engines and refrigeration systems.

5. What are the applications of reversible adiabatic expansion?

Reversible adiabatic expansion has various practical applications, including in heat engines, refrigeration systems, and gas turbines. It is also used in the study of thermodynamics and is essential in understanding the behavior of gases and their energy transfer processes.

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