How Does Adiabatic Expansion Affect an Ideal Gas in a Closed Cycle?

[member 392791]

Homework Statement

A 4.00-L sample of a diatomic ideal gas with specific heat
ratio 1.40, confined to a cylinder, is carried through a
closed cycle. The gas is initially at 1.00 atm and at 300 K.
First, its pressure is tripled under constant volume.
Then, it expands adiabatically to its original pressure.
Finally, the gas is compressed isobarically to its original
volume. (a) Draw a PV diagram of this cycle. (b) Determine
the volume of the gas at the end of the adiabatic
expansion. (c) Find the temperature of the gas at the
start of the adiabatic expansion. (d) Find the temperature
at the end of the cycle. (e) What was the net work done on the gas for this cycle?

Homework Equations

The Attempt at a Solution

Part A is a graph of a typical adiabatic expansion. Part B I used PV^γ is constant and found V=8.77 L. Part C I used PV = nRT and got 900K.

I am stuck on part D, I know it should be 300 K, but I want to know why the equation TV^(γ-1) = constant isn't working. Lastly for part E, I don't know how to find the area under the curve without P as a function of V.

 

[ehild]

The Attempt at a Solution

Part A is a graph of a typical adiabatic expansion. Part B I used PV^γ is constant and found V=8.77 L. Part C I used PV = nRT and got 900K.

Correct!

I am stuck on part D, I know it should be 300 K, but I want to know why the equation TV^(γ-1) = constant isn't working.

that equation gives the temperature at the end of the adiabatic expansion. It is not the end of the cycle: You have one isobaric compression left.

Lastly for part E, I don't know how to find the area under the curve without P as a function of V.

No need to integrate. It is a cycle, the gas returns back to its original state, so the change of internal energy is zero. According to the First Law, the heat Q and W, the work done on the gas, add up to zero. You can calculate the heat exchange for each process: It is a diatomic gas, what are Cv and Cp? You also can determine the amount of gas.

ehild

 

[Andrew Mason]

Homework Statement

A 4.00-L sample of a diatomic ideal gas with specific heat
ratio 1.40, confined to a cylinder, is carried through a
closed cycle. The gas is initially at 1.00 atm and at 300 K.
First, its pressure is tripled under constant volume.
Then, it expands adiabatically to its original pressure.
Finally, the gas is compressed isobarically to its original
volume. (a) Draw a PV diagram of this cycle. (b) Determine
the volume of the gas at the end of the adiabatic
expansion. (c) Find the temperature of the gas at the
start of the adiabatic expansion. (d) Find the temperature
at the end of the cycle. (e) What was the net work done on the gas for this cycle?

Is the adiabatic expansion quasi-static?

AM

 

[member 392791]

Thanks guys!

I completely overlooked that it was asking for the temperature at the original spot...I was thinking at the end of the adiabatic expansion.

I'll give part E a try.

Yes it is quasi-static.

 

[Nickie]

Can you please provide more information on how to approach this problem?

I would first like to commend you on your attempt at solving this problem and your use of the relevant equations. Your solution for parts A-C seems to be correct.

For part D, the reason why the equation TV^(γ-1) = constant may not be working is because it is only valid for adiabatic processes, where there is no heat exchange with the surroundings. In this problem, the gas undergoes an isobaric process (constant pressure) after the adiabatic expansion, which means there is heat exchange and the gas is no longer adiabatic. Therefore, you cannot use the adiabatic equation for this part of the cycle.

To find the temperature at the end of the cycle, you can use the ideal gas law, PV = nRT, and the fact that the number of moles of gas (n) remains constant throughout the cycle. This means that the product of pressure and volume will be the same at the beginning and end of the cycle. You can set up an equation using this information and solve for the temperature at the end of the cycle.

For part E, to find the net work done on the gas for the entire cycle, you need to consider the area under the curve on the PV diagram. However, since the curve is not a simple function, you cannot find the area using integration. Instead, you can divide the area into smaller sections, calculate the work done in each section, and then sum them up to find the total work done. For example, you can divide the area into a rectangle and a triangle, and then use the equations for work done in each shape to find the total work done.

I hope this helps guide you towards finding a complete solution for this problem. Remember to always carefully consider the type of process and the relevant equations before attempting to solve a problem. Good luck!