Thermally isolated cylinder - work done and temperature changes in each chamber

In summary: Classical equipartition theory states that the total energy in a gas is constant no matter the temperature. This is due to the fact that the atoms in a gas have a fixed number of protons in their nucleus. This means that the total energy in a gas is equal to the sum of the individual energy of the atoms.
  • #1
davidray
7
0

Homework Statement



There is a closed, thermally insulated cylinder, separated into two chambers by a moveable, frictionless, gas-tight, thermally insulating piston.

Initial, the volume of each chamber is 3 liters, and each contains an identical monoatomic ideal gas at pressure of 1 atmosphere and a temperature of 20C. You may assume that all relevant properties of the gas are temperature independent, and that classical equipartition theory is valid.

A heating coil inside chamber A is used to slowly supply heat to the gas in this chamber. This gas expands until equilibrium is reached and the pressure in chamber B is 3.5 atmospheres.

Calculate:
1. The work done on the gas in chamber B.
2. The final temperature of the gas in chamber B.
3. The final temperature of the gas in chamber A.
4. The amount of heat supplied by the heating coil.

Any help with this would be greatly appreciated!
I really have no idea where to start! I've been reading over all of my notes and books, but I can't work out which equation or bit of the question to start with. Thank you so much if you can help with this!
 
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  • #2
Ok, what can you say about the pressures in the two cylinders once the heating is done?
 
  • #3
It's until equilibrium is reached, so they'll be the same?
I wasn't sure because the piston is thermally insulating
 
  • #4
Yes the pressures will be the same. If you calculate the change in volume needed to create a pressure of 3.5 atmospheres in chamber B, a bunch of the answers should fall out. But be careful because both pressure and temperature are dependent on volume.
 
  • #5
P1V1/P2=V2=8.79 litres

I can then use W=-(P2xV1)-(P2vV2)
=2227.55J?
 
  • #6
Following this, how do I work out the change in temperature in chamber B?
 
  • #7
davidray said:
P1V1/P2=V2=8.79 litres

I can then use W=-(P2xV1)-(P2vV2)
=2227.55J?

I'm pretty sure you can't do that. You will need a differential equation.

EDIT: The problem is that nothing is constant. Chamber B experiences a change in pressure volume and temperature. The one thing that you have on your side is that it is an adiabatic process. You will need to use the fact that there is no heat added to the system to relate all of these variables. The key things are that you have a final pressure, and its adiabatic. with these two parameters you should be able to draw a connection.
 
Last edited:
  • #8
It's from w=-integral between V1 and V2 (p dV)
That then leads to -[pv]between v1 and v2
which then leads to -[pv1]-[pv2]
?
 
  • #9
see here is the problem. You want to integrate across all volumes. But the pressure is changing with the volume. in other terms, as you change V, P changes as well. Your equation is one for a constant pressure.
 
  • #10
davidray said:
You may assume that all relevant properties of the gas are temperature independent

I don't know what this means.
 
  • #11
"You may assume that all relevant properties of the gas are temperature independent."



Teacher is talking about specific heats.
 

Related to Thermally isolated cylinder - work done and temperature changes in each chamber

1. How does a thermally isolated cylinder work?

A thermally isolated cylinder is a closed system that is insulated from its surroundings. This means that no heat can enter or leave the cylinder. As a result, any changes in temperature or work done within the cylinder are solely due to internal processes.

2. What factors affect the work done in a thermally isolated cylinder?

The work done in a thermally isolated cylinder depends on the initial and final temperatures of the gas in each chamber, the volume change of the gas, and the type of gas used. Additionally, the type and efficiency of the piston used to compress or expand the gas can also impact the work done.

3. How do the temperature changes in each chamber of a thermally isolated cylinder occur?

The temperature changes in each chamber of a thermally isolated cylinder occur due to the expansion or compression of the gas within the chamber. When the gas is compressed, its molecules become more tightly packed, increasing the temperature. Conversely, when the gas expands, its molecules become more spread out, resulting in a decrease in temperature.

4. What is the relationship between work done and temperature changes in a thermally isolated cylinder?

The work done in a thermally isolated cylinder is directly proportional to the temperature changes in each chamber. This means that as more work is done, the temperature of the gas will increase, and vice versa. This relationship is described by the First Law of Thermodynamics, which states that energy cannot be created or destroyed, only transferred or converted.

5. How is a thermally isolated cylinder used in real-world applications?

Thermally isolated cylinders are commonly used in heat engines, such as steam engines and internal combustion engines. The cylinder allows for the controlled expansion and compression of gas, which is used to power the engine and perform work. These types of engines are used in various industries, including transportation, power generation, and manufacturing.

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