Air machine and compressed air

AI Thread Summary
The discussion focuses on calculating the entrance temperature of air in an air machine undergoing an adiabatic process, with air entering at 20 atm and exiting at 1 atm at 0°C. The calculations show that the entrance temperature must be 369°C to achieve this condition, assuming 1 mol of air. Participants clarify that during adiabatic expansion, the gas cools down as it does work against external pressure, and the assumption of 1 mol is valid since heat capacity scales proportionally with mass. The conversation also touches on the difference between adiabatic expansion into a vacuum versus against pressure, noting that real gases can exhibit temperature changes due to intermolecular interactions. The discussion concludes that the calculations and assumptions made are correct for the scenario presented.
Karol
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Homework Statement


Air at pressure 20[atm] enters an air machine and leaves at 1[atm]. what must be the temperature of the air at the entrance so that when it leaves it is 00C.
Assume adiabatic process.

Homework Equations


PV=nRT
Adiabatic: ##P_1V_1^\gamma=P_2V_2^\gamma##

The Attempt at a Solution


The air expands and only the mass per second is unchanged. i take 1[mol] at the exit:
$$PV=nRT\rightarrow 1[atm]\cdot V_2=1\cdot 0.08208\cdot 273\rightarrow V_2=22.4[liter]$$
$$P_1V_1^\gamma=P_2V_2^\gamma \rightarrow 20\cdot V_1^{1.4}=22.4^{1.4}\rightarrow V_1=2.63[liter]$$
$$P_1V_1=nRT_1\rightarrow 20\cdot 2.63=1\cdot 0.08208\cdot T_1\rightarrow T_1=642^0K=369^0C$$
Is it correct?
 
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Karol said:
an air machine
Sound like it might be doing work? Is there any mention of a heat source? Could this be an expansion process?
 
Bystander said:
Sound like it might be doing work? Is there any mention of a heat source? Could this be an expansion process?
The answer looks right to me. During an Adiabatic expansion (q = 0) a gas will cool down. Since Delta E is negative (Delta E = Cv Delta T), w <0 -- work is done by the gas.

If you have ever deflated a bicycle tire quickly before, you may have noticed this. The gas coming out of the valve stem is cold, and will actually cool the valve stem.
 
Thanks
 
Quantum Defect said:
If you have ever deflated a bicycle tire quickly before, you may have noticed this. The gas coming out of the valve stem is cold, and will actually cool the valve stem.
Is it because the expansion is against pressure of 1[atm]? since in adiabatic free expansion to vacuum there should be no change in temperature, right?
 
Why can you just simply assume that you are working with 1 mol of air? This doesn't seem quite right to me...

Edit: Well, I completed it without assuming 1 mol, and had the same answer as you did, so apparently you can. Is this since the only important thing here are ratios and not actual values?
 
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Karol said:
Is it because the expansion is against pressure of 1[atm]? since in adiabatic free expansion to vacuum there should be no change in temperature, right?

Yes.

For ideal gases, expansion into a vacuum (the familiar two bulb 'experiment', one with gas, the other with vacuum, separated by a stopcock) the change in temperature is zero when the gas expands into the empty container. For real gases, there can be a change in temperature (due to non-zero intermolecular interactions).

In some physical chemistry experiments, researchers use a supersonic expansion of a gas from high pressure to essentially vacuum through a small orifice to create samples with very low temperatures. This is more complicated, but you can read about it here: http://www.chem.utah.edu/_documents/faculty/morse/84.pdf
 
Fede Aguilera said:
Why can you just simply assume that you are working with 1 mol of air? This doesn't seem quite right to me...
You can assume 1 mol, because the heat capacity is proportional to the mass. Increasing the amount of mass in the sample increases the heat capacity by the same proportion. The final temperature will be the same, regardless of how much material you deal with.
 
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