Thermal Physics Adiabatic Heating Problem

[Borntofly123]

Greetings all,

My first post here on this forum. I'm currently revising for exams and have got stuck on a question where I'm not sure where I'm going wrong...Question:
1g of iron filings at 500 °C are inserted into a sealed 20 litre vessel containing 1 mol of an ideal monatomic gas at a pressure of 10^5 Pa.

1. Assuming the walls of the vessel are of negligible heat capacity and the process is adiabatic, what is the final temperature of the system?
(specific heat capacity of iron = 0.45 kJ kg–1K–1.)The attempt at a solution:
I start by arguing that as this is a sealed environment, then the energy lost by the iron equals the energy gained by the gas.

Therefore for the iron Q=mc*deltaT where deltaT=T(final) - (500+273K)

As this is adiabatic, then the potential gained by the gas is given by dU=C(v)dT where C(v) is the heat capacity of constant volume for the gas given by nRf/2 where f is the number of degrees of freedom, in this case 3 for a monatomic gas.

Then equating mc(T(final)-773)=1.5R(T(final)-(pV/nR)) where pV/nR is used to find the initial temperature of the gas, leads to T(final) found to be 221K, which is clearly wrong. I'm told the answer is 260K.

Any pointers or corrections would be greatly appreciated.

 

[Chestermiller]

It looks like you are missing a minus sign in your equation. The change in the combined internal energy of the filings and the gas should be zero. Otherwise, everything else looks OK.

Chet

 

[Borntofly123]

So simple, yet so frustrating. Thank you for your help, much appreciated!

 

[Chestermiller]

So simple, yet so frustrating. Thank you for your help, much appreciated!

I hope you're not too frustrated. You got the hard part completely right. You just made a minor error at the end.

 

[HalfLostAndSearching]

Hello,

It seems like you are on the right track with your approach to this problem. However, there are a few things that may be causing your answer to differ from the correct answer.

Firstly, when calculating the energy gained by the gas, you should use the specific heat capacity at constant pressure (Cp) instead of the specific heat capacity at constant volume (Cv). This is because in an adiabatic process, the pressure remains constant and therefore Cp is the appropriate value to use.

Secondly, when calculating the initial temperature of the gas, you should use the ideal gas law (pV = nRT) instead of pV/nR. This is because the ideal gas law accounts for both pressure and volume, whereas pV/nR only accounts for pressure.

With these corrections, the equation should look like this:

mc(T(final)-773)=1.5R(T(final)-(nRT/pV))

Solving for T(final) will give you the correct answer of 260K.

I hope this helps and good luck on your exams!