# Constant volume or constant pressure?

Constant volume or constant pressure??

In a question I tried to solved: "A house has well-insulated walls. It contains a volume of 100 m3 of air at 300 K. Calculate the energy required to increase the temperature of this diatomic ideal gas by 1.00°C." The solution start with the assumption by saying "consider heating it at constant pressure....." But why? wouldn't be a more accurate assumption if we said the volume is constant but not the pressure as -albeit slight- all temperature increase will result an increase in pressure as well due to P=nRT/V? Obviously house's volume is not changing!
Results is highly effected as which way you go either by Cp or Cv.

Q_Goest
Homework Helper
Gold Member

Interesting question. The house's volume isn't changing, but can't we also say the same about the pressure in the house (ie: the pressure in the house remains constant)? When we heat the air in the house at constant pressure, the density drops, so where does the air go? Can the air be doing any PdV work?

What is dV then? What would you put as "initial" and "final" volumes in W=Integral(PdV)?

On the other hand, if we calculate the Q by Q=nCvΔT or Q=nCpΔT we'd get totally different results.

Interesting question. The house's volume isn't changing, but can't we also say the same about the pressure in the house (ie: the pressure in the house remains constant)? When we heat the air in the house at constant pressure, the density drops, so where does the air go? Can the air be doing any PdV work?

Q_Goest
Homework Helper
Gold Member

The best way to look at the Cv versus Cp issue is not so much constant volume versus constant pressure. That's fine, but it's a bit misleading as you can see in this example.

Consider putting a control volume around the air in the house instead of putting it around the house. A control volume in which there is no mass that crosses the control surface is called a "control mass". When you heat the air, it expands at constant pressure, so air goes out the windows and doors. Otherwise the house would need to be sealed and pressure inside the house would rise according to the ideal gas law. Consider that if the temperature rose from 270 K to 300 K, the pressure in the house would increase accordingly. But it doesn't. Pressure in the house stays the same. So your control mass is actually expanding and getting bigger as the temperature increases. What this is saying is the air is doing work, so there is additional energy needed to increase not just the temperature but to do the work in expanding the air. The final volume of air can then be determined from the equation of state (ideal gas law as applicable).

What is dV then? What would you put as "initial" and "final" volumes in W=Integral(PdV)?
Can you figure out what the initial and final volumes are now and how to integrate to get work done by the air in the control mass as it is heated?

I understand but I think it's a bit misleading the way the question asked saying "A house has well-insulated walls..." as if the house is "sealed" as you put it. Or it's just me as I immediately thought the air inside is trapped and it's getting hotter and the pressure is rising.

As for the integral, I don't know. I understand due to control mass air is doing a work out and loses energy. But on what boundaries? Would it be correct if we say Vi is 100m3 and Vf is Vf=nRTf/P ??

HallsofIvy