I'd like to preface this by stating that I'm by no means a physicist or even a physics student, but recently a sibling of mine had a homework assignment in which they had to calculate the amount of energy needed to heat up a house using a very basic formula given by their teacher.

Because of this I was wondering, if there was any way to calculate precisely the amount of energy required to heat up the amount of air (510 m_3), given that the density of air is different depending on the temperature.

The formula I came up with was this:

The idea being that I'd calculate the energy required to heat the air at x degrees using its density at that specific temperature and summing them all etc.

The specific heat capacity for air seems to be the same from 0 to 20 degrees celcius, hence why I just used 1005 flat for it - http://www.engineeringtoolbox.com/air-properties-d_156.html

So is this correct? Why/why not?

Edit: Forgot to add I used the density formula for air from Wikipedia /Density_of_air (can't post link because of rep)

  • $\begingroup$ Are the formula and the calculation yours or did you find them somewhere else? $\endgroup$ – J. Shupperd Sep 22 '16 at 20:20
  • $\begingroup$ energy = $C_vM(\delta T)$, where mass $M ~\rou V=1.225 kg/m^3 \times 510 m^3=625 kg$, $C_v= 0.720 kJ/kg-K$. Therefore, energy = 8996kJ. In your calculation, the universal gas constant should be divided by air molecular weight and the heat capacity may need to use that at constant volume condition. $\endgroup$ – user115350 Sep 22 '16 at 20:21

It is unnecessary to calculate the density of air for each step as if this is an isolated system like many of these problems are then mass will not be added or taken away. The density that you would want to use is that of the starting temperature to get the mass of the system. if you were to include changing densities then you would want to do an integral instead of a summation as an integral covers all the steps in between specific temperatures. A summation will give a good approximation but an integral is more accurate.

Since you are given a specific volume and it does not say anything about the gas expanding, you would want to use the volume constant specific heat like user115350 said in their comment. You would use the pressure constant specific heat if you had the gas expanding. And also like user115350 said, using change in energy = $m*c_{v}*change in T$ where m = density*Volume.

And just one side note $m^3$ is the correct notation for cubic meters where $m_{3}$ is most the time interpreted as the mass of a third object. Not trying to be mean just informative.

  • $\begingroup$ Oh okay, that makes sense! And no worries about the cubic meter thing, I appreciate you trying to be informative. I didn't consider the fact that it was an isolated system, that really does simplify the problem. Thank you for such a detailed answer! $\endgroup$ – Arcads Sep 22 '16 at 20:46

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