How to obtain temperature from a PV vs P diagram using ideal gas law?

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Discussion Overview

The discussion revolves around how to obtain temperature values from a PV vs P diagram using the ideal gas law, particularly in the context of specific volume and varying pressures. Participants explore the implications of negative slopes in the plotted data and the applicability of the ideal gas law under different conditions.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant expresses confusion about how to derive temperature from the PV vs P graph, noting the negative slopes and the relationship to the ideal gas law.
  • Another suggests selecting any point along the isotherm to calculate temperature using the formula T=PV/nR, but this is challenged by others.
  • Concerns are raised that different points along the isotherm would yield varying temperature values, indicating a potential misunderstanding of the graph's representation.
  • A participant questions whether the ideal gas law applies, suggesting that the gas might behave differently and proposing the Van der Waals equation as an alternative.
  • Discussion includes the observation that temperature appears to decrease with increasing pressure, prompting speculation about real-world data implications.
  • One participant proposes using the lowest pressure point for temperature approximation, while another questions the validity of this approach given the gas's behavior.
  • Another participant suggests that slopes on the graph could be proportional to respective temperatures, leading to a potential method for comparison.
  • Discrepancies in calculated temperatures are noted, with one participant reporting values for superheated steam and considering the assumption of ideal gas behavior at low pressure.

Areas of Agreement / Disagreement

Participants express differing views on the applicability of the ideal gas law and the interpretation of the graph. There is no consensus on the best method to derive temperature values, and multiple competing ideas remain unresolved.

Contextual Notes

Participants highlight limitations in their understanding of the graph's representation and the assumptions underlying the ideal gas law. The discussion reflects uncertainty regarding the behavior of the gas under varying conditions.

trelek2
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As stated in the question: I have PV(T) [J/kg] vs P [Pa] plotted for 2 different temperatures. I'm to approximate the the temperatures using the lines and the ideal gas law. Note: V is the specific volume. I have no clue how to do this:
I know that PV vs P is just as if I had nRT vs P. From this the temperature should be obtainable, however the slopes of the lines are negative, so I don't see how I should approach this. The pressure varies from 0 to 10^6 Pa and the PV varies from about 310600 to 306600 J giving a slope of about -0.004. I also had been given the information that the weight is 0.018kg/mole.
 
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Here's a guess: pick anyone point, out of the infinite number of point available along one of the isotherms, read off the values of P and V from the graph, and then calculate T=PV/nR.
 
I don't think its that easy: The y-axis corresponds to different values of PV. That would mean that if I take any two different points and calculate the temperature this way, I'll get a different value of T for each point along the isotherm which is clearly wrong...
 
Not sure if I understood this correctly. The problem is that pV_m is not constant along the line that's supposed to be the isotherm? Then the only solution is, that the ideal gas law doesn't apply?
Maybe it can be fitted to a Van der Waals gas?
I'll think about that...
 
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trelek2 said:
The pressure varies from 0 to 10^6 Pa and the PV varies from about 310600 to 306600 J giving a slope of about -0.004.


So your temperature drops as you increase pressure? This just means that you have actual real world data... Take the ideal gas law, pV=NT. Then

\frac{d}{dp} (pV) =T \frac{dN}{dp}.

So, as you increase your isothermal systems pressure, unless it is perfectly sealed, you expect to see a small downward slope on the pV-p graph as your molecules are leaking outside (dN/dp < 0).
 
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Hey, you're right. I just got the hint that "all gases behave as ideal gases in the low pressure limit". Do you think I should take the point for the lowest pressure and approximate the temperature from that?

clamtrox: Are you sure about this? Or is it what Grenuk pointed out: The gas doesn't behave exactly as an ideal gas and therefore the line is not horizontal. I don't see how could I find the temperature with the information I have, using your formula...
 
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trelek2 said:
clamtrox: Are you sure about this?

I certainly am not, I just said the first thing that popped into my head :D But you can check: just assume that dN/dp is independent of T (as it probably is). Then the slopes on the graphs should be proportional to the respective temperatures, so you can just check if k_1 / k_2 = T_1/T_2.
 
That's not really what I get:
For the highter temperature the slope is -0.003, and taking the point at lowest pressure (0.1MPa) I get the temp = 772 K = 500C.
For the lower temp the slope is -0.004 and I get the temperature to be 672K = 400C...
These values are pretty reasonable considering we're dealing with superheated steam. Maybe I should just say I had taken the point measured at lowest pressure and at that point we can assume that the gas behaved as an ideal gas.
 

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