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

In summary, at high pressure the gas does not behave as an ideal gas and the line slopes downwards, while at lower pressure the gas behaves as an ideal gas and the line slopes upwards.
  • #1
trelek2
88
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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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  • #2
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.
 
  • #3
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...
 
  • #4
Not sure if I understood this correctly. The problem is that [itex]pV_m[/itex] 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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  • #5
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, [tex] pV=NT [/tex]. Then

[tex] \frac{d}{dp} (pV) =T \frac{dN}{dp}. [/tex]

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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  • #6
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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  • #7
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 [tex] k_1 / k_2 = T_1/T_2 [/tex].
 
  • #8
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.
 

1. How do I calculate temperature from a PV vs P diagram using the ideal gas law?

To calculate temperature from a PV vs P diagram using the ideal gas law, you will need to use the formula T = PV/nR, where T is temperature, P is pressure, V is volume, n is the number of moles, and R is the ideal gas constant. Simply plug in the known values for P, V, n, and R and solve for T.

2. Can I use the ideal gas law to calculate temperature for any gas?

Yes, the ideal gas law can be used to calculate temperature for any gas as long as the gas behaves ideally and the temperature, pressure, and volume are within the appropriate range of values.

3. What units should I use for each variable when using the ideal gas law to calculate temperature?

The ideal gas law equation uses the following units: pressure (P) in Pascals (Pa), volume (V) in cubic meters (m^3), number of moles (n) in moles (mol), and the ideal gas constant (R) in Joules per mole Kelvin (J/mol*K). Make sure to use consistent units for accurate results.

4. Do I need to account for any other factors when using the ideal gas law to calculate temperature from a PV vs P diagram?

In most cases, the ideal gas law will provide an accurate calculation of temperature from a PV vs P diagram. However, in real-world scenarios, there may be some deviations from ideal gas behavior due to factors such as intermolecular forces or non-constant temperature or pressure. In these cases, a more advanced equation, such as the Van der Waals equation, may be necessary.

5. Can I use the ideal gas law to calculate temperature if I only have a PV vs P graph and not numerical values?

No, the ideal gas law requires numerical values for pressure, volume, number of moles, and the ideal gas constant. Without these values, it is not possible to accurately calculate temperature using the ideal gas law.

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