- #1
RJLiberator
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Homework Statement
A constant volume gas thermometer contains a gas whose equation of state is
[tex](p+\frac{a}{V^2_m})(V_m-b)=RT[/tex]
and another, of identical construction, contains a different gas which obeys the ideal gas law, [tex]pV_m = RT[/tex]. The thermometers are calibrated at the ice and steam points. Show that they will give identical values for a temperature.
[Assume that the thermometers are constructed so that all the gas is at the temperature being measured.]
Homework Equations
V is constant, we must consider the variation of T with p.
The Attempt at a Solution
Since we have to consider the variation of T with p, I want to take the derivative [tex]\frac{dT}{dp}[/tex].
We start with the ideal gas law, and end up with:
[tex]\int \frac{pV_m}{R}dp=\int dT[/tex]
This ends up being
[tex]T=\frac{V_m}{R}\frac{p^2}{2}[/tex]
With the same strategy on the more complicated equation of state we see
[tex]\int( \frac{pV_m}{R}-\frac{pb}{r}+\frac{a}{RV_m}-\frac{ab}{RV^2_m}) dp=\int dT[/tex]
This turns out to be
[tex]\frac{p^2V_m}{2R}-\frac{p^2b}{2R}+\frac{ap}{RV_m}-\frac{abp}{RV^2_m}=T[/tex]
Here, we see there is a similar term [tex]\frac{p^2V_m}{2R}[/tex] on both sides.
I'm not sure if what I did was right...
Could I set the constants equal to some value to make the equations equal?