Two students in a physics lab have to construct a constant-volume gas thermomete

Question

Two students in a physics lab have to construct a constant-volume gas thermometer as part of their end of year project. They use liquid nitrogen (-196°C) and boiling allyl alcohol (96.9°C) to calibrate the thermometer. The corresponding values of pressure are 0.380 atm and 1.425 atm. The students plot a pressure versus temperature graph with pressure on the y-axis and temperature on the x-axis.

(a) Determine the y intercept and slope of the linear graph that relates the pressure to the temperature.

(b) What value of absolute zero do the students get from this calibration?

(c) What pressure readings would this thermometer give for the freezing and boiling points of water?

Explanation / Answer

I will assume that you are assuming the ideal gas law and have not yet been introduced to more complex gas laws. Then we have:

P*V = n*R*T {ideal gas law}

where
P = the absolute pressure of the gas
V = the volume of the gas
T = the absolute temperature of the gas
n = the number of moles of gas
R = the universal gas constant = 8.314472 J/mol/K

The thermometer is constant volume and is (of course) sealed, so that both V and n are constant.
The ideal gas law must apply for the gas at each temperature, so we get:
Pa*V = n*R*Ta
Pb*V = n*R*Tb

However, the temperatures given are in degrees Celsius, not in an absolute temperature scale. We must have that:
Ta = A*ta + B
Tb = A*tb + B
where
ta = -196 Celsius
tb = 96.9 Celsius
and A and B are unknown constants that relate the Celsius scale to the absolute scale.

Then we can write the ideal gas law using the Celsius scale as follows:

P*V = n*R*(A*t+B)
P = (n*R*A/V)*t + (n*R*B/V)
P = C*t + D

where we have defined two new unknown constants:
C = n*R*A/V
D = n*R*B/V

Then for the two calibration conditions we have:

Pa = C*ta + D
Pb = C*tb + D

and we now have two equations in the two unknowns C & D, since we know:
ta = -196 Celsius
Pa = 0.380 atm
tb = 96.9 Celsius
Pb = 1.425 atm

So we solve for C and D:
1.425 = C*(96.9) + D
0.380 = C*(-196) + D

Equation solving a 2x2 system , the values of " C " are obtained and " D "

C= 0.00356 atm/Celsius

D= 1.07928 atm/Celsius

So our equation for P and t becomes:

P = (0.00356)*t + 1.07928

Since you cannot have negative pressure, we must have that at P=0 we have the lowest possible temperature:

0 = (0.00356)*tzero + 1.07928
-1.07928 = 0.00356*tzero
tzero = -303.1685 Celsius

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