The Zeroth Law and the Ideal-Gas Temperature Scale

Establish a thermodynamic temperature scale with a constant-volume gas thermometer.

Objectives

On completing this experiment you should be able to:

State the zeroth law of thermodynamics and explain why it makes temperature measurable.
Operate a constant-volume gas thermometer and calibrate it at the triple point of water.
Show that the apparent temperature indicated by a real gas depends on its density, and that the dependence vanishes as the density tends to zero.
Determine the ideal-gas (Kelvin) temperature of three unknown baths by extrapolation to zero filling pressure.
Determine the absolute zero of temperature on the Celsius scale from a constant-volume pressure-temperature plot.
Show that the zero-pressure limit is independent of the working gas, and hence that it defines a universal scale.

Theory

If two bodies are each in thermal equilibrium with a third, they are in thermal equilibrium with one another. This is the zeroth law, and it is what allows a single number - the temperature - to be attached to a body, and a third body (the thermometer) to be used to compare any two others.

A thermometer needs a thermometric property. For a constant-volume gas thermometer that property is the pressure. Defining the temperature by $T = a P$ and fixing the constant a at the triple point of water,

T(P) = 273.16\, \mathrm{K} \, \frac{P}{P_{tp}} .

Different gases, and the same gas at different filling pressures, give slightly different answers, because no real gas is ideal. Expanding the equation of state as a virial series in the molar density,

P V_m = R T (1 + B(T)/V_m + \dots) ,

the apparent temperature becomes

T_{apparent} = T \left[1 + (B(T) - B(T_{tp})) \rho + O(\rho^2)\right] ,

which is linear in the filling pressure to first order. Extrapolating to zero filling pressure removes the whole correction, and every gas gives the same limit. That limit is the ideal-gas temperature, and it can be shown from the second law to be identical to the thermodynamic temperature defined by a Carnot cycle.

Key relations

$T = 273.16\,\mathrm{K}\;\lim_{P_{tp}\to 0}\frac{P}{P_{tp}}$
Definition of the ideal-gas (Kelvin) temperature.
$P V_m = RT\left(1 + \frac{B(T)}{V_m}\right)$
Virial equation of state truncated after the second coefficient.
$T_{\mathrm{app}} \simeq T\left[1 + \frac{B(T)-B(T_{tp})}{RT_{tp}}P_{tp}\right]$
First-order dependence of the apparent temperature on filling pressure.

Procedure by part

Part A - Calibration and the zero-pressure limit

Fill the bulb to a chosen pressure at the triple point of water, then read the pressure with the bulb in the bath of interest.

Select the working gas and the filling pressure. The apparatus automatically records the pressure $P_{tp}$ at the triple point of water when it is filled.
Select the bath and take the reading. The mercury is always returned to the fiducial mark, so the volume is constant.
Repeat for at least five filling pressures spanning the available range, keeping the gas and bath fixed.
Plot the apparent temperature 273.16 K x $\frac{P}{P_{tp}}$ against $P_{tp}$ and extrapolate the straight line to $P_{tp} = 0$ .
Repeat the whole sequence for at least two more gases and confirm that the intercepts agree.

Part B - Absolute zero from a constant-volume plot

At a single filling pressure, record the pressure at each of the fixed points and plot pressure against Celsius temperature.

Choose helium and a filling pressure near the middle of the range.
Record the pressure at every fixed point whose Celsius temperature you know.
Plot P against the Celsius temperature and fit a straight line.
Extrapolate to $P = 0$ . The intercept on the temperature axis is the absolute zero of the Celsius scale.
Repeat at a second, much lower filling pressure and comment on the change in the intercept.

Questions to answer in your report

State the zeroth law of thermodynamics and explain precisely which step of this experiment relies on it. [5]
Your extrapolated intercepts for different gases agree, but the values at finite filling pressure do not. Explain the physical origin of the difference and why the sign of the discrepancy differs between helium and nitrogen. [6]
Why is the ideal-gas temperature defined by an extrapolation rather than by a measurement with a single, very dilute filling? Answer with reference to your uncertainties. [5]
The ideal-gas scale is defined operationally, whereas the Kelvin scale is defined from the second law using a Carnot engine. Outline the argument that the two coincide. [6]
ITS-90 does not use gas thermometry directly for routine work. What is used instead over the range of your unknown baths, and why? [4]