SBU Intro Physics Labs, PHY 133 Ideal Gas Law Lab

Here, $P$ is the pressure of the gas, $V$ is the volume in which the gas is contained, $n$ is the number of moles of gas in that volume, and $T$ is the temperature of the gas. The quantity $R=8.314\frac{\text{J}}{\text{mol}\cdot\text{K}}$ is the gas constant.

It is worth noting that we can express it in a similar way in terms of the number of particles $N$, using the Boltzmann constant $k_B=1.38\times 10^{-23}\frac{\text{J}}{\text{K}}$:

$$PV=NkT$$

This formulation is equivalent up to a unit conversion (from moles to particles).¹

In this lab, we will only be varying two of these parameters at once.¹

Procedure

Part I: Boyle's and Avogadro's Laws

Begin by connecting both your pressure and temperature probes to the LabPro box and opening LoggerPro (the program, not any particular file) on the computer. They should both connect automatically, and you should see measurements of room temperature and pressure in the lower-left. Record these values, with uncertainty.

Take your syringe, set the volume to 10mL, and connect it to your pressure sensor by screwing them together. Record the pressure and volume (with uncertainty).

Expand to 20mL, and record the new pressure (with uncertainties¹). Then repeat for 30mL and 40mL.² Do not disconnect the pressure sensor from the syringe at any point during these measurements: we want to keep $n$ constant, so we cannot break the seal.

Now: disconnect the pressure sensor, and set the volume to 20mL (while open). Reconnect the pressure sensor, and record the pressure at volumes 20mL, 30mL, 40mL, and 50mL (with uncertainties).

Part II: Determining an Unknown Volume

Now, we are going to use the ideal gas law to determine the volume of the gas cell.

Set the syringe to 0mL, and use the T-piece to connect the gas cell, pressure sensor, and syringe.

Expand the syringe sequentially to 10mL, 20mL, 30mL, and 40mL. At each volume, record the pressure (and the volume itself, of course) and corresponding uncertainties. As before, do not separate the pressure sensor during this process.

Part III: Determining Absolute Zero

In this part, we're going to want to make a plot of temperature vs. pressure. You can change the axes on the plots that appear in the LoggerPro by clicking on them. Set up one of the plots to have temperature on the y-axis and pressure on the x-axis.

Disconnect your setup, and connect your pressure sensor to your volume. (Do this at room temperature, and as usual, don't disconnect after this.)

Turn on your kettle, and let it heat the water inside to a boil.³

Once it reaches a boil, turn off your kettle (or it may turn off automatically). Place your gas cell and temperature sensor in the water.

Watch both the pressure and temperature measurements: they will start out increasing as the gas and temperature probe heat up to equilibrium with your sample.

Once they reach their maximum meeasurement, give them a little stir to make sure they're in equilibrium with the whole pot. If you still observe them decreasing, you're set to take measurements.

Click "collect" on LoggerPro to take data as it cools. As it takes data, autoscale your axes (right-click on the plot) and prepare a linear fit (as usual, click the "R=" button up above).

Once the range of temperature change is 5 degrees or so (should be about 3-5 minutes, ~200 measurements), stop recording data. Record the slope and intercept of your fit line.⁴

Start recording again (no need to re-heat), overwriting your previous data. Re-autoscale, delete the old fit line and make a new one.

Once it has cooled another 5 degrees, stop recording, note your slope and intercept again.

Repeat one more time for a third slope and intercept.

Finally, take your gas cell out of the hot water, disconnect it from the pressure sensor, and cool it briefly in the water in the front of the room before returning it to your table. (This is to make sure it has reached close to room temperature for the next class.)

Analysis

Part I: Boyle's and Avogadro's Laws

Convert your room temperature measurement to K (assuming the "correct" value for absolute zero).

Calculate $1/P$ for each pressure measurement. Propagate uncertainty.

For each initial volume, make a plot of $V$ vs. $1/P$.

From each slope and your measurement of room temperature (plus the known value for $R$), calculate the number of moles $n$ in your container for each initial volume. Propagate uncertainties.

Then, for each volume, calculate $n/V_0$, where $V_0$ is your initial volume. Propagate uncertainties and answer the question about whether your results agree with expectation.

Part II: Determining an Unknown Volume

Calculate $1/P$ for each pressure measurement. Propagate uncertainty.

Make a plot of $V$ vs. $1/P$. Orient your axes so that your $y$-intercept is $V_0$, the volume of the container.

Use your slope (and knowledge of room temperature) to extract a measurement of $n$, the number of moles in the unknown volume at room temperature and pressure.

Part III: Determining Absolute Zero

Take the average of your slopes and intercepts, and calculate the uncertainty in this average.

Answer the question about the agreement of your intercept with expectation.

Using the slope and the volume $V_0$ you determined in the previous part, calculate the number of moles in the container.

Recall that this number of moles was determined by sealing the container at room temperature and pressure. Hence, it should be the same as the previous part. On your data sheet, answer the question: was it?

Discussion Topics

Your TA will ask you to discuss some of the following points (they will tell you which ones):

Systemic Error: Volume of the Pressure Sensor: Presumably, there is a small volume of gas actually contained in our pressure sensor (or at the base of the syringe, etc.) at any particular time. What is the impact of this volume on each part of our experiment?
Volume Change from Sealing: You may have noticed sealing the container slightly increased the pressure. Why would this be?
Systemic Error: Uniformity of Temperature: In the last part, only our container was submerged underwater. There was also a small amount of gas in the tube and pressure sensor, which might be at a somewhat cooler temperature. How would this impact our results?
Thermal Equilibrium: In the last part, the temperature probe determines the temperature of the water, not of the cell. In assuming they match, we assume the cell is in thermal equilibrium with the water (and the probe, for that matter). What assumption do we make on the time it takes the water to reach equilibrium, relative to the rate of cooling, does this rely on?²

References and Tools

Guide to Uncertainty Propagation & Error Analysis (Quick Reference)

PHY133/134 Plotting Tool

Hovering over these bubbles will make a footnote pop up. Gray footnotes are citations and links to outside references.

Blue footnotes are discussions of general physics material that would break up the flow of explanation to include directly. These can be important subtleties, advanced material, historical asides, hints for questions, etc.

Yellow footnotes are details about experimental procedure or analysis. These can be reminders about how to use equipment, explanations of how to get good results, or clarifications on details of frequent confusion.

For a review of ideal gasses, see (...).

As a point of historical interest, there are various two-variable laws that were discovered individually leading up to the early 19th century.

The constant-temperature law (stating that $PV$ is a constant if $T$ is a constant) is called Boyle's Law.

The constant-pressure law (that $V$ and $T$ are proportional at constant $P$) is called Charles's Law.

The constant-volume case (that $P$ and $T$ are proportional at constant $V$) is called Gay-Lussac's Law.

All of those assume a constant number of moles. There is also Avogadro's Law, which states that the number of moles of the gas is proportional to the volume (at constant density and pressure).

Since we had a temperature vs. time plot from the LoggerPro, we actually could have done this measurement. A first estimate is that it takes about 1s for the container to reach equilibrium, and a similar-ish time for the temperature probe. Since their "lag times" are about equal, by coincidence, it actually works out nicely even without the assumption we want to make! That said, we can also determine (with those numbers and your observed rate of cooling) that our assumption is acceptable regardless.

Consider: should your volume uncertainty be the same for all these measurements? Can you "hold it steady" as precisely as you initially set it? (The answer may be "yes," but you should at least think about this!)

If at any point you are having significant physical difficulty performing this experiment, you may adjust the volumes you use.

It will make noise before it is actually boiling. Wait for it to actually boil.

The data will be noisy, but you should still observe the net change pretty clearly over this temperature range, and hence the best fit line should extract the data well enough for our purposes.