PHYS111Q : Labs

12 The Ideal Gas Law

Introduction

The ideal gas law, relating the pressure $P$, volume $V$, and absolute temperature $T$ of $N$ molecules of an ideal gas, is given by

$$PV = Nk_BT$$ (59)

where $k$ is the Boltzmann constant

$$k = 1.38 \times 10^{-23} \mathrm{J/K}$$ (60)

An ideal gas is one in which the volumes of individual molecules are small compared with the total volume occupied by the gas so that interactions between molecules are insignificant. You will have an opportunity to test the relationships between pressure, volume, and temperature for air. You will also attempt to determine a value of zero absolute temperature on the Celsius scale.

Experiments

$P$ and $V$ at Constant $T$

For this experiment, you will use a low-friction cylinder-piston system and digital pressure and temperature sensors.

1. Disconnect the tube from the cylinder so that the system is open to the atmosphere. Then, place the piston at a position of 45 mm and reconnect the tube.

2. Connect the pressure sensor to the port labeled CH 1 of the LabPro interface and the temperature sensor to the port labeled CH 2.

3. Run Logger Pro. It should recoginze the sensors and automatically give you graphs of pressure vs. time and temperature vs. time.

4. Begin acquiring data. After 5 seconds at a position of 45 mm, compress the piston to a position of 40 mm, and hold it there for 5 seconds. Then, de-compress the piston to 50 mm and hold it there for 5 seconds. Then, compress the piston to 35 mm and hold for 5 seconds. Finally, decompress to 55 mm for 5 seconds. By alternating compressions and decompressions in this way, you partially balance any leakage that might occur.

5. Measure and record the total length of tubing connecting the pressure sensor to the piston chamber.

$P$ and $T$ at Constant $V$ : Absolute Zero

For this experiment, you will use the spherical copper pressure cell and an alcohol thermometer.

1. Record the pressure and temperature of the system at room temperature.

The following three steps may be performed in any order. Give the cell time to cool/warm up between steps. Handle the cell only by its plastic handle. Do not touch the metal parts until it returns to room temperature.

1. Record the pressure and temperature of the system submerged in boiling water.

2. Record the pressure and temperature of the system submerged in ice water.

3. Your lab instructor will submerge your cell in in liquid nitrogen. Record the pressure. The temperature is approximately $-196^\circ \mathrm{C}$. Do not put the alcohol thermometer into the liquid nitrogen!

Analysis

$P$ and $V$ at Constant $T$

1. Use the Statistics button on the toolbar to determine the pressure corresponding to each cylinder position. You may see evidence of leakage (a gradual trend back toward atmospheric pressure) during compressions and decompressions. In that case, only average over a short time interval immediately after each compression/decompression.

2. Put your pressure and piston position data into two columns in Excel.

3. Since you don't have a precise value for the total volume of the piston chamber and tubing, use only the volume of the piston chamber. You recorded the length of the cylinder under the piston in millimeters. You'll need to convert this into volume using
   

   $$V_\mathrm{cylinder} = \pi r^2 h$$ (61)

   
   
   The radius of the cylinder is $r = 0.01625 \pm 0.00005$ m.

4. Use Excel to create a graph of $V$ vs. $\frac{1}{P}$.

5. If your data are compatible with a linear model, then apply a linear regression using the LINEST() function.

6. The absolute value of the y-intercept of your graph corresponds to the volume of the tubing.

7. Use your measured dimensions of the tubing in Eq. 61 to calculate its total volume. The inner diameter of the tubing is about 0.003 m.

$P$ and $T$ at Constant $V$ : Absolute Zero

1. Use Excel to create a graph of $T$ vs $P$. Add a linear fit to the graph.

2. Apply a linear regression to your data using the LINEST() function. The ``y-intercept'' of this fit corresponds to absolute zero on the Celsius scale according to an extrapolation of your measurements.

Before You Leave Lab

Show your work to your instructor and discuss preliminary answers to the questions below.

Group Assignment

Hand in your spreadsheet and answers to the following questions.

1. Compare the two volumes you determined in your analysis of the $P$-$V$ data. Are they consistent with each other within uncertainty?

2. Are your $P$-$V$ data consistent with the ideal gas law? Explain.

3. Report your result for absolute zero temperature on the Celsius scale. Is your result compatible with the accepted value of $-273.25^\circ\mathrm{C}$?

Copyright © 2003-2010, Lewis A. Riley

Updated Fri Aug 27 11:05:11 2010

This work is licensed under a Creative Commons License.