Gay-Lussac’s Law Temperature-Pressure Relationship in Gases and the Determination of Absolute Zero

By Chuck Roser, Retired Chemistry Instructor
North Carolina School of Science and Mathematics

### Next Generation Science Standards

 Science and Engineering Practices Disciplinary Core Ideas Crosscutting Concepts Practice: Planning and carrying out investigations Students will investigate the relationship between temperature and pressure of a closed container. DCI: Matter and its interactions • Students will determine the relationship between pressure and temperature and relate temperature to kinetic energy and motion of particles. Concept: Scale, proportion, and quantity • Students will establish a proportional relationship between temperature and pressure.

### Objectives

This experiment’s objectives are to (a) observe the relationship between the pressure and temperature of a constant number of moles of gas in a constant volume container, and (b) to experimentally determine an estimated value for absolute zero.

### Introduction

When a gas is heated, its molecules’ average speed and kinetic energy are increased. If the container has a constant volume, the molecules will strike its sides with greater frequency. This creates greater force on the container’s walls per unit area, increasing pressure in the container. It is one of the bases for the warning against heating an aerosol spray can.

A graph may be plotted to show how the pressure of a fixed mass of gas varies as the temperature is changed. The temperature at which the pressure of an ideal gas would, in theory, reach zero can be determined by extrapolating the pressure vs. temperature graph to zero pressure. This temperature is referred to as absolute zero and is the zero point for the Kelvin temperature scale. An extrapolation to zero pressure is necessary because real gases condense to liquids and solidify before reaching absolute zero. The relationship between the pressure of an ideal gas and its Kelvin temperature is expressed in Gay-Lussac’s law:

In this experiment, the pressure within the Absolute Zero Demonstrator apparatus is measured at several different temperatures. A graph of pressure vs. temperature is then prepared to establish the relationship between pressure and temperature and to estimate a value for absolute zero. The apparatus consists of a copper bulb having a fixed volume (copper expands and contracts only slightly with temperature), a pressure gauge, and a fixed mass of gas. Gas pressure is measured with the pressure gauge. Before a measurement is taken, the apparatus is allowed to equilibrate to ensure that the gas and bulb are at the same temperature.

• 2 2-L Beakers (for boiling-water baths)
• 2 Stirring Hot Plates (with large magnetic stir bars)
• 2 Celsius Thermometers (alcohol thermometers are preferable)
• 2 Ring Stands (with large 3-prong clamps and clamp holders to hold the Absolute Zero Demonstrator)
• 2 Small 3-Prong Clamps and Clamp Holders (to hold the thermometers)
• 4 2-L Beakers (for the 50/50 boiling water/room temperature water bath; the room temperature water bath; the ice water bath; and the dry ice-ethanol, or acetone, bath)
• Absolute Zero Demonstrator
• 1 Pair Insulated Thermal Gloves (for handling the water baths, dry ice, liquid nitrogen, and Absolute Zero Demonstrator)
• Wide-Mouth Dewar Flask (with liquid nitrogen) (optional)

### Safety

1. Caution: Do not use open flames during the experiment. Ethanol and acetone are very flammable.
2. Caution: Dry ice and liquid nitrogen should be handled very carefully (wear safety glasses and insulated thermal gloves) due to the risk of frostbite.
3. Caution: Never put dry ice or liquid nitrogen in a closed container because each will build up pressure and explode the container.
4. Wear safety glasses during the experiment.
5. Use insulated thermal gloves and appropriate care when handling the water baths, dry ice, liquid nitrogen, and Absolute Zero Demonstrator.

### Procedure

1. All participants put on safety glasses. Individuals responsible for handling the Absolute Zero Demonstrator apparatus, water baths, dry ice, and liquid nitrogen put on insulated thermal gloves.
2. Immerse the apparatus’s bulb in the boiling-water bath. Support the apparatus with a large 3-prong clamp and clamp holder and support the thermometer with a small 3-prong clamp and clamp holder. Allow the water to return to a full boil.
3. Wait a few minutes for the apparatus to equilibrate. The pressure reading should stabilize at a constant value.
4. Record the pressure to the nearest mm Hg and temperature to the nearest 0.1° C.
5. Remove the apparatus from the bath.
6. Carefully pour out about one-half of the boiling water and replace it with room temperature water to create a 50/50 boiling-water/room temperature water bath. Immerse the apparatus’s bulb in the bath. Support the apparatus and thermometer as described in step 2. Repeat steps 3 to 5.
7. Immerse the apparatus’s bulb in the room temperature water bath. Support the apparatus and thermometer as described in step 2. Repeat steps 3 to 5.
8. Immerse the apparatus’s bulb in the ice water bath. Support the apparatus and thermometer as described in step 2. Repeat steps 3 to 5.
9. Immerse the apparatus’s bulb in the dry ice-ethanol (or acetone) bath. Support the apparatus as described in step 2. Do not use a thermometer to measure the temperature. The dry ice bath is assumed to be at the sublimation temperature of carbon dioxide, 1 atm pressure and –78.5° C. Wait a few minutes for the apparatus to equilibrate. Record the pressure gauge reading and –78.5° C.
10. Remove the apparatus from the bath.
11. Optional: Purge the apparatus with helium. Immerse the apparatus’s bulb in a wide-mouth Dewar flask containing liquid nitrogen. Support the apparatus as described in step 2. Do not use a thermometer to measure the temperature. The liquid nitrogen bath is assumed to be at its boiling point, 1 atm pressure and –195.7° C. Wait a few minutes for the apparatus to equilibrate. Record the pressure gauge reading and –195.7° C.
12. Remove the apparatus from the bath.

### Data analysis

1. Prepare a graph of pressure vs. temperature in °C. This graph should have labeled axes with units as follows: scale the x-axis from –300 to 100° C (or a value negative enough to clearly show the x-intercept) and the y-axis from 0 to
1,000 mm Hg. Draw the best straight line through the data. Extend the line until it intersects the x-axis. The x-intercept (pressure = 0) gives the extrapolated value for absolute zero.
2. If you have access to a graphing calculator or graphing software, graph the data and perform a linear regression analysis.
3. Record the slope, y-intercept, and correlation coefficient (r). The closer the value of r is to 1.00 or –1.00, the better the data fit a straight line.
4. Follow your instructor’s directions about printing the graph with the linear regression line and regression statistics.
5. To solve for the value of absolute zero, use the equation for a line, y = mx + b. Absolute zero is the temperature at which the gas’s pressure equals zero. This is the line’s x-intercept. To calculate this value, set y = 0, substitute in the value of the slope, and solve for x.

### Data table

 Reading No. Type of Bath Used Temperature (°C) Temperature (°K) Pressure (mm Hg) 1 2 3 4 5 6

Attach a graph of pressure vs. temperature in °C. This graph should have labeled axes with units as follows: scale the x-axis from –300 to 100° C (or a value negative enough to clearly show the x-intercept) and the y-axis from 0 to 1,000 mm Hg. Show the linear regression line through your data with the slope, y-intercept, and correlation factor.

### Calculations and questions

1. Regression analysis:
Linear regression statistics: slope: _________ y-intercept: _______ correlation factor (r): _____.
Equation of the regression line:
Use the slope and a y value of zero to calculate a value for absolute zero:___________. Show your calculations.
2. Calculate your relative percent error for the value of absolute zero based on an accepted value of –273° C.
4. Discuss any sources of error that occurred or any improvements that could be made in this experiment.

### Instructor’s notes

1. Caution: Do not use open flames during the experiment. Ethanol and acetone are very flammable.
2. Caution: Dry ice and liquid nitrogen should be handled very carefully (wear safety glasses and insulated thermal gloves) due to the risk of frostbite.
3. Caution: Never put dry ice or liquid nitrogen in a closed container because each will build up pressure and explode the container.
4. Sample data: Reading No. Type of Bath Used Temperature (°C) Temperature (°K) Pressure (mm Hg) 1 Boiling water 100.0 373.1 945 2 Boiling water + room temp. water 53.0 326.1 840 3 Room temp. water 21.0 294.1 765 4 Ice water 0.0 273.1 720 5 Dry ice/ethanol (or acetone) –78.5 6 Liquid nitrogen –195.7 77.4 225
5. It is easier to have 1 beaker available for each bath. Have the boiling-water, room temperature water, ice water, and dry ice/acetone baths already prepared. Have an extra-large beaker of boiling water available for making the boiling-water/room temperature water bath. Check the approximate amount of water needed to cover the bulb in the bath. Note: Be careful not to overfill the beakers or they will overflow when the bulb is immersed.
6. Different pairs of students can read the pressure and temperature values for each data point. Note: A mercury thermometer cannot be used with the dry ice/acetone or liquid nitrogen baths since mercury freezes at –39° C. An alcohol thermometer will freeze in liquid nitrogen. The dry ice/acetone bath is assumed to be at the sublimation temperature of 1 atm and –78.5° C, and the liquid nitrogen at its boiling point of 1 atm and –195.7° C.
7. A linear relationship between pressure and temperature can be demonstrated with the 4 water bath temperatures. A good value for absolute zero requires a low-temperature data point to reduce the range over which the extrapolation is done. The dry ice/acetone bath provides a good low-temperature data point; however, the liquid nitrogen value works better since it is closer to absolute zero.
8. The device should be purged with helium if liquid nitrogen is used since the oxygen in air will condense at the temperature of liquid nitrogen, causing the pressure reading to be too low.
9. Denatured ethanol and acetone can be obtained from a hardware store. Dry ice can be obtained from grocery or party stores. A Dewar flask and liquid nitrogen can sometimes be obtained from a local college.
10. The dry ice is allowed to sublime at the end of the lab and the ethanol or acetone can be reused. The liquid nitrogen is allowed to evaporate.

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