# 7. Equation of State

## Equation of State Experiment -- Temperature and Pressure Measurements of an Ideal Gas During Heating Within a Closed Container

### Purpose

The ideal gas equation of state (pV = mRT) provides a simple relationship between pressure, temperature, mass, and volume for gases at relatively low pressures and high temperatures. This experiment illustrates the application of the ideal gas equation and other equations of state to relate the properties of several gases contained in a rigid vessel with changes in temperature.

### Assignment

A stainless steel pressure vessel with a volume of 1 liter will be instrumented with a pressure transducer and thermocouple to measure the gage pressure and temperature, respectively, of the air inside the vessel. You will need to select your vessel and confirm its volume. How do you determine the volume of the vessel? Use the Pasco™ temperature and pressure transducers to connect to the vessel and secure so that it is air-tight. Check with your instructor for how to determine if a fitting is air tight. Please consult the laboratory instructor regarding the appropriate compression fittings. What is a compression fitting? How does it differ from other types of pipe fittings?

The pressure transducer produces a voltage signal that can be related to the gage pressure in the cylinder by a calibration curve that is supplied with the instrument. Note that atmospheric pressure in Washington, DC is approximately 13.6 psia. The calibration curve for the pressure transducer is given by the following equation:

$p = 4.3087(V·V) - (13.1176\,V) + 10.7276$

where V equals the voltage output (volts) from pressure transducer, and p equals the absolute pressure (kPa).

The thermocouple in the pressure vessel is connected in series with a second thermocouple that is located in an ice bath that provides a reference temperature. The thermocouple pair produces a voltage signal that is related to the temperature (T) of the air in the pressure vessel by the following expression:

$T = T_{ref} + \dfrac{V}{S}$

where $$T_{ref}$$ equals the ice bath reference temperature (0°C), V equals the voltage (volts) measured across the thermocouple pair, and S equals the thermocouple constant, 42.4 µV/°C.

A signal processor displays and records the voltage signals from both the pressure transducer and the thermocouple during the course of an experimental trial. In addition, the signal processor records the data in a data file that can be downloaded to other programs for data analysis and plotting.

The steps you should follow to carry out the experiment are as follows:

1. Turn on the signal analyzer. Identify the location of the "start" and "pause" buttons.
2. Make sure the pressure readings and ranges are appropriate. The signal analyzer should display two different plots of pressure and time and temperature and time. You will need to combine these data to plot pressure versus temperature at the same times for your system over a range of different temperatures.
3. Check to verify that your hot plate is operating.
4. Temperature profiles (repeat as needed in order to generate reliable statistics)
• Press "start" to begin reading data, and then heat the pressure vessel by placing it on the hot plate. Continue heating the vessel until the temperature plateaus (The pressure will reach approximately 400 kPa).
• Create a slush using crushed dry ice or a liquid nitrogen slurry to cool the vessel to temperatures of at least -40oC. Record the data as above.
NOTE: Review the handling of cryogenics below before proceeding.
• Repeat for at least three gases
NOTE: DO NOT place a hot vessel into the cryogen or an extremely cold vessel into the heat bath.
1. Save the data appearing on the display onto a disk (ask the instructor for help if needed). Choose a meaningful name such as "yourname_LabTemp.dat" for the files. You may have to save two sets of data--one for temperature and another for pressure.
2. Using the data obtained in both the cooling and heating experiments, construct a P-T curve for the entire range of temperatures.

### Handling of Cryogens

Cryogenic Materials will be defined here as any material used for cooling baths that has a melting point, boiling point or sustained temperature below -20°C. This includes dry ice (solid CO2), dry ice-isopropanol mixture, frozen slush baths prepared from liquid nitrogen and organic liquids (methanol, ethanol, hexane, toluene, etc.) and liquid nitrogen.

#### Handling Procedure:

When cryogenic liquids are transferred to containers located above waist-height of the operator the liquid must first be poured from the storage Dewar into a small, easily handled, insulated container (such as a glass Dewar flask) and then poured into the final container. The operator must wear a face shield in addition to safety glasses when pouring cryogenic liquids above waist-level. All containers with evacuated wall space (Dewar flasks) and evacuated round-bottom flasks that have volume of greater than 500 ml. must be wrapped with tape or surrounded by protective net to contain flying glass in case the container should break while being cooled with cyrogenic liquid. Only round bottom flasks should be used for freezing and thawing of organic liquids. Evacuated flasks containing liquids frozen at cryogenic temperatures must be contained in another vessel (a beaker or open dish capable of holding the entire contents of the flask at all times).

#### Preparation of a Dry Ice Slurry:

The best low temperature baths are made with as a high a percentage as possible of the coolant (dry ice here, or ice in general). Approximately the amount of dry ice that will be needed should be crushed and placed in the container (usually a dewar or equivalent). To this you should add the acetone or isopropanol slowly. This is most conveniently done with a squeeze bottle. Add only enough to cover the dry ice. If you need a looser slurry, this can be accomplished by addition of more solvent.

Loose slurries can be made in the following way:

A suitable container, usually a Dewar flask is filled about one-fourth to one-third full of the solvent. Acetone and isopropanol are both often used, but isopropanol is less volatile, less toxic, and is less flammable. A little crushed dry ice is added to the solvent and vigorous bubbling will result. As the dry ice sublimes and the solvent cools, the bubbling will become less violent. A little more dry ice can be added at this point. It should be remembered that the temperature of this bath will not be as low as with the previous recipe.

Dry ice can be added to either bath as needed to maintain the low temperature. When finished using the slurry, allow the dry ice to sublime completely, pour the solvent into a bottle indicating the solvent is for cold baths only. Loosely cap the container for about 24 hours or overnight to prevent a degassing mishap.

### Analysis

• Using the initial values of the measured pressure and temperature, the volume of the vessel, calculate, from the ideal equation of state, the mass of air in the vessel. Assuming the volume and mass are constant during the heating process, calculate the temperature of the air (from the ideal gas equation) for each of the pressure values--remember that the recorded values of the pressure are absolute. Plot the experimental and calculated temperatures as a function of the measured pressure. Clearly label the graphs. Compare the experimental results with the theoretical results. What might be responsible for the differences?
• Plot the P vs T curves for each graph and compare the slopes to the value nR/V. Perform an error analysis based on the comparison of the theoretical and observed values. What differences to you see? Which gas or molecular characteristics might be responsible for the differences?
• Which of your gases might best be described by a van der Waals equation of state? Compare your P vs T curve to a the van der Waals P vs T curve over the same range of T and P for each potential van der Waals gas. Discuss the similarities and differences. What might be responsible for the differences?
• Extract an estimate of the van der Waals constant from your graphs. How do they compare to the literature values? How might you improve your results?