ChE E3820y: CO_{2} COMPRESSIBILITY FACTOR AND VIRIAL COEFFICIENT (CO2)

A. SAFETY

1. Wear eye protection at all times.

2. Wear jeans or slacks, a long-sleeved shirt, and sturdy shoes that give good traction on possibly wet floors.

3. Guard against electrical hazards by making sure that all

equipment is well grounded using three-wire plugs and other means.

4. Do not allow hands, hair, clothes to come close to the vacuum pump belt.

5. **Think first of safety in any action you take**. If not certain, ask the TA or faculty member before you act. You are dealing with a large cylinder of compressed CO_{2}.

B. INTRODUCTION

The compressibility factor Z for a gas is defined by the equation

PV = ZRT

where P is pressure, V is molar volume, R is the gas constant, and T is absolute temperature. For an ideal gas Z = 1. If an ideal gas expands isothermally from a volume V to a volume twice as large, the pressure will be halved. If a real gas expands to twice the initial volume, the final pressure will differ from half the initial pressure by an amount that depends on the compressibilty factor Z, at the two pressures. Thus it seems possible that data from isothermal expansion experiments can be used to measure Z. This experiment is designed to make that conjecture precise, obtain expansion data, and calculate values for Z for carbon dioxide and helium, as functions of pressure.

C. THEORY

Consider an isothermal expansion of a pure gas from volume V_{A} into volume V_{A} + V_{B}, with initial pressure P_{0} and final pressure P_{1}. We have

P_{0} = (1/V_{A})Z_{0}RT

and

P_{1} = [1/(V_{A} + V_{B})]Z_{1}RT

It follows that

P_{1}/P_{0} = (1/N)(Z_{1}/Z_{0})

where the volume ratio, a constant of the apparatus, is given by

N = (V_{A} + V_{B})/V_{A}

If the expansion is repeated we have

P_{2}/P_{0} = N^{-2}Z_{2}/Z_{0})

and, in general, for n expansions

P_{n}/P_{0} = N^{-n}(Z_{n}/Z_{0})

As n becomes large, the pressure of the gas approaches zero, the gas behaves more and more as an ideal gas, and Z_{n} approaches 1. Thus, for large n,

P_{n}/P_{0} = N^{-n}/Z_{0}

Solving for Z_{0} gives

Z_{0} = N^{-n}(P_{0}/P_{n})

Rearranging, we have an equation from which Z_{0} can be determined

P_{0}/Z_{0} = P_{n}N^{n}

or

Z_{0} = P_{0}/P_{n}N^{n}

Since n cannot be infinite, one way to process data from a series of expansions is to plot P_{n}N^{n} versus P_{n}, and extrapolate to

P = 0. The y-axis intercept is P_{0}/Z_{0} and, since P_{0} is known, Z_{0} can be calculated. Note that in this process no volume had to be measured. However the volume ratio N is needed, and this comes from the relation

N = lim (P_{0}/P_{1})

P->0

evaluated at the lowest possible pressure using a gas as close to ideal as possible. Note also that, for large n, a small error in N may affect the results significantly.

Once Z_{0} = Z(P_{0}) has been determined, it seems possible that Z at other pressures P_{1}, P_{2}, ... can also be determined from the same data. In fact this can be done on the basis of the equation

P_{n}/P_{0} = N^{-n}(Z_{n}/Z_{0})

or

Z_{n} = Z_{0}N^{n}(P_{n}/P_{0})

D. APPARATUS

The apparatus is shown schematically in the attached Figure. Study it carefully, and make sure you understand the sequential valve operations needed to accomplish an expansion.

E. PROCEDURE

1. Starting the Experiment

You are dealing with gases at high pressure. BE CAREFUL. Have your teaching assistant approve your procedure in the initial expansions.

Turn on the constant temperature bath and heat the water to just above room temperature, say 25 to 30 C, as indicated by the bath thermometer. Do not continue until this temperature has been reached. Open all valves except the cylinder regulator valve and the cylinder valve itself. Evacuate the system including the Heisse gauge. Zero the Heisse gauge dial (using the upper chrome knob) on the Heisse gauge. Note that the needle position is read when the needle coincides with its reflection in the gauge mirror.

2. Evacuation of the System

Close the valve upstream of the upstream cylinder, open all other system valves, begin pumping down the system until a pressure of less than 1 torr is obtained.

3. Purging the System with Working Gas

Close the valve leading to the vacuum pump and begin feeding gas into the system by slowly opening the CO_{2} cylinder valve. After reaching a pressure of about 50 psig, discontinue feeding gas and open the valve at the vacuum pump inlet until the gas in the system is removed. Repeat steps B and C once more.

4. Compressibility Factor and Virial Coefficient Data

Note that during the runs the Heisse gauge will be connected to the first cylinder, so that the effective upstream cylinder volume is that of the cylinder itself, the valve, the gauge, and the connecting tubing.

The basic procedure is to evacuate downstream cylinder to less than 1 torr (1 torr = 1000 microns of Hg), close the valve downstream of this cylinder, expand the upstream cylinder (plus the Heisse gauge and tubing) into the downstream cylinder, read the pressure on the Heisse gauge, close the valve between cylinders, and iterate until the pressure is close to 30 psia.

This procedure can be done at room temperature (25 to 30 C) and at intermediate temperatures up to 60 C. Note that cold water can be run through a coil in the water bath to rapidly lower the temperature, if needed.

F. VIRIAL COEFFICIENT DETERMINATION

The function Z = f(P) is usually assumed to have the linear form

Z = a + bP (1)

or a parabolic form

Z = a' + b'P + c'P^{2} (2)

For the linear form, a best linear plot of Z vs. P gives the constants a and b.

For a parabolic relationship, a least-squares linear plot of

(Z_{n} - a)/P_{n} vs. P_{n} gives a line of the form (with a' = a)

(Z - a')/P = b' + c'P (3)

with slope c' and y-intercept b'.

G. REPORT

1. For carbon dioxide, calculate and plot compressibilty factor Z as a function of P, based on the theory above. If data for helium are available, repeat the calculations.

2. Fit the Z(P) data for carbon dioxide, using both the linear and quadratic relationships. Which gives the better fit? Compare the virial constants with literature values (see Perry's Handbook).

3. How would you change or improve this experiment? Comment on its overall educational value.

H. COMMENTS

1. If a vacuum gauge is not available, it can be assumed that 2 minutes of pumping will reduce the system pressure to less that 1 torr, if all valves in the path are well open. The vacuum pump will make a characteristic flat cracking noise when the pressure is low.