Equation of State_guide

7/27/2019 Equation of State_guide

http://slidepdf.com/reader/full/equation-of-stateguide 1/4

1

Physics Department

Thermodynamics Laboratory

EQUATION OF STATEAND CRITICAL POINT

1 . A i m

Obtain two isotherms on a P-V diagram for a gas.

Obtain the vaporization curve of the gas.

Determine the critical point.

2 . Ov e r v i e w

The state of a gas (in a closed system) is determined by three variables: pressure, volumeand temperature. But only two of them can be independent. The other variable will be

determined by these two, according to the equation of state.

The simplest equation of state for gases is the well-known PV=nRT , which is met by allgases when they are rarefied enough. To deduce it, ignore the size of the gas molecules andthe interactions among them. This way the equation cannot predict the phase transitionsthat occur in all substances.

The simplest equation that takes into account the size of the particlesand their interaction is Van der Waals' equation:

( ) nRT =nbV V

an+P −⎟⎟

⎠

⎞⎜⎜⎝

⎛ 2

2

[1]

where n is the mole number, b is the size of a molecule, and a is a parameter of attractiveinteraction among particles. This equation predicts the liquid-gas phase transition, and it

also predicts some of its main features:

-Some isotherms in the P-V diagram have two regions: High compressibility region

(gas) and low compressibility region (liquid).-Only below a given temperature (critical temperature T c ) it is possible to produce thegas-liquid phase transition at a constant temperature by increasing the pressure onthe system.- If T c is the value of the critical temperature and P c and V c are the values of pressureand volume at the critical isotherm's inflexion point, then:

T c=

8a

27 R b [2]



2

V

c= 3b n [3]

Pc=

a

27 Rb2 [4]

Also, using the principles of thermodynamics we can deduce that the phase transition (at aconstant temperature) happens at constant pressure: as we reduce the volume of thesystem, liquid and gas phases coexist in vapor form without changing pressure bydecreasing the volume of gas and increasing the amount of liquid (which takes up lessvolume, see Figure 1).

Figure 1: P-V Diagram

There is also a relationship between the value of pressure where liquid and gas coexist (vapor pressure) and the absolute temperature. This relationship, Clapeyron's equation, can besimplified and yields:

P=P0el / RT

[5]

where P 0 is a constant, R the gas constant, and l the latent heat of vaporization: the heat

necessary to cause the phase transition.



3

3 . L e a r n m o r e . ..

O n t h e I n t e r n e t

4 . Eq u i p m e n t

1. Manometer and thermometer.2. Graduated gas tube.

3. Water bath.

5 . Ex p e r i m e n t a l P r o c ed u r e

5.1 Measuring equipment

The gas under study will be inside a gas tube immersed in a water bath, the temperatureof which will be controlled by a thermostat. Also, the gas volume can be decreased bypushing mercury up inside the tube. As a consequence, the pressure inside the tube will rise.

5.2 Measurements

Pressure and volume will be measured for two isotherms, one below the criticaltemperature and one approximately at critical temperature.

The values of vapor pressure will be measured for several temperatures (always below thecritical temperature) in order to obtain the value of the latent heat of vaporization.

5.2.1 Isotherm below critical temperature

Set the bath thermostat at minimum temperature and start the circulation of water,making sure that water does not go over the top of the gas tube.

Write down the value of the temperature of the tube thermometer. Starting at a 4cm3 volume, start writing down the values of pressure and volume, in 0.2 cm3 steps,at constant temperature.

Do not increase pressure beyond 50x105 Pa.

When a condensate starts to appear, continue with the procedure more slowly.

Draw these data on the isotherm graph.

5.2.2 Measurements at a constant volume

Keep a constant volume of approximately 0.5 cm3. Leave the mercury column in this position and start increasing the temperature of the

water bath in steps of one Celsius degree.

Write down the values of pressure for each temperature. For this, wait for about oneminute for the temperature to reach a stable value on each increment.

When the liquid dissapears completely, increase pressure slightly and wait for it tobecome stable. If the condensate reappears, then resume the measurementprocedure.

If no condensate appears, then the temperature of the gas is approximately equal tothe critical temperature.

Do NOT increase the temperature any further.

Calculate ln P and 1/T . Draw a graph representing these experimental data.

Perform a least-squares line fit of these data. From equation [5], discuss the meaning of the parameters of this least-squares fit,



4

and obtain the latent heat of vaporization of the gas, l .

5.2.3 Isotherm near critical temperature

Keep the water bath thermostat at the temperature reached in the previousexperiment.

Write down the value of the temperature of the tube thermometer. Starting at a 4cm3 volume, start writing down the values of pressure and volume, in 0.2 cm3 steps,at constant temperature.

Do not increase pressure beyond 50x105 Pa.

When a condensate starts to appear, continue with the procedure more slowly.

Draw these data on the isotherm graph.

5.2.4 Questions

Estimate the number of moles and the values of the a and b constants in the Van derWaals equation. Apply equations [2], [3] and [4].

The gas in the column can either be ethane or sulfur hexafluoride. Deduct which oneof them it is.

Documents

Equation of State_guide