1C Real Gases

Question 1

What is the key idea of the chapter?

Answer 1

Attractions and repulsions between gas molecules account for modifications to the isotherms of a gas and critical behaviour.

Question 2

When deviations from the perfect gas law are the most important?

Answer 2

At high pressures and low temperatures.

Question 3

Why real gases show deviations from the perfect gas law?

Answer 3

Because molecules in real gases interact with one another.

Question 4

Repulsive forces between molecules assist …., attractive forces assist ….

Answer 4

Repulsive forces assist expansion.

Attractive forces assist compression.

Question 5

When are repulsive forces most significant?

Answer 5

Only when molecules are almost in contact.
(when the average separation of the molecules is small -
especially at high pressure, when many molecules occupy a small volume)
BECAUSE repulsive forces are short-range interactions.

Question 6

When are attractive forces most significant?

Answer 6

When the molecules are fairly close together but not necessarily touching.
(when the separation of the molecules is intermediate)
BECAUSE attractive forces are long-range interactions.
They are ineffective if the molecules are too far apart.

Question 7

When are the intermolecular forces also important?

Hint: temperature

Answer 7

When the temperature is so low that the molecules travel with such low mean speeds that they can be captured by one another.

Question 8

When are the consequences of intermolecular interactions shown?

Answer 8

They can be seen by shapes of experimental isotherms.

Question 9

When the attractive forces dominate the repulsive forces?

Answer 9

At moderate pressures, when the average separation of the molecules is a few molecular diameters.
(The gas is expected to be more compressible than a perfect gas because the forces help draw the molecules together).

Question 10

When the repulsive forces dominate the attractive forces?

Answer 10

At high pressures, when the average separation of the molecules is small.
(The gas is expected to be less compressible than a perfect gas because the forces help to drive the molecules apart).

Question 11

What is the vapour pressure?

at the TEMPERATURE of experiment

Answer 11

The pressure corresponding to the horizontal line in a graph of experimental isotherms, when both liquid and vapour are present in the equilibrium

Question 12

What is the compression factor Z?

Answer 12

It is the ratio of the measured molar volume of a gas (Vm=V/n) to the molar volume of a perfect gas V°m
at the SAME temperature and pressure.

Question 13

The formulas of the compression factor Z.

Answer 13

Z = Vm/V°m
and
pVm = RTZ
(since pV=RT, then Z= RT/pV°m)

Question 14

What factor is the measure of departure from perfect gas behaviour?

Answer 14

The compressibility factor, Z.
Z=1 when it is a perfect gas (or when pressure is very low)
Z>1 when pressure is high (repulsive forces are dominant, larger Vm)
Z

Question 15

When the real-gas isotherms do not differ greatly from from perfect-gas isotherms?

Answer 15

At large molar volumes and high temperatures.

Question 16

Virial equation of state

two forms

Answer 16

1. pVm = RT (1 + B’p + C’p^2 + …)
2. pVm = RT (1 + B/Vm + C/Vm^2 + …)
   (here the term in parentheses is Z)
   * 1, B, C, etc - virial coefficients

Question 17

What is the virial equation?

Answer 17

It is an empirical extension for the perfect gas equation that summarizes the behaviour of real gases over a range of conditions.

Question 18

What are the virial coeficients?

Answer 18

The terms 1, B, C, etc (first, second, third…) in the virial equation of state.
They depend on the temperature.
The first virial coefficient is always 1.

Question 19

A fact: although the equation of state of a real gas may coincide with those of a perfect gas as p→0, not all the properties of a real gas do so in that limit.

Answer 19

dZ/dp = B’ + 2pC’ + … → B’
(when p→0)
But B is not always zero, so the slope might not approach the 0 in the same angle.

Question 20

What is the Boyle temperature?

Answer 20

The Boyle temperature (Tb) is the temperature at which Z→1 with 0 slope (Z versus pressure graph) at low pressure or high molar volume.
At Boyle temperature, the properties of a real gas coincide with those of a perfect gas as p→0.
B=0 at the Boyle temperature.

Question 21

What are the critical coefficients (constants) of a substance?

Answer 21

Tc - the critical temperature
pc - the critical pressure
Vc - the critical MOLAR volume

Question 22

At an isotherm at critical temperature (Tc) …

Answer 22

…the surface separating two faces does not appear and the volumes at each end of the horizontal part of the isotherm merge to a single point - the CRITICAL POINT of a gas.

Question 23

The liquid phase of a substance does not form above the critical temperature and the single phase that fills the entire volume when T>Tc can be …

Answer 23

… much denser than typical of gas and is named SUPERCRITICAL FLUID.

Question 24

The van der Waals equation of state.

two forms

Answer 24

1. p = nRT/(V - nb) - an^2/V^2
2. In terms of the molar volume Vm = V/n:
   p = RT/(Vm - b) - a/Vm^2

Question 25

What are the van der Waals coefficients?

Answer 25

a - represents the strength of attractive interactions between molecules.
b - represents the strength of repulsive interactions.
They are independent of temperature.
They ARE NOT precise molecular properties!

Question 26

Which part of van der Waals isotherms does NOT resemble experimental isotherms?

Answer 26

The oscillations (the van der Waals loops) below the critical temperature.
They wrongly suggest that under some conditions an increase of pressure results in an increase of volume.

Question 27

When can a gas be liquefied by pressure alone?

Answer 27

Only when the temperature is AT or BELOW its critical temperature.

Question 28

How van der Waals loops are replaced?

Answer 28

MAXWELL CONSTRUCTION: they are replaced by horizontal lines drawn so that the loops define equal areas above and below the lines.

Question 29

What are the principal features of the van der Waals equation?

Answer 29

1. Perfect gas isotherms are obtained at high temperatures and large molar volumes.
2. Liquids and gases coexist when the attractive and repulsive effects are in balance.
3. The critical constants are related to the van der Waals coefficients.

Question 30

A fact: the properties of real gases are coordinated by expressing their equations in terms of DIMENSIONLESS reduced variables.

Answer 30

Divide the actual variable by the corresponding critical constant:
Vr = Vm/Vc
pr = p/pc
Tr = T/Tc

Question 31

How to calculate the real pressure of a gas if critical pressure and reduced pressure are given?

Answer 31

p = pr x pc

Question 32

What is the principle of corresponding states?

Answer 32

The observation that real gases at the same Vr and Tr exert the same pr.
The principal is only an approximation.
It fails badly if molecules are non-spherical or polar.