Ideal (perfect) gas and Real gas, Virial Equation

Ideal (perfect) gas and Real gas, Virial Equation:

The state of a pure substance is specified by giving the values of n, V, p, and T, it has been established experimentally that it is sufficient to specify only three of these variables, for then the fourth variable is fixed. That is, it is an experimental fact that each substance is described by an equation of state, an equation that interrelates these four variables. The general form of an equation of state is

  P= f (T, V, n)            General form of an equation of state                          (1A.3)

This equation tells us that if we know the values of n, T, and V for a particular substance, then the pressure has a fixed value. Each substance is described by its own equation of state, but cases. One very important example is the equation of state of a ‘perfect gas’, which has the form p = nRT/V, where R is a constant independent of the identity of the gas. The equation of state of a perfect gas was established by combining a series of empirical laws.

(a) The empirical basis:

We assume that the following individual gas laws are familiar:

Boyle’s law: pV =constant, at constant n, T                                     (1A.4a)

Charles’s law: V= constant x T, at constant n, p                            (1A.4b)

p= constant x T, at constant n, V                                                           (1A.4c)

Avogadro’s principle: V= constant x n, at constant p, T            (1A.4d)

Boyle’s and Charles’s laws are examples of a limiting law, a law that is strictly true only in a certain limit, in this case p →0.

The perfect gas law is of the greatest importance in physical chemistry because it is used to derive a wide range of relations that are used throughout thermodynamics. However, it is also of considerable practical utility for calculating the properties of gas under a variety of conditions. For instance, the molar volume, Vm= V/n, of a perfect gas under the conditions called standard ambient temperature and pressure(SATP), which means 298.15 K and 1 bar (that is, exactly 105 Pa), is easily calculated from Vm = RT/p to be 24.789 dm3 mol−1.

Molecular explanation of Boyle’s and Charles’s Law:

The molecular explanation of Boyle’s law is that if a sample of gas is compressed to half its volume, then twice as many molecules strike the walls in a given period of time than before it was compressed. As a result, the average force exerted on the walls is doubled. Hence, when the volume is halved the pressure of the gas is doubled, and pV is a constant. Boyle’s law applies to all gases regardless of their chemical identity (provided the pressure is low) because at low pressures the average separation of molecules is so great that they exert no influence on one another and hence travel independently. The molecular explanation of Charles’s law lies in the fact that raising the temperature of a gas increases the average speed of its molecules. The molecules collide with the walls more frequently and with greater impact. Therefore they exert greater pressure on the walls of the container.

(b) Mixtures of gases:

When dealing with gaseous mixtures, we often need to know the contribution that each component makes to the total pressure of the sample. The partial pressure, pJ, of a gas J in a mixture (any gas, not just a perfect gas), is defined as

                                       pJ=xJp                                 Definition Partial pressure                              (1A.8)

where xJis the mole fraction of the component J, the amount of J expressed as a fraction of the total amount of molecules, n, in the sample:

                                              xJ=nJ/n n=nA+nB+……..                 Definition Mole fraction          (1A.9)

When no J molecules are present, xJ= 0; when only J molecules are present, xJ= 1. It follows from the definition of xJ that, whatever the composition of the mixture, xA+ xB+ … = 1 and therefore that the sum of the partial pressures is equal to the total pressure:

                                            pA+ pB+……= (xA+ xB+……)p = p                                                  (1A.10)

This relation is true for both real and perfect gases.

When all the gases are perfect, the partial pressure as defined in eqn 1A.9 is also the pressure that each gas would exert if it occupied the same container alone at the same temperature. The latter is the original meaning of ‘partial pressure’. That identification was the basis of the original formulation of Dalton’s law:

The pressure exerted by a mixture of gases is the sum of the pressures that each one would exert if it occupied the container alone.

Real gases do not obey the perfect gas law exactly except in the limit of p→0. Deviations from the law are particularly important at high pressures and low temperatures, especially when gas is on the point of condensing to liquid.

Deviations from perfect behaviour:

Real gases show deviations from the perfect gas law because molecules interact with one another. A point to keep in mind is that repulsive forces between molecules assist expansion and attractive forces assist compression.

Repulsive forces are significant only when molecules are almost in contact: they are short-range interactions, even on a scale measured in molecular diameters. Because they are short-range interactions, repulsions can be expected to be important only when the average separation of the molecules is small. This is the case at high pressure when many molecules occupy a small volume. On the other hand, attractive intermolecular forces have a relatively long range and are effective over several molecular diameters. They are important when the molecules are fairly close together but not necessarily touching.

Attractive forces are ineffective when the molecules are far apart. Intermolecular forces are also important when the temperature is so low that the molecules travel with such low mean speeds that they can be captured by one another. At low pressures, when the sample occupies a large volume, the molecules are so far apart for most of the time that the intermolecular forces play no significant role, and the gas behaves virtually perfectly. At moderate pressures, when the average separation of the molecules is only a few molecular diameters, the attractive forces dominate the repulsive forces. In this case, the gas can be expected to be more compressible than a perfect gas because the forces help to draw the molecules together. At high pressures, when the average separation of the molecules is small, the repulsive forces dominate and the gas can be expected to be less compressible because now the forces help to drive the molecules apart.

The pressure when both liquid and vapour are present in equilibrium is called the vapour pressure of the liquid at the temperature.

The Compression Factor (Z):

The compression factor, Z, the ratio of the measured molar volume of a gas, Vm= V/n, to the molar volume of a perfect gas, V0m, at the same pressure and temperature:

                          Z= V m/V0m                         Definition Compression factor

Because the molar volume of a perfect gas is equal to RT/p, an equivalent expression is

                    Z= RT/pV0m, which we can write as pVm= RTZ

Because for a perfect gas Z = 1 under all conditions, deviation of Z from 1 is a measure of departure from perfect behaviour. Some experimental values of Z are plotted in Fig. 1C.3. At very low pressures, all the gases shown have Z ≈ 1 and behave nearly perfectly. At high pressures, all the gases have Z > 1, signifying that they have a larger molar volume than a perfect gas. Repulsive forces are now dominant. At intermediate pressures, most gases have Z < 1, indicating that the attractive forces are reducing the molar volume relative to that of a perfect gas.

 

The van der Waals equation:

The van der Waals equation is

 The equation is often written in terms of the molar volume Vm=V/n as
The constants a and b are called the van der Waals coefficients where a represents the strength of attractive interactions and b that of the repulsive interactions between the molecules. They are characteristic of each gas but independent of the temperature. Although a and b are not precisely defined molecular properties, they correlate with physical properties such as critical temperature, vapour pressure, and enthalpy of vaporization that reflect the strength of intermolecular interactions.

The van der Waals equation of state:

The repulsive interactions between molecules are taken into account by supposing that they cause the molecules to behave as small but impenetrable spheres. The non-zero volume of the molecules implies that instead of moving in volume V they are restricted to a smaller volume Vnb, where nb is approximately the total volume taken up by the molecules themselves. This argument suggests that the perfect gas law p = nRT/V should be replaced by

when repulsions are significant.

Calculation of the excluded volume (nb):

To calculate the excluded volume we note that the closest distance of two hard-sphere molecules of radius r, and volume V molecule = 4 /3 π r3, is 2r, so the volume excluded is 4/3 π(2r)3or 8Vmolecule. The volume excluded per molecule is one-half this volume or 4Vmolecule, so b≈4VmoleculeNA.

Reduction of pressure:

The pressure depends on both the frequency of collisions with the walls and the force of each collision. Both the frequency of the collisions and their force are reduced by the attractive interaction, which act with a strength proportional to the molar concentration, n/V, of molecules in the sample. Therefore, because both the frequency and the force of the collisions are reduced by the attractive interactions, the pressure is reduced in proportion to the square of this concentration. If the reduction of pressure is written as a(n/V)2, where a is a positive constant characteristic of each gas, the combined effect of the repulsive and attractive forces is the van der Waals equation of state as expressed in eqn 1C.5.

Virial equation of state:

At large molar volumes and high temperatures, the real-gas isotherms do not differ greatly from perfect-gas isotherms. The small differences suggest that the perfect gas law pVm=RT is, in fact, the first term in an expression of the form.

A more convenient expansion for many applications is

 

These two expressions are two versions of the virial equation of state. By comparing the expression with eqn 1C.2 we see that the term in parentheses in eqn 1C.3b is just the compression factor, Z. The coefficients B, C, …, which depend on the temperature, are the second, third, … virial coefficients; the first virial coefficient is 1. The third virial coefficient, C, is usually less important than the second coefficient, B, in the sense that at typical molar volumes C/ V 2m <<B /V m. The values of the virial coefficients of a gas are determined from measurements of its compression factor.

Although the equation of state of a real gas may coincide with the perfect gas law as p→0, not all its properties necessarily coincide with those of a perfect gas in that limit. Consider, for example, the value of dZ/dp, the slope of the graph of compression factor against pressure. For a perfect gas dZ/dp=0 (because Z=1 at all pressures), but for a real gas from eqn 1C.3a we obtain

 
However, B′ is not zero, so the slope of Z with respect to p does not approach 0 (the perfect gas value). Because several physical properties of gases depend on derivatives, the properties of real gases do not always coincide with the perfect gas values at low pressures.

 Because the virial coefficients depend on the temperature, there may be a temperature at which Z→1 with zero slope at low pressure or high molar volume (as in Fig. 1C.4). At this temperature, which is called the Boyle temperature, TB, the properties of the real gas do coincide with those of a perfect gas as p→0. According to eqn 1C.4b, Z has zero slope as p→0 if B=0, so we can conclude that B=0 at the Boyle temperature. It then follows from eqn 1C.3 that pVmRTBover a more extended range of pressures than at other temperatures because the first term after 1 (that is, B/Vm) in the virial equation is zero and C/Vm2and higher terms are negligibly small.

 
 
 
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