1.4 Ideal gas. Ideal gas laws

 

In the seventeenth to nineteenth centuries, investigators studying the behavior of gases under pressures close to atmospheric elaborated a number of important empirical regularities.

In 1662 R. Boyle, and in 1676 E. Mariotte, independently from Boyle, found that at a constant temperature the product of the pressure of a gas by its volume is constant, i.e. that in the isothermal process of expansion or compression of a gas,

 

                                                                                                                (1.8)

 

The equation (1.8) is known as Boyle's law.

In 1802 J. L. Gay-Lussac found that if the pressure of a gas is maintained constant in the course of heating (i.e. if an isobaric process is accomplished), the volume of the heated gas will increase with temperature, the dependence being linear and of the form

 

                                                                                                           (1.9)

 

This equation is referred to as Gay-Lussac's law. Here V0 is the volume of the gas at 0 °C, V the volume of the gas at a temperature of t °C, and α the coefficient of volume expansion of the gas. It was shown that at sufficiently low pressures the magnitude of the coefficient of volume expansion happens to be the same for all gases and equal to about α = 1/273 = 0.00366 °C-1; precision measurements have shown the volumetric coefficient of expansion α to be equal to 0.003661 °C-1.

If V1 and V2 are volumes of a gas, respectively, at temperatures t1 and t2 with the same pressure p = const, it follows from equation (1.9) that

 

                                                                                                          (1.10)

 

Let us consider now the process of heating a gas in a vessel of constant volume (isochoric process). This process is also described by Gay-Lussac's law in the following form:

 

                                                                                                        (1.11)

 

where p0 and p are respectively gas pressures at temperatures 0 and t °C.

If p1 and p2 are the pressures of the gas at temperatures t1, t2 with the volume of the vessel V = const, Eq. (1.11) states that

 

                                                                                                          (1.12)

 

Let us introduce the following notation:

 

                                                                                                              (1.13)

 

Since, as it was stated above, α = 0.003661 °C-1,

 

                                                                                                     (1.14)

 

The dimension of T is temperature and it should be regarded as the tem­perature read on a scale differing from the Celsius (Centigrade) scale in that the scale's zero is located at a temperature of -273.15 °C. The temperature read on this scale is known as absolute temperature and is denoted K (kelvin). As it will be shown later on, in Chapter 3, the concept of absolute temperature has a profound physical meaning and is one of the fundamental concepts of thermodynamics.

It is clear that using the concept of absolute temperature, Eqs. (1.10) and (1.12) can be represented in the following form:

 

                                                                                                                   (1.15)

 

and

 

                                                                                                                  (1.16)

 

Let us consider now the sequence of two thermodynamic processes, the isothermal process 1-m and the isobaric process 2-m (Fig. 1.2). In the isother­mal process 1-m the compression of gas is described by Boyle's law

 

                                                                                                                           (1.17)

 

(where v1 and vm are specific volumes), and the subsequent heating of the gas in the isobaric process 2-m satisfies Gay-Lussac's law

 

                                                                                                                                    (1.18)

 

Since the process 1-m is an isothermal one, it is clear that T1 = Tm and, consequently, from Eq. (1.18) we get:

 

                                                                                                                            (1.19)

 

Since the process 2-m is an isobaric one, pm = p2. Taking this fact into account, from Eqs. (1.17) and (1.19), we get:

 

                                                                                                                         (1.20)

 

By analogy, transferring the gas into any third state with properties p3, v3, and T3, it can be shown that

 

                                                                                                 (1.21)

 

 

 

Fig. 1.2

 

Thus, for a gas in any state with properties p, v and T obeying the Boyle and Gay-Lussac laws,

 

                                                                                                                           (1.22)

 

The constant quantity in Eq. (1.22) does not depend on the state of the gas. It only depends on the properties of the gas and is individual for each gas. It is known as the gas constant. Denoting the gas constant by R, we find for Eq. (1.22) the following form:

 

                                                                                                                               (1.23)

 

In this way we obtained an equation which relates unambiguously the properties p, v, and T of a gas, i.e. the equation of state of an ideal, or perfect, gas. This equation is known as the Clapeyron equation.

It was mentioned above that the Boyle and Gay-Lussac laws, used to derive Eq. (1.23), were formulated after experimental studies of low-pressure processes in gases.

It is understood that the instruments used by Boyle, Mariotte and Gay-Lussac were less accurate than modern pressure gauges and thermometers. In this connection a question arises of how accurate the above formulated gas laws are.

Precision experimental work has shown the actual behaviour of gases, even at low pressure, to deviate somewhat from that described by Eq. (1.23). However, the lesser the density of a gas, the more accurate Eq. (1.23) describes its behaviour.

The gas which strictly obeys Eq. (1.23) is referred to as an ideal, or perfect, gas, and Eq. (1.23) is known as the equation of state of an ideal (perfect) gas.

With respect to properties, a real gas is the closer to an ideal one, the smaller the density of the gas.

Equation (1.23) can be derived by the methods of the kinetic theory of gases, assuming the molecules of gas to be material points without molecular attraction. This is usually done in courses of general physics.

Thus, the concept of an ideal gas is based on the following assumptions:

an ideal gas strictly obeys Clapeyron's equation (the basic definition of an ideal gas);

an ideal gas is the limiting state of a real gas at p → 0;

an ideal gas is a gas whose molecules are considered to be material points, the interaction between them being restricted to collisions.

A real gas differs the more from an ideal gas, the greater its density. From the molecular-kinetic point of view, the "non-ideality" of a gas is due to the fact that molecules have their own volume and also to the existence of intermolecular interaction of a rather intricate nature.

What is then the sense of introducing the concept of the ideal, or perfect, gas? Firstly, in practice, we have to deal with gases at low pressures, at which various gas processes can be calculated with sufficient accuracy with the aid of the equation of state of an ideal gas.

Secondly, the concept of the ideal gas and the laws governing its behavior happen to be useful as the limit of the laws governing the behavior of real gases. This is important from the methodological and mainly from the practical point of view: later on it will be shown that it is expedient to regard a number of quantities characterizing the properties of a real gas (its heat capacity, for instance) as the similar quantities for an ideal gas and some correction factor, accounting for the non-ideality of the real gas. Such an approach proves to be highly fruitful in a number of cases.

Consider two equal volumes V of two ideal gases I and II (Fig. 1.3). Assume that the first volume, filled with gas I, contains NI molecules of this gas under a pressure pI and at a temperature TI; also assume that the mass of one molecule of gas I is equal to mI. Accordingly, the number of molecules in the second volume, filled with gas II, is NII, the pressure is pII, the temperature is TII, and the mass of one molecule of gas II is mII. Now assume that the pressure and the temperature of the gases in the two volumes are the same.

 

 

Fig. 1.3

 

In 1811 A. Avogadro suggested a hypothesis, now known as Avogadro's law:

Equal volumes of ideal gases at the same temperature and pressure contain equal numbers of molecules.

Avogadro's law permits the following important conclusion. The mass of gas I in the first volume can, obviously, be determined from the formula

 

                                                                                                                           (1.24)

 

and the mass of gas II in the second volume from

 

                                                                                                                              (1.25)

 

hence,

 

                                                                                                                            (1.26)

 

It is clear that the ratio between the masses of molecules mI / mII is equal to the ratio between the molecular masses of these gases μI/μII. In accordance with Avogadro's law, NI = NII and, consequently, it follows from equation (1.26) that

 

                                                                                                                             (1.27)

 

i.e. the ratio between the masses of various ideal gases contained in equal volumes at equal pressure and temperature is equal to the ratio between the molecular masses of these gases.

Let us introduce the concepts of the mole and kilomole. A mole is defined as the amount of a substance expressed in grams and numerically equal to its molecular mass, and a kilomole is the amount of the substance expressed in kilograms and numerically equal to its molecular mass[1]. So, for instance, a kilomole of oxygen (O2) equals 32 kg; a kilomole of carbon dioxide (CO2), 44 kg; etc. It is clear that 1 kmol is equal to 1000 mol.

Let us formulate the second conclusion that follows from Avogadro's law and which is the reciprocal of the first conclusion, viz. the masses of various gases at same temperature and pressure, when related like the molecular masses, have the same volumes. This gives a basis for the conclusion that the molar volumes of different gases at the same temperature and pressure are equal to each other. If v is the specific volume of a gas, and μ its molecular mass, then the volume of a mole (the so-called molar volume) is equal to μv. Thus, for various ideal gases at the same temperature and pressure,

 

                                                                                                                           (1.28)

 

In accordance with Avogadro's law, it is clear that a kilomole of any ideal gas contains a fixed number of molecules, and this number is called the Avogadro (or Avogadro's) number, NA. The numerical value of Avogadro's number has been found experimentally to be NA = 6.022045 × 1026 kmol-1.

Let us determine now the volume of one kilomole of an ideal gas under the so-called standard physical conditions, which as it is known from the course of general physics correspond to a pressure of p = 760 mm Hg = 101.325 kPa and a temperature of t = 0 °C. Since with given p and T the product μv does not depend on the kind of gas, its magnitude can be deter­mined using the data on the specific volume of any ideal gas, of oxygen, for instance. The molecular mass of oxygen μ = 32, and its specific volume, calculated by Clapeyron's equation, v = 0.700 m3/kg. Hence,

 

                                                                                                             (1.29)

 

Thus, the volume of one mole, which is the same under similar conditions for all ideal gases, is equal to 22.4 m3/kmol under standard conditions.

It is sometimes convenient to express the mass of a gas in moles or in kilomoles. Denoting the number of moles (or kilomoles) of a gas by M, we find that

 

                                                                                                                               (1.30)

 

Let us turn to the problem of determining the gas constant R in Eq. (1.23). The magnitude of R can be easily calculated knowing the properties of the gas at any state.

Let us assume that the state of the gas is known under standard conditions. Then, substituting in Clapeyron's equation (1.23) p = 101.325 kPa and T = 273.15 K, we get:

 

                                                                                              (1.31)

 

Substituting the specific volume from Eq. (1.29) in Eq. (1.31), we obtain

 

                                                                                                                             (1.32)

 

Substituting the above value of the gas constant in equation (1.23), we get:

 

                                                 or                                            (1.33)

 

Formula (1.33) is the equation of state of an ideal gas for one kilomole, and the number 8314 is the gas constant reduced to one kilomole of the gas. This quantity is the same for all gases; it is called the universal gas constant and denoted by μR. The unit of measurement of the universal gas constant is J/(kmol·K).[2]

It follows from Eq. (1.33) that the gas constant of individual gases, R, is determined by their molecular or molar mass. So, for instance, for nitrogen (μN2 = 28)

 

                                                   

 

In Clapeyron's equation the individual properties of each given ideal gas are determined from the magnitude of its gas constant.

It is clear from the above that the equation of state of an ideal gas (Clapeyron's equation) can take the following forms:

for 1 kg of gas (1.23)

 

                                                                

 

for G kg of gas, having in mind that Gv = V,

 

                                                               

 

for one mole of gas

 

                                                              

 

Let us consider now how isotherms, isobars and isochors of an ideal gas are represented on the p-v, p-T and v-T diagrams (Fig. 1.4).

 

 

Fig. 1.4

 

Since for an ideal gas pv = const when T = const, on the p-v diagram the isotherm will, obviously, be represented by an equilateral hyperbola (Fig. 1.4a), and the higher the temperature T, the higher is the isotherm in the p-v plane.

It follows from Eq. (1.23) that for an ideal gas the isochors are represen­ted on the p-T diagram and the isobars on the v-T diagram by straight lines initiating at the origin of coordinates, as illustrated in Fig. 1.4b, c. It follows from Eq. (1.23) that

 

                                                                 

 

i.e. the slope of an isochor on the p-T diagram is equal to R/v, and, consequently, the greater v, the lesser the slope of the isochor. Similarly, from

 

                                                                       

 

it follows that on the v-T diagram the slope of an isobar is equal to R/p, indicating that the greater p, the lesser the slope of the isobar.

 

 

 


[1] More strictly, a mole is defined as the amount of a substance containing a number of molecules equal to the number of atoms contained in 0.012 kg of the carbon isotope 12C (accordingly, a kilomole, in 12 kg of this isotope).

[2] More exactly, the universal gas constant μR = 8314.41 J/(kmol·K). The universal gas constant μR is sometimes expressed in other units: μR = 847.83 kgf·m/(kmol·K), or μR = 1.98719 kcal/(kmol·K). The quantity k = μR/ NA is called Boltzmann’s constant (k = 8314.41/6.022045×1028 = 1.380658×10-23 J/K = 1.380662×10-16 erg/K).