Chapter 1, Problem 1.42E

Interpretation Introduction

Interpretation:

Table 1.5 is to be used to list the gases from most ideal to least ideal. The trend or trends obvious from this list are to be found out.

Concept introduction:

An ideal gas is denoted as the one in which there are no intermolecular attractive forces or repulsive forces and in which all collisions between the particles such as atoms or molecules are superlatively elastic. Besides, ideals gases can be visualized as a group of perfectly hard spheres which can collide with each other, but which otherwise will not interact with each other. In contrast, real gases are non-hypothetical gases and its molecules can occupy space and have interactions with each other by adhering the gas laws.

Answer to Problem 1.42E

The gases can be listed as most ideal to least ideal as follows;

$\underset{\begin{array}{l}\text{Lower}\text{Higher}\\ \text{ideality}\text{ideality}\end{array}}{\overset{{\text{CO}}_{\text{2}}\text{<}{\text{CH}}_{\text{4}}\text{<}\text{Ar}\text{<}{\text{O}}_{\text{2}}\text{<}{\text{N}}_{\text{2}}\text{<}\text{Ne}\text{<}{\text{H}}_{\text{2}}\text{<}\text{He}}{\to }}$

The gases are arranged based on increasing trend of their Boyle temperature.

Explanation of Solution

An ideal gas is denoted as the one in which there are no intermolecular attractive forces or repulsive forces and in which all collisions between the particles such as atoms or molecules are superlatively elastic. Besides, ideals gases can be visualized as a group of perfectly hard spheres which can collide with each other, but which otherwise will not interact with each other. At STP most real gases behave like an ideal gas such as nitrogen, hydrogen, oxygen, noble gases, carbon dioxide. The term ideal is applicable for gas at higher temperature and lower pressure. In these conditions the potential energy due to intermolecular forces becomes less important as compared with the particle’s kinetic energy. Besides, the size of the gas molecules is less significant as compared to the empty space between them. Thus, one mole of an ideal occupies a volume of $22.7 L$ at standard temperature and pressure.

The main drawback of ideal gas it is unsuccessful at lower temperatures or higher pressures, where intermolecular forces and molecular size of gases plays a significant role. Importantly, it fails for most heavy gases (ex: refrigerants) and gases having strong intermolecular forces (ex: water vapor) and ideal gas does not elucidate phase transitions. Thus, the deviations from the ideal gas behavior can be best described by the ‘compressibility factor Z’. The ideal gas equation is $PV=nRT\cdot \cdot \cdot \text{(1)}$

$P = \mathrm{P}\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{e}\mathrm{ }\mathrm{o}\mathrm{f}\mathrm{ }\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{ }\mathrm{g}\mathrm{a}\mathrm{s}$

$V = \mathrm{V}\mathrm{o}\mathrm{l}\mathrm{u}\mathrm{m}\mathrm{e}\mathrm{ }\mathrm{o}\mathrm{f}\mathrm{ }\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{ }\mathrm{g}\mathrm{a}\mathrm{s}$

$N =\mathrm{ }\mathrm{N}\mathrm{u}\mathrm{m}\mathrm{b}\mathrm{e}\mathrm{r}\mathrm{ }\mathrm{o}\mathrm{f}\mathrm{ }\mathrm{m}\mathrm{o}\mathrm{l}\mathrm{e}\mathrm{s}$

$R =\mathrm{ }\mathrm{U}\mathrm{n}\mathrm{i}\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{s}\mathrm{a}\mathrm{l}\mathrm{ }\mathrm{g}\mathrm{a}\mathrm{s}\mathrm{ }\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t}$

$T =\mathrm{ }\mathrm{T}\mathrm{e}\mathrm{m}\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{u}\mathrm{r}\mathrm{e}\mathrm{ }\mathrm{o}\mathrm{f}\mathrm{ }\mathrm{g}\mathrm{a}\mathrm{s}$

In contrast, real gases are non-hypothetical gases and its molecules can occupy space and have interactions with each other by adhering the gas laws. Under most conditions, they gas going with low temperatures and high pressures are called non-ideal gases. In terms of volume, the compressibility of non-ideal gases can be written as;

$\left(p+\frac{a{n}^2}{V^2}\right)\left(V-nb\right)=nRT\cdot \cdot \cdot (2)$

In terms of volume, the compressibility of non-ideal gas is expressed as

$Z=\frac{p\overline{V}}{RT}=1+\frac{B}{\overline{V}}+\frac{C}{{\overline{V}}^2}+\frac{D}{{\overline{V}}^3}+\cdot \cdot \cdot \cdot \cdot \cdot (3)$

Where B, C, D virial coefficient and the equation is called virial equation of state.

The temperature at which the virial coefficient B becomes zero is called Boyle temperature.

$T_B=\frac{a}{bR}$

a and b are van der Waals constant. This Boyle temperature is used to arrange the gases based on ideal to non-ideal behavior. The order of gases are as follows;

$\underset{\begin{array}{l}\text{Lower}\text{Higher}\\ \text{ideality}\text{ideality}\end{array}}{\overset{{\text{CO}}_{\text{2}}\text{<}{\text{CH}}_{\text{4}}\text{<}\text{Ar}\text{<}{\text{O}}_{\text{2}}\text{<}{\text{N}}_{\text{2}}\text{<}\text{Ne}\text{<}{\text{H}}_{\text{2}}\text{<}\text{He}}{\to }}$

At lower ideality values the gases behaves as ideal gas and undergoes various reactions that an ideal gas will undergo.

Conclusion

Thus, Table 1.5 is used to list the gases from most ideal to least ideal. The trend or trends obvious from this list are found out.