Gaseous State
8.0 Real gas (deviation from ideality)
8.0 Real gas (deviation from ideality)
Gases deviate from ideality due to two faulty assumptions in the kinetic theory of gases .
1. Volume occupied by gaseous molecules is negligible with respect to volume of container i.e., Real gas molecules have a finite volume.
2. Gaseous particles do not exert force on each other or on the wall of the container except during collision i.e., Inter-molecular attractive forces between real gas molecules is not zero.
If these assumptions were to be true then the gases would have never been liquefied . But we know that gases liquefy on decreasing temperature and increasing pressure. Therefore, ideal gas equation is not applicable here. So, ideal gas equation is slightly modified for real gases.
Its modified form is known as 'Vander waal's equation'. Mathematically, $$\left[ {P + \left( {\frac{{{n^2}a}}{{{V^2}}}} \right)} \right](V - nb) = nRT$$ Here, $a$ and $b$ are constants known as Vanderwaal's constants.
$a$ is known as coefficient of attraction. Greater is the value of $a$ greater is the force of attraction among molecules. $b$ is known as effective volume of the one mole of gaseous molecules.
Before discussing Vanderwall's equation, let us discuss how this deviation of real gases from ideal behaviour is measured.
Compresibility factor: It is the ratio of product of $PV$ and $nRT$. Mathematically, $$Z = \frac{{PV}}{{nRT}}$$ For an ideal gas, the ratio $\frac{{PV}}{{nRT}}$ is always equal to 1 ($\because $ $PV=nRT$ for ideal gas). For real gases, $Z \ne 1$.
At high pressure, all the gases have $Z > 1$. These are more difficult to compress.
At intermediate pressures, most gases have $Z < 1$.
At low pressure, all gases have $Z \approx 1$ .
Thus, gases show ideal behaviour when the volume occupied is large so that the volume of the molecules can be neglected in comparison to it. For real gases, $Z$ can also be expressed as, $$\begin{equation} \begin{aligned} Z = \frac{{P{V_{real}}}}{{nRT}} \\ Z = \frac{{{V_{real}}}}{{(nRT/P)}} \\ Z = \frac{{{V_{real}}}}{{{V_{ideal}}}} \\\end{aligned} \end{equation} $$
Pressure correction term: At high pressure, molecules do not strike the walls of the container with full impact because these are dragged back by other molecules due to molecular attractive forces. This affects the pressure exerted by the molecules on the walls of the container. Thus, the pressure exerted by the gas is lower than the pressure exerted by the ideal gas.
$${P_{ideal}} = {P_{real}} + \frac{{{n^2}a}}{{{V^2}}}$$
Volume correction term: When the volume of container is comparable to volume of gas particles then the compressible volume if the container contains n mole of gas is $(V-nb)$. Volume used in ideal gas equation is total compressible volume. In case of ideal gas it is equal to the volume of container but in case of real gas it is equal to $(V-nb)$. Hence, the volume correction term is '$nb$'.
Boyle's temperature: The temperature at which a real gas obeys ideal gas law over an appreciable range of pressure is called Boyle's temperature or Boyle's point. Mathematically, $${T_b} = \frac{a}{{Rb}}$$
Question 8. Calculate compressibility factor when (a) pressure is high (b) temperature is low.
Solution: (a) Since pressure is high, pressure correction term can be neglected. Therefore, Vanderwaal's equation becomes$$\begin{equation} \begin{aligned} P(V - nb) = nRT \\ PV - Pnb = nRT \\\end{aligned} \end{equation} $$On dividing both sides by $nRT$, we get$$\begin{equation} \begin{aligned} Z - \frac{{Pb}}{{RT}} = 1 \\ Z = 1 + \frac{{Pb}}{{RT}} \\\end{aligned} \end{equation} $$
(b) Since pressure is high, volume correction term can be neglected. Therefore, Vanderwaal's equation becomes$$\begin{equation} \begin{aligned} (P + \frac{{{n^2}a}}{{{V^2}}})V = nRT \\ (PV + \frac{{{n^2}a}}{V}) = nRT \\\end{aligned} \end{equation} $$On dividing both sides by $nRT$, we get$$\begin{equation} \begin{aligned} Z - \frac{{na}}{{RTV}} = 1 \\ Z = 1 - \frac{{na}}{{RTV}} \\\end{aligned} \end{equation} $$
