Question

One way of writing the equation of state for real gases is, $PV = RT\left[ {1 + {B}{V} + ........} \right]$ where $B$ is constant. An approximately expression for $B$ in terms of van der Waals constant $a$ and $b$ is:
A.$b + \dfrac{a}{{RT}}$
B.$b - \dfrac{a}{{RT}}$
C.$b + \dfrac{a}{{2RT}}$
D.$a + \dfrac{b}{{RT}}$

Answer 1

Hint:
Ideal gases are the gases that obey the laws of kinetic theory of gases. It also follows the ideal gas equation $(PV = nRT)$ . But gases do not always show ideal behavior. Such gases are called real gases.

Complete step by step answer:
Ideal gases are the gases that obey the ideal gas equation at all conditions of temperature and pressure. In reality this is not the case. Gases deviate from their ideal behavior and are called real gases. It is observed that deviation from gas laws are very high when the pressure is high and temperature is low.
Van der Waals suggested that such large deviations are due to the incorrect assumptions of kinetic theory of gases. Kinetic theory of gases assumes that-
Actual volume of individual gas molecules is negligible as compared to the total volume of the gas.
Intermolecular attractions are not present in gases.
Van der Waals pointed out that in case of real gases, molecules do have a volume and also exert intermolecular attractions especially when the pressure is high and temperature is low. He applied two corrections- one for volume and one for pressure.
After applying correction factor, Van der Waals gave a new equation for the behavior of real gases-
$\left( {P + {{\dfrac{n^2}a}}{{{V^2}}}} \right)\left( {V - nb} \right) = nRT$
Where $P$ is the pressure of real gas, $n$ is the number of moles of gas, $V$ is the volume, $R$ is the gas constant, $T$ is the temperature, $a$ is the constant depending on the nature of the gas and $b$ is the measure of volume occupied by molecules.
This equation is obeyed by real gases at all ranges of temperature and pressure and hence this equation is called an equation of state for the real gases. The constants $a$ and $b$ are called Van der Waals constants and are characteristic of each gas.
Now considering one mole of real gas, the above equation can be written as-
$\left( {P + \dfrac{a}{{{V^2}}}} \right)\left( {V - b} \right) = RT$
On rearranging,
$P =\dfrac{RT}{(V - b)} - \dfrac{a}{V^2}$
Now,
$PV = \dfrac{RTV}{(V - b)} - \dfrac{a}{V}$
Or $PV = RT{\left(\dfrac{V}{{V - b}}\right)} - \dfrac{a}{V}$
Rearranging the equation-
$PV = RT{\left(\dfrac{1}{{1 - \dfrac{b}{V}}}\right)} - \dfrac{a}{V}$
$PV = RT(1 + {b}{V}) - {a}{V}$
Neglecting higher powers of $a$ and $b$
$PV = RT{\left(1 + \dfrac{b}{V} - \dfrac{a}{{RTV}}\right)}$
$PV = RT{\left[1 + \dfrac{1}{V}(b - \dfrac{a}{{RT}})\right]}$
Comparing this equation with the given equation-
$B = b - \dfrac{a}{{RT}}$
Thus, the correct option is (B).

Note:All real gases follow the Van der Waals equation. The unit of $a$ is $N{m^4}mo{l^{ - 2}}$ and that for $b$ is ${m^3}mo{l^{ - 1}}$ .
-At low pressure and moderately high temperature, real gases approach ideal behavior.
-$PV = nRT$ is called the ideal gas equation.