Question

The van der waals constant for ${O_2}$ are a= 1.642 atm\[{L^2}mo{l^{ - 2}}\]and b = 0.04 \[Lmo{l^{ - 1}}\]. Calculate the temperature at which${O_2}$ gas behaves ideally for a longer range of pressure.

Answer 1

Hint:Look at the second part of the question it says to find out the temperature at which ${O_2}$ gas (a real gas) behaves ideally for a longer range of pressure, this statement is the very meaning of Boyle's temperature. Hence, we are going to use the concept of Boyle's temperature.

Formula used: ${T_B} = \dfrac{a}{{bR}}$
Where, ${T_B}$= Boyle's temperature
a = correction factor for attractive forces
b = correction factor for the volume of moles
R = gas constant( 0.082 L atm ${\text{mo}}{{\text{l}}^ - }{\text{ }}{{\text{K}}^ - }$)

Complete answer:
Now, simply using the mentioned formula of Boyle's temperature we can determine the required temperature,
So, let's put up the values and calculate,
$ \Rightarrow $${T_B}$= $\dfrac{{1.642}}{{0.04 \times 0.082}}$
We will use the basic arithmetic operations(multiplication followed by division) to solve this,
$ \Rightarrow $ ${T_B}$= $\dfrac{{1.642}}{{0.00328}}$
$ \Rightarrow $${T_B}$= 500 K

Therefore, we can say that at around 500 K ${O_2}$ gas will behave as an ideal gas.

Additional information:
1. An ideal gas is a theoretical gas, which follows the ideal gas equation(or law). i.e, PV = nRT , here, P = pressure(force per unit area) of the gas, V = Volume of the gas, n = number of moles, R = Gas constant, T = temperature of the ideal gas.
2. A real gas does not follow the ideal gas equation(or law). However, they obey van der waals gas law, which also gives us the a and b correction factors
3. Van der waals gas equation is $\left( {{\text{P + }}\dfrac{{{\text{a}}{{\text{n}}^2}}}{{{{\text{V}}^2}}}} \right)\left( {{\text{V - nb}}} \right) = {\text{nRT}}$ , where terms have their usual meaning.

Note:
As the question mentions van der wall and the values associated with this equation, a student might get confused about which formula to use. It is advised to read up the whole question before attempting it.