Boyle's temperature `(T_(B))` : It is the temperature at which a real gas exhibits ideal behavior in low pressure range. Different gases have different value of `T_(B)`, which is related to the van der Wall's constant `a` and `ab` as follows: `T_(B) = (a)/(Rb)`
The compressiblity factor for a real gas is given as `Z = 1 + 0.35 P - (168)/(T)P`. Where `P` is in bar and `T` is in Kelvin. What is the Boyle's temperature of the gas?

Options

A. Compressiblity factor `(Z)` for `1` mol of a van der Wall's get at Boyle temeprature is
`1 + (b^(2))/(V(V - b))`
B. Compressiblity factor `(Z)` for `1` mol of a van der Wall's gas at Boyle temperature is `1 + (b^(2))/(V^(2))`
C. Critical temperature `(T_(C))` of a real is gas is less than its's Boyle temperature `(T_(B))`
D. Boyle's temperture `(T_(B))` represents a temperature range.

Answer

A, C

Explanation

For `1` mol gas
`(P + (a)/(V_(2)))(V - b) = RT. , (T_(B) = (a)/(Rb))`
`(P + (a)/(V^(2))) = (R)/(V - b).(a)/(Rb)`
`P = (a)/(b(V - b)) - (a)/(V^(2))`……..(1)
Equation `(1) xx (V)/(RT)`
`(PV)/(RT) = ((a)/(b(V-b)) - (a)/(V^(2))) xx (V)/(RT)`
`(PV)/(RT) = ((a)/(b(V - b))-(a)/(V^(2))) xx (V)/(R) xx (Rb)/(a)`
`z = ((1)/(b(V-b))-(a)/(v^(2)))V.b`
`z = (V)/(V - b) - (b)/(V)`
`z = (V^(2) - b(V - b))/(V(V - b))`
`z = (V^(2) - Vb + 2)/(V(V - b))`
`z = (V(V - b) + b^(2))/(V(V - b)) rArr z = 1 + (b^(2))/(V(V - b))`