Here's the question: For what temperatures are the atoms in an ideal gas at pressure [itex] P [/itex] quantum mechanical? Hint: Use the idea gas law [tex] PV = N k_B T [/tex] to deduce the interatomic spacing. Answer: [tex] T < \left( \frac{1}_{k_B} \right) \left( \frac{h^2}_{3m} \right)^{\left( \frac{3}_{5} \right)} \left( P^\frac{2}_{5} \right) [/tex] ------------- Now, I have been given the formula for the typical de Broglie wavelength: [tex] \lambda = \frac{h}_{\sqrt{3 m k_B T}} [/tex] Further, I know I am trying to determine when [tex] \lambda > d [/tex] where d is the interatomic spacing. However, what I don't understand is how I can calculate a value for d given the idea gas law in the question. Any ideas?
More generally, is there some approach that can be used to calculate the interatomic spacing using the ideal gas law outside of the quantum mechanics context?