# Calculating Spacing Using Ideal Gas Law

1. ### cyberdeathreaper

46
Here's the question:

For what temperatures are the atoms in an ideal gas at pressure $P$ quantum mechanical?

Hint: Use the idea gas law
$$PV = N k_B T$$
to deduce the interatomic spacing.

$$T < \left( \frac{1}_{k_B} \right) \left( \frac{h^2}_{3m} \right)^{\left( \frac{3}_{5} \right)} \left( P^\frac{2}_{5} \right)$$

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Now, I have been given the formula for the typical de Broglie wavelength:

$$\lambda = \frac{h}_{\sqrt{3 m k_B T}}$$

Further, I know I am trying to determine when
$$\lambda > d$$
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?

2. ### cyberdeathreaper

46
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?

3. ### krab

905
Just use $d=V^{1/3}$. Doesn't that make sense to you?