Atomic Structure problem regarding interatomic spacing

[rxy152]

Homework Statement

In a gas of hydrogen under normal conditions, the interatomic spacing is 1.56 × 10^−8 m . Assume the gas is made of atomic, not molecular, hydrogen. For what n-value of the hydrogen atoms is the size of the atom comparable to the interatomic spacing?The value of h bar is 1.05457 × 10^−34 J · s, the Bohr radius is 5.29177 × 10^−11 m , and the Rydberg constant for hydrogen is 1.09735 × 10^7 m−1 .

Homework Equations

r= n^2 hbar^2 / Z k q^2 m

n = energy level
Z=number of protons
hbar = Planck's /2/pi
k = Coulombs Constant
q = charge of electron
m = mass

The Attempt at a Solution

Please don't roast me, but i tried using the only equation I know that has both r and n in it. I plugged in my values and got the wrong answer ( 17.18) . Is there another equation that can be applied to correctly solve this problem? Again, I really have no knowledge of questions in this field. Thanks!

 

[TSny]

Can n be any real number, or is it restricted in any way?

 

[rxy152]

Can n be any real number, or is it restricted in any way?

It didn't mention any restrictions in the problem, so I guess it can be any real number.

 

[TSny]

It didn't mention any restrictions in the problems, so I guess it can be any real number.

If n can be any number, then doesn't that mean that r could be any number?

 

[rxy152]

If n can be any number, then doesn't that mean that r could be any number?

I think r in this case is the interatomic spacing. I do not believe it changes. The questions simply asks for n based on the given "r", if I am not mistaken.

 

[TSny]

What is the meaning of r in the equation r= n^2 hbar^2 / Z k q^2 m? What is the meaning of n in this equation?

 

[rxy152]

What is the meaning of r in the equation r= n^2 hbar^2 / Z k q^2 m? What is the meaning of n in this equation?

Not entirely sure the equation is applicable to this questions, but n pertains to which energy level basically has a radius of the interatomic spacing.

 

[TSny]

Not entirely sure the equation is applicable to this questions, but n pertains to which energy level basically has a radius of the interatomic spacing.

What equation did you use to get the answer of n = 17.18?

 

[rxy152]

The one above

 

[TSny]

The one above

This equation: r= n^2 hbar^2 / Z k q^2 m ?

 

[rxy152]

yep

 

[TSny]

yep

OK. This equation gives the radius of the atom for a given value of n. In the Bohr model, can the radius of the atom have any value, or is the radius restricted to only certain values?

 

[rxy152]

OK. This equation gives the radius of the atom for a given value of n. In the Bohr model, can the radius of the atom have any value, or is the radius restricted to only certain values?

Any value I believe. As I mentioned before, the equation probably doesn't apply here. I want to find another one that works.

 

[TSny]

Any value I believe. As I mentioned before, the equation probably doesn't apply here. I want to find another one that works.

You want the "size of the atom" to be comparable to the interatomic distance. You will need to interpret the meaning of "size of the atom". Does it refer to the radius or to the diameter?

I assume you are working with the Bohr model of the hydrogen atom rather than the full quantum mechanical description of the atom. The equation r= n^2 hbar^2 / Z k q^2 m gives the radius in the Bohr model for a given value of n. The n value that occurs in the formula for r is also the n value that occurs in the energy level formula for hydrogen. One of the important features of the Bohr model is that not all classical orbits are allowed. That is, not all values of radius are allowed and not all values of energy are allowed. So, the value of n must be restricted in some way. Refer to your text or notes to see what values of n are allowed.