# Homework Help: Minimum Kinetic energy of the hydrogen atom

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1. May 19, 2017

### Rahulrj

1. The problem statement, all variables and given/known data
A hydrogen atom collides with another hydrogen atom at rest. If the electrons in both atoms are in the ground state, what is the minimum kinetic energy of the hydrogen atom such that the hydrogen atom at rest will have its electron in the first excited state after collision?

2. Relevant equations
$E=h \nu$
$E_n=E_1/n^2$

3. The attempt at a solution
I know that the first excited state mean n=2 however, I am not sure how to write the energy conservation here as the question does not mention the energy of the colliding atom after the collision.

2. May 19, 2017

### Staff: Mentor

Not sure how this is relevant.

3. May 19, 2017

### Rahulrj

I don't think I see what you wanted me to see. Need a hint more than that. The only formula I know for KE is the relativistic one $(\gamma - 1) mc^2$ and $1/2mv^2$ but I don't see how to get it from the available info given in the question.

Last edited: May 19, 2017
4. May 19, 2017

### Staff: Mentor

Before the collision, where is the energy? After the collision, where can the energy be?

5. May 19, 2017

### Rahulrj

The colliding atom has the energy before collision and after collision, some energy from the colliding atom is imparted to the atom at rest.

6. May 19, 2017

### Staff: Mentor

I think it would help if you gave a more detailed answer: what kind of energy has each atom before and after?

7. May 19, 2017

### Rahulrj

The colliding atom has kinetic energy and the atom at rest has potential energy. After collision, the kinetic energy of the colliding atom lowers and that amount is compensated by the change in energy of the atom at rest(to kinetic). Since its been given that the electron in the atom at rest excites to n=2, it absorbs photon of equivalent energy?

8. May 19, 2017

### Staff: Mentor

I'm not sure what potential energy you are considering (none is mention in the problem). Let me rephrase what you wrote as (calling atom 2 the atom initially at rest)

before: kinetic energy of atom 1
after: kinetic energy of atom 1 + kinetic energy of atom 2 + electronic energy of atom 2

As you said in the OP, you need to use conservation of energy. As you are to find the minimum energy for which this "after" is possible, what is the minimum value of energy after the collision?

There are no photons involved here.

9. May 19, 2017

### Rahulrj

I only know how to find the electronic energy here and I have not much idea on how to find the KE as I don't know what formula to use there.

10. May 19, 2017

### Staff: Mentor

You already gave the equation for the kinetic energy. But it is not that relevant. Again, what is the minimum kinetic energy you can have after the collision?

11. May 19, 2017

### Rahulrj

Minimum energy the colliding atom 'can' have after collision is zero? It is possible that all of it can be imparted to atom 2 right?
since the colliding atom has minimum KE, am I right in saying that the energy is just enough to get the electron in atom 2 to be excited and not impart KE to the atom 2 itself to experience a momentum and therefore minimum KE of atom is $E_1/n^2$?

Last edited: May 19, 2017
12. May 19, 2017

### haruspex

If the colliding atom comes to rest, what does momentum conservation say about the final speed of the other atom? What energy will that leave for the excitation?

13. May 19, 2017

### Rahulrj

Using Momentum conservation I can say $p_1=p_2$ so the speeds are same for both? but that doesn't make sense because
If the speeds are same then from energy conservation $KE_1 = KE'_1+KE_2+E_{electronic}$, $KE'_1 =0$ and $KE_1-KE_2 =E_{electronic}$ (1 is for colliding atom and 2 is for the other atom) the two kinetic energies have to be zero right?

14. May 19, 2017

### haruspex

Define p1 and p2 there. If you mean total momentum before and after then yes, that is conservation of momentum, but it does not mean they would have the same final speed. (They just might, though.)
Right.
No, why?

15. May 19, 2017

### Rahulrj

$KE'_1 = 0$ is the KE of colliding atom after collision which you said it comes to rest and that means KE is zero right?
$p_1$ is the momentum of colliding atom before collision and $p_2$ is the momentum of the atom at rest after collision

16. May 19, 2017

### haruspex

No, I did not say it comes to rest. I asked, IF it comes to rest (as you had assumed), what would conservation of momentum tell you about the final speed of the other atom? I asked this inorder to demonstrate that it certainly does not come to rest.

You have an equation for conservation of energy and another for conservation of momentum. They involve two unknown final velocities and one unknown initial velocity. Two equations with three unknowns leaves one degree of freedom. As you vary one velocity, the other two velocities vary in consequence. Your task is to find how to minimise the unknown initial velocity.

17. May 19, 2017

### Rahulrj

I still didn't get how you came to the conclusion that the colliding atom does not come to rest. According to my thinking, to minimize unknown initial velocity just enough to get the electron in atom 2 to be excited means the kinetic energy of the colliding atom after collision has to be zero as the question is asking only for a minimum KE.

18. May 19, 2017

### haruspex

If the incoming atom comes to rest, conservation of momentum says the other atom, being of the same mass, must acquire the speed that it lost. The would mean that it also acquires all the KE, leaving no energy for the excitation.

Please, write out the equations as I suggested.

19. May 20, 2017

### Rahulrj

Okay, so writing down the equations
$mv_{1i} = mv_{1f}+mv_{2f}$
$1/2mv_{1i}^2 = 1/2mv_{1f}^2+1/2mv_{2f}^2+E_{electronic}$ substituting $v_{1i}$ from momentum equation to energy equation
$mv_{1f}v_{2f}= E_{electronic}$
Not sure what I should be looking for in these equations to get minimum velocity.

20. May 20, 2017

### ehild

You know the product of two numbers, when is their sum minimum? (Think of the relation between the geometric mean and the arithmetic mean.)

21. May 20, 2017

### haruspex

You can use ehild's excellent hint, or use your last equation to substitute for v2f in your first equation. That would give you one equation with two unknowns, v1i and v1f. How do you minimise one variable with respect to another?

22. May 21, 2017

### Rahulrj

I am not very familiar solving using AM and GM concepts, it has been a while but I would love to know how it can be done. So I choose to go with the substitution method and the equation I get is $v_{1i}=v_{1f}+E_{electronic}/mv_{1f}$. So now do I need to take a derivative w.r.t $v_{1f}$ to find minimum?

23. May 21, 2017

Yes.