# Homework Help: Control rods nuclear reaction equation, moles liberated, pressure

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1. Oct 31, 2016

### moenste

1. The problem statement, all variables and given/known data
A control rod to limit the rate at which a nuclear reactor is working is made from boron which is sealed in a casing. A boron atom (105B) is able to capture a neutron; an atom of lithium (Li) and an alpha particle being produced in the process. As a result helium gas is produced which occupies the spaces between the atoms in the rods which may be assumed to have a crystalline structure.

Each cubic metre of the control rod can absorb 1.5 * 1027 neutrons before it must be replaced.

(a) Write down an equation for the nuclear reaction which takes place in the control rods.

(b) How many moles of helium are liberated in each cubic metre of control rod?

(c) The boron atoms themselves occupy 75 % of the total volume occupied by the rods. Calculate the pressure inside the casing, at a temperature of 300 K, just before the rod is replaced.

(Molar gas constant R = 8.3 J mol-1 K-1, Avogadro constant NA = 6.0 * 1023 mol-1.)

Answers: (b) 2.5 * 103 mol, (c) 2.5 * 107 Pa.

2. The attempt at a solution
(a) 105B + 10n → 63Li + 42α + 42He. Is it correct?

(b) No idea where to begin.

2. Oct 31, 2016

### TSny

Is electric charge conserved in your equation?
Are the total number of nucleons conserved in your equation?

How is an alpha particle related to a Helium nucleus?

3. Oct 31, 2016

### moenste

It should be like this then: 105B + 10n → 63Li + 42α.

4. Oct 31, 2016

### TSny

That's close. But the number of nucleons on the left doesn't match the number of nucleons on the right.

5. Oct 31, 2016

### moenste

105B + 10n → 63Li + 42α + 10He? We have helium included, as in the problem statement.

6. Oct 31, 2016

### TSny

What do the 1 and 0 stand for in the symbol 10He?

Also, think about what an alpha particle is made of.

Does the lithium isotope have to be 63Li? Could it be a different isotope of lithium?

7. Oct 31, 2016

### moenste

1 is nucleon number and 0 is proton number.

2 electrons and 2 protons and 4 - 2 neutrons.

Why shouldn't it be? When an alpha-particle is added, 4 and 2 should be subtracted.

8. Oct 31, 2016

### TSny

Does a Helium nucleus have 0 protons?

An alpha particle does not have any electrons.

Can you fill in the blanks in the folowing?
(a) An alpha particle has __________ protons.
(b) An alpha particle has __________ neutrons.
(c) A helium nucleus has _________ protons.
(d) A helium nucleus has _________ neutrons.

Last edited: Nov 1, 2016
9. Nov 1, 2016

### moenste

I'm not sure what "Helium nucleus" means...

The element from the periodic table is 42He, where 4 is the nucleon number = protons + neutrons and 2 is the proton number = electrons. So He from the periodic table has 2 protons and electrons and 4 - 2 = 2 neutrons.

(a) 2 protons.
(b) 4 - 2 = 2 neutrons.
(c, d) What is a helium nucleus?

---

Maybe the equation is like this: 105B + 10n → ??Li + 42α + 42He. We have 11 as the nucleon number and 5 as the proton number on the left with 8 as the nucleon number and 4 as the proton number respectively. So we are lacking 3 and 1. So: 105B + 10n → 31Li + 42α + 42He.

10. Nov 1, 2016

### Staff: Mentor

The nucleus of a helium atom.
A neutral helium atom has 2 electrons, 2 protons and 2 neutrons, right. Which particles are in the nucleus, which particles are not?

11. Nov 1, 2016

### moenste

Nucleus has neutrons and protons, while electrons are flying around the nucleus.

12. Nov 1, 2016

### TSny

The purpose of asking you how many protons and neutrons are in an alpha particle and in a $_2^4$He nucleus was to have you compare the alpha particle and the nucleus of the $_2^4$He atom.

https://en.wikipedia.org/wiki/Alpha_particle

13. Nov 1, 2016

### moenste

Hm, it says that they are identical.

105B + 10n → 73Li + 42He?

14. Nov 1, 2016

### TSny

Yes. That's good.

15. Nov 1, 2016

### moenste

Great!

How to approach (b)?
I know that in 42He -- 4 g of He have 6 * 1023 atoms (NA number). But how to find moles I'm not sure.

16. Nov 1, 2016

### TSny

How many alpha particles will be produced in 1 m3 of the control rod material before the rod must be replaced?

17. Nov 1, 2016

### moenste

1 m3 can absorb 1.5 * 1027 neutrons and an α-particle has 4 - 2 = 2 neutrons.

1.5 * 1027 / 2 = 7.5 * 1026 α-particles.

18. Nov 1, 2016

### TSny

No. The neutrons that get absorbed by the boron nucleus do not necessarily end up in the alpha particle. They could end up in the lithium nucleus.

According to the reaction equation, how may alpha particles are produced for each neutron that gets absorbed.

19. Nov 1, 2016

### moenste

One alpha particle is produced for each neutron?

20. Nov 1, 2016

### TSny

Right.

21. Nov 1, 2016

### moenste

So 1.5 * 1027 α-particles are produced in 1 m3?

22. Nov 1, 2016

### TSny

Yes.

23. Nov 1, 2016

### moenste

How many moles does an alpha particle have?

In that case multiply that number by 1.5 * 1027, the total number of alpha particles in 1 m3, and that will be the answer.

24. Nov 1, 2016

### TSny

A better question would be how many alpha particles does a mole of alpha particles have?

25. Nov 1, 2016

### moenste

1 mol = number of atoms in 4 g of 42He.

So since we have 1.5 * 1027 He, therefore we have 1.5 * 1027 moles.

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