# Mole balance issue with volume division

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1. Apr 19, 2017

### williamcarter

1. The problem statement, all variables and given/known data

Here we divide the mole balance by the volume $4*\pi$ $*$ r2$*$dr and take lim as dr->0(standard procedure)

1) How exactly he makes the transition from equation from 6 to 7?

Solution for Problem#2

2)Why does he divide here by $4*\pi$ $*Δr$? and not by $4*\pi$ $*$ r2$*$dr ??

Thank you very much in advance.

2. Apr 19, 2017

### Staff: Mentor

What is your problem with this?
Because the r^2 is inside the parenthesis.

3. Apr 19, 2017

### williamcarter

In the first problem he didnt subsitute S(r) with 4pi*r2, he let it just S(r), but why in second problem he put it inside the ( ) as 4pi*r2?

I don't get why he divided in 2nd example just by 4pi*Δr, because I can't see the difference in procedures, between the 2 examples.

4. Apr 19, 2017

### Staff: Mentor

In Eqn. 7, with $S=4\pi r^2$, he would have:$$\frac{d(4\pi r^2)N}{dr}=0$$ Factoring out the $4\pi$ leaves: $$\frac{d(r^2N)}{dr}=0$$

5. Apr 19, 2017

### williamcarter

But why does he factor out the 4*pi? I want to understand the principle

6. Apr 19, 2017

### Staff: Mentor

Are you saying that you don't think it is valid to factor out the 4 pi, or are you asking "why would anyone want to factor out the 4 pi?"

7. Apr 19, 2017

### williamcarter

I am going for the 2nd choice ''why would anyone want to factor out the 4 pi?'', how to know when to factor it out or when to leave it there?

8. Apr 19, 2017

### Staff: Mentor

Factoring out the constant (4 pi) simplifies the equation. It's analogous to what we do when we reduce a fraction to its least common denominator.

9. Apr 19, 2017

### williamcarter

Thank you for your answer, but why in the first example he did not factor the 4pi out?

How to know when to factor it out and when not to?

10. Apr 19, 2017

### Staff: Mentor

I have no answer for this. It is not addressing any specific problem, so there is no one right way of doing it. It is up to the taste of the writer.