Mole balance issue with volume division

[williamcarter]

Homework Statement

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?

Exercise #2(see below please)

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.

 

[Chestermiller]

Homework Statement

View attachment 195716 View attachment 195717

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?

What is your problem with this?

Exercise #2(see below please)
View attachment 195718

Solution for Problem#2
View attachment 195720

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

Thank you very much in advance.

Because the r^2 is inside the parenthesis.

 

[williamcarter]

What is your problem with this?
I don't get exactly how he got equation 7 from 6?

Because the r^2 is inside the parenthesis.

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.

 

[Chestermiller]

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.

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: $$\frac{d(r^2N)}{dr}=0$$

 

[williamcarter]

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: $$\frac{d(r^2N)}{dr}=0$$

Thank you for your reply !
But why does he factor out the 4*pi? I want to understand the principle

 

[Chestermiller]

Thank you for your reply !
But why does he factor out the 4*pi? I want to understand the principle

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?"

 

[williamcarter]

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?"

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?

 

[Chestermiller]

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?

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.

 

[williamcarter]

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.

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?

 

[Chestermiller]

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?

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.