Is this a printing error? (book example problem -- radius of a molecule)

[Buffu]

Question :- The critical volume of a gas is 0.072 L mol^-1. What will be the radius of the molecule in cm ?

Answer in the book :-

$V_c = 3b \implies b = 0.024 L mol$
$\therefore$ for every molecule $b = {24 \text{cm}^3\over 6 \times 10^{23}} = 4 \times 10^{-23}\text{cm}^3$ per molecule.

$\therefore 4 \times 10^{-23}\text{cm}^3 = \color{red}{4} \times \frac43 \pi r^3$
$r^3 = \left({3\over 4\pi} \times 10^{-23}\right)^{1\over3} = 1.336 \times 10^{-8}$ cm

I did not get from where that 'red' 4 came in the third line ? I think volume of a sphere is just 4/3 pi r^3.
Mostly likely a mistake on behalf of the author.

 

[mathman]

What is b?

 

[Buffu]

What is b?

Van der waal constant.

 

[mathman]

I can't help you much, because the physics involved is something I have never studied. Wikipedia description doesn't help.

 

[Buffu]

I can't help you much, because the physics involved is something I have never studied. Wikipedia description doesn't help.

No problem, thanks for taking the time to consider my question.

 

[Buffu]

@Borek Can you help me with this question ?

 

[256bits]

I think volume of a sphere is just 4/3 pi r^3.

That's right.

Vc=3b⟹b=0.024LmolVc=3b⟹b=0.024LmolV_c = 3b \implies b = 0.024 L mol

Are the dimensions of b correct here ?

Why the 4?
Draw a picture and do some simple math.
Take 2 molecules of r.
The volume of one molecule would be as from the formula of a sphere.
Have the second molecule approach and touch the first.
Of course their centres cannot overlap but are separated by distance 2r.
From no matter what direction the second molecule approaches the first it can get no closer than 2r.
What volume is that?

So for your answer, what volume is now excluded for movement of the two (2) molecules.

 

[Buffu]

Are the dimensions of b correct here ?

Yes the dimension is correct.
$\left(P+{n^2a\over V}\right)(v-nb) = nRT$

Why the 4?
Draw a picture and do some simple math.
Take 2 molecules of r.
The volume of one molecule would be as from the formula of a sphere.
Have the second molecule approach and touch the first.
Of course their centres cannot overlap but are separated by distance 2r.
From no matter what direction the second molecule approaches the first it can get no closer than 2r.
What volume is that?

So I have to find the volume of this figure and divide it by 2 because there are 2 molecules. Am I correct ?

 

[256bits]

So I have to find the volume of this figure and divide it by 2 because there are 2 molecules. Am I correct ?

It looks as if the green sphere has penetrated the blue sphere. Only the surfaces should physically just touch.
Divide by 2 for 2 molecules is correct.

 

[256bits]

 

[Buffu]

View attachment 110745

So the total volume is volume is of green sphere by 2.
i.e $\displaystyle{{4\over 3}\pi \times8r^3\over 2} = {{4\over 3}\pi \times4r^3}$

Ok I did get the answer but I don't know why we did this ?

 

[256bits]

Perhaps that is more intuitive, as how molecules with volume occupy space mathematically definitely is not.

The second picture:
Using the centre of the molecules.
Second picture shows the black molecule approaching the pink molecule from several directions ( the grey ).
The green ( sphere ) shows the volume within which the black molecules centre cannot penetrate since the pink molecule does not allow due to its presence.

The first picture:
shows the volume within which both molecules can occupy if they would swirl around one another.

By the way, this is not close packing of solids or liquids, where one has several molecules touching one another at the same time.
For gases, I guess the assumption is made that contact time is brief, and very rarely involves more than 2 molecules.

You can also consider the green circle in the second picture, as the sphere inside of which the two molecules will collide, and outside of which they will not ( which by the way is related to the mean free path of the molecules ) as they move along their trajectory path. May be the better of the explanations.

Does that help @Buffu?

 

[Buffu]

@256bits Oh, I get the solution now. Really nice explanation with pictures.

I have a side question, What part of physics is this ? statistical mechanics ?