Relative velocity equation in Atkins physical chemistry

[mccoy1]

Hi folks, can someone please point out how Atkin got the following relative velocity A-B equation: see attached file. It's in atkin pchem 9e page 834.
Thank you all.

 

[tiny-tim]

Hi folks, can someone please point out how Atkin got the following relative velocity A-B equation: see attached file. It's in atkin pchem 9e page 834.
Thank you all.

hi mccoy1!

v_{rel A-B} is the component of v_rel in the AB direction, ie v_relcosθ

by Pythagoras, cosθ = adj/hyp = √(d² - a²)/d

 

[mccoy1]

hi mccoy1!

v_{rel A-B} is the component of v_rel in the AB direction, ie v_relcosθ

by Pythagoras, cosθ = adj/hyp = √(d² - a²)/d

Thank you very much Tiny-Tim. Yes that's what I got before. And if substitute the value of cos(theta) into vrel,A-B = vrel/cos(theta) equation, then you get
Vrel,A-B =vrel/cos(theta) = v^rel[d²/(d²-a²)]^1/2 , which is not the equation in the attached file or am I doing something wrong?
Thanks again.

 

[tiny-tim]

hi mccoy1!

(have a square-root: √ and a theta: θ )

… then you get
Vrel,A-B =vrel/cos(theta) = v^rel[d²/(d²-a²)]^1/2 , which is not the equation in the attached file or am I doing something wrong?

oh yes, i didn't notice the book got it wrong!

the book is definitely wrong … it has cosθ > 1, which is impossible!

 

[mccoy1]

Thanks again.
As an aside, in 8e of the same book, the authors have multiplication instead of a division between cos(θ) and vrel (which I think is impossible!), what do you think of that as well? I can't believe that the authors got it wrong in both editions! However, what surprised me is that they end up getting correct collision cross-section using that equation.

Cheers for the symbol tips.

 

[tiny-tim]

As an aside, in 8e of the same book …

sorry, i don't have the book (i was going on your picture)

 

[mccoy1]

As an aside, in 8e of the same book …

sorry, i don't have the book (i was going on your picture)

Yes I know. I copied that image from Atkin 9e. I've also attached an image and a text copied from 8e.
Cheers.

 

[tiny-tim]

no, i can't see what I'm supposed to be looking at

btw, it should be multiplication …

… the authors have multiplication instead of a division between cos(θ) and vrel (which I think is impossible!) …

v_rel is the whole vector, and v_relcosθ is the component

 

[mccoy1]

no, i can't see what I'm supposed to be looking at

btw, it should be multiplication …v_rel is the whole vector, and v_relcosθ is the component

Ok there's a section (almost 1/2 way down the page) where it says "Justification 22: The collision cross-section"

Anyway don't worry, I think the fact that you have figured out that it should be a multiplication is enough. You get correct equation if it's multiplication. The only trouble I've with that is v_rel,A-Bcos(θ) is supposed to be scalar component of v_rel,A-B along v_rel. To me , it seems like v_relcos (θ ) is a scalar component of vrel along 'a' in the diagram.
I'm lost!

Edit: okay i think I'm wrong on the last bit. vrelcos (θ) is a scalar component of vrel on vrel,A-B, but the equation vrel,A-B=vrel*cos (θ) suggests that it's a vector .

 

[tiny-tim]

Ok there's a section (almost 1/2 way down the page) where it says "Justification 22: The collision cross-section"

ah, i see it now!

yes, that section is virtually word-for-word the same as the other one, except for the misprint of dividing instead of multiplying

The only trouble I've with that is v_rel,A-Bcos(θ) is supposed to be scalar component of v_rel,A-B along v_rel. To me , it seems like v_relcos (θ ) is a scalar component of vrel along 'a' in the diagram.
I'm lost!

Edit: okay i think I'm wrong on the last bit. vrelcos (θ) is a scalar component of vrel on vrel,A-B, but the equation vrel,A-B suggests that it's a vector .

i think you're over-thinking this

the "real" vector is the actual relative velocity, v_rel,

multiply by cosθ and you get a component (btw, no need to add "scalar") …

usually you would write v and v_x, but instead of "x" the direction is called "AB", so it's written v_AB …

and in this case it's confused by the fact that there are two different suffices, v_rel,AB, with entirely different significances!

 

[mccoy1]

Thank you very much for your help. Yes you are right, that's a reckless claim actually because i need to say abs(vrel)cos(θ) for it to make sense as a scalar. I think i got it now though.
As for the suffices, vrel, A-B (with minus between A and B) is supposed to be velocity along the internuclear axis/line 'd') while vrelAB (no minus is a velocity at an angle θ to the internuclear axis)...as on the second diagram.
Cheers buddy...it seems there's no kudos button here...but still > kudos for you.