Can anyone identify the approximation used in this solution?

[foldylocks]

I am trying to follow the reasoning of the last problem in the set linked below. I can't figure out what approximation they used in step 24. Thanks.

http://www.physics.fsu.edu/courses/spring08/phy5524/sol1.pdf

It looks like it might be an approximation based on a geometric series, but there is no summation.

 

[D H]

They're using the approximation $\frac 1 {1-x} \approx 1+x$, or even more appropriately, $1+x \approx \frac 1 {1-x}$ where $x=\frac {N-1}2 \frac {v_0} V$.

 

[Simon Bridge]

There is a summation at step 20.
The approximation comes in at 21-22

But you want to know how you go from $$\frac{P}{k_BT}\simeq \frac{1}{V-(N-1)v_0/2}$$ to $$\frac{P}{k_BT}\simeq \frac{1}{V-Nv_0/2}$$ ... well $(N-1)x \simeq Nx$ if $x<<Nx$ right? Presumably N is a very big number?

[edit] DH has it formally :)

 

[foldylocks]

Thanks for your help! DH, I assume that approximation is only valid for small x correct? It seems as if N is very large, then the approximation wouldn't be a good one.

 

[Simon Bridge]

Hmmm... I didn't check it properly.
We seem to be saying different things after all - I am pointing out that for large $N$, $N - 1 \approx N$.

You can get some very hokey-looking approximations in physics.

 

[D H]

Thanks for your help! DH, I assume that approximation is only valid for small x correct? It seems as if N is very large, then the approximation wouldn't be a good one.

Yes, N is large, which makes step 25 valid, where N-1 is simplified to N. The assumption that makes step 24 valid is that $Nv_0 \ll V$. In other words, this assumption says the volume excluded by all of the gas molecules is much less than the total volume of the system.