How to derive partial gas equation? [closed]








**Closed.** This question is [off-topic](/help/closed-questions). It is not currently accepting answers.
 
 






---








 **Homework questions** must demonstrate some effort to understand the underlying concepts. For help asking a good homework question, see: [How do I ask homework questions on Chemistry Stack Exchange?](https://chemistry.meta.stackexchange.com/questions/141/how-do-i-ask-homework-questions-on-chemistry-stack-exchange)




Closed 3 years ago.








[Improve this question](/posts/2753/edit) 




> 
> If the following gasses are to mix, what is the partial pressure of neon?
> 
> 
> ![https://i.stack.imgur.com/8Pn7w.png](https://i.stack.imgur.com/PZ8Kb.png)
> 
> 
> 


The correct answer is **0.1**. I cannot figure out how my professor derived this equation. From the ideal gas law, all I can arrive to is this:


$P\_x = \frac{P\_tV\_t}{V\_x}$


This gives me an answer of 4.4, which is obviously incorrect.


I've determined the equation he used is as follows:


$P\_x = \frac{P\_xV\_x}{V\_t}$


which gives the correct answer of **0.1**. How did he arrive to this answer?




Prepare for an amazing feat of algebra! I sincerely hope there is a shorter way to the endpoint. (there is! See the end of my answer.)


To calculate partial pressure, $P\_{\ce{He}}$, we need the total pressure $P\_t$ and the fraction of the gas that is He $X\_{\ce{He}}$: $$P\_{\ce{He}}=X\_{\ce{He}} P\_t$$


You have $PV=nRT$. $T$ is not given, so assume it is constant. $n$ is unknown, but the system is closed, so $n\_x$ for each gas is constant. Thus we have: 


$$PV=\text{ constant}$$



> 
> Total Pressure
> 
> 
> 


It is tempting to write variations of $$P\_1 V\_1 = P\_2 V\_2$$


However, for each gas, **both $P$ and $V$ are changing**, so we need to consider their products: $$(PV)\_1 = (PV)\_2$$


Thus, as trb456 suggests, $$P\_t V\_t = (PV)\_t \implies P\_t =\frac{\sum{(PV)\_i}}{\sum{V\_i}}$$



> 
> Fraction of He
> 
> 
> 


The fraction of the mixture that is helium is determined by the ratio $\dfrac{n\_{\ce{He}} }{n\_t}$. Since $n=PV/RT$, and $R$ and $T$ are constant, we can write: $$X\_{\ce{He}}=\frac{n\_{\ce{He}}}{\sum{n\_i}}=\frac{(PV)\_{\ce{He}}}{\sum{(PV)\_i}}$$



> 
> At last!
> 
> 
> 


$$P\_{\ce{He}}=X\_{\text{He}} P\_t = \left( \frac{(PV)\_{\ce{He}}}{\sum{(PV)\_i}}\right) \left( \frac{\sum{(PV)\_i}}{\sum{V\_i}}\right)$$
$$P\_{\ce{He}\_2}=\frac{(PV)\_{\ce{He}\_1}}{V\_t}$$



> 
> Or, as it struck me as I finished:
> 
> 
> 


Helium expands to fill the total volume. Now we can use $P\_1 V\_1 = P\_2 V\_2$ and completely ignore the other gasses (there is a lot empty space for the He atoms to fit in).


$$P\_{\ce{He}\_2}=\frac{P\_{\ce{He}\_1}V\_{\ce{He}\_1}}{V\_t}$$


This equation is very similar to your second equation, except the subscripts are added to denote initial and final pressure of helium. 


 3

The total pressure of the system is:


$\frac{1.0\times0.5+1.0\times0.4+2.0\times0.2}{1.0+1.0+2.0}=1.3/4=0.325$


i.e. just the sum of the $PV$s divided by the sum of the $V$s. Then $0.325$ is the sum of the partial pressures by the Law of Partial Pressures. So this at least suggest that the individual partial pressures are near $0.1$.


Can you figure out how to parcel out the individual pressures from here?


 3