Partial Pressures of an Ideal Gas Mixture Containing Water Vapor

[kilgorethecat]

Homework Statement

A gas mixture of 0.13 mol NH₃, 1.27 mol N₂, and 0.025 mol H₂O vapor is contained at a total pressure of 830 mm Hg and 323 K. Calculate the following:
(a) Mole fraction of each component.
(b) Partial pressure of each component in mm Hg.
(c) Total volume of mixture in m³ and ft³.

Homework Equations

X_i = n_i/n_total
P_total = P₁ + P₂ + P₃ ... + P_n
P_i = X_iP_total
PV = nRT
perhaps others...

The Attempt at a Solution

OK, so I have been working on this for a while now and can't seem to find the answer. Since water vapor only behaves like an ideal gas at pressures below 75 mm Hg, it cannot be treated as so in this problem.

I calculated the mole fractions no problem:
(0.13 mol NH₃)/1.425 mol total = 0.0912
(1.27 mol N₂)/1.425 mol total = 0.891
(0.025 mol H₂O)/1.425 mol total = 0.0175

The problem comes when I try to calculate partial pressures of each gas and the water vapor. Since vapor pressure is a function of temperature alone, I was able to reference a table of values which says at 323 K (50 C), water vapor has a saturation pressure of 92.5 mm Hg. Subtracting this from 830 mm Hg gives 737.5 mm Hg, which should be the remaining combined pressures of ammonia and nitrogen gas. Next, I multiplied each gas's mole fraction by 737.5 mm Hg to obtain their partial pressures. However, according to the ideal gas law, these partial pressures should add up to the total pressure in the container, but they don't.

Can someone please help me understand how to approach and complete this problem? Thank you so much!

 

[Borek]

Why do you assume water vapor is saturated?

 

[RTW69]

V=n_tot*R*T/P_tot

P_i=n_iR*T/V

Am I missing something?

 

[Borek]

V=n_tot*R*T/P_tot

P_i=n_iR*T/V

Am I missing something?

No need to calculate volume - molar fractions are enough.

 

[kilgorethecat]

I'm sorry, I didn't mean saturation pressure. I meant vapor pressure. So, should I recalculate mole fractions of the two ideal gases excluding water vapor and use that to calculate their partial pressures?

 

[Borek]

I'm sorry, I didn't mean saturation pressure.

I am afraid you meant it - you have read saturated vapor pressure from the tables and used it in calculations. This is equivalent to assume water vapor was saturated.

I meant vapor pressure. So, should I recalculate mole fractions of the two ideal gases excluding water vapor and use that to calculate their partial pressures?

Use all gases in exactly the same way - that is, ignore water vapor pressure tables. THEN, after you know partial pressure of water, check if it is saturated - IF it is, you have to check how much water would condense and become a liquid. If not, water vapor behaves exactly as every other gas.

 

[rude man]

Doesn't NH₃ dissolve rapidly in water to give NH₄OH? Can we really consider these components (phases?) as not reacting with each other?

 

[kilgorethecat]

I am afraid you meant it - you have read saturated vapor pressure from the tables and used it in calculations. This is equivalent to assume water vapor was saturated.

Crap, you're right! Sorry, my brain gets jumbled from the numbers flying around in my head all day with these engineering classes... and I didn't do so great in psychrometrics or thermodynamics. But I'm in biothermodynamics now, so I need to understand this or it's fail for me.

Use all gases in exactly the same way - that is, ignore water vapor pressure tables. THEN, after you know partial pressure of water, check if it is saturated - IF it is, you have to check how much water would condense and become a liquid. If not, water vapor behaves exactly as every other gas.

So, I calculated the partial pressure of water vapor from the original mole fraction, and I got 14.56 mm Hg. The saturation pressure of water at 50 C from the table is 92.5 mm Hg. Since the value I calculated is less than the value of saturation pressure at that temperature, this means that it is NOT saturated, correct? In that case, if it behaves like an ideal gas, why does almost every source I come across say that the pressure conditions must be below 75 mm Hg? This is the part that confused me in the first place, because I didn't think my given conditions allowed water vapor to behave as the other gases...

On a side note, this is a 300 level class in Biosystems Engineering and I didn't think they would give us that simple of a problem.

 

[Borek]

Doesn't NH₃ dissolve rapidly in water to give NH₄OH? Can we really consider these components (phases?) as not reacting with each other?

You don't have a liquid water here. No idea if they react in gaseous phase - but I strongly doubt.

 

[Borek]

So, I calculated the partial pressure of water vapor from the original mole fraction, and I got 14.56 mm Hg.

Looks OK to me.

The saturation pressure of water at 50 C from the table is 92.5 mm Hg. Since the value I calculated is less than the value of saturation pressure at that temperature, this means that it is NOT saturated, correct?

Yes.

In that case, if it behaves like an ideal gas, why does almost every source I come across say that the pressure conditions must be below 75 mm Hg?

Must be below 75 mmHg for what?

 

[kilgorethecat]

Must be below 75 mmHg for what?

From my original post:

Since water vapor only behaves like an ideal gas at pressures below 75 mm Hg, it cannot be treated as so in this problem.

In my last post when I said I didn't understand how I could treat water vapor as an ideal gas, I was referring to the total pressure (given) of 830 mm Hg. Perhaps to be more clear: I am confused as to whether or not this rule is referring to the TOTAL pressure of a gaseous mixture or the partial pressure of the water vapor itself.

I cannot seem to find a better explanation from a reputable source in my text nor on the web...Either we still don't understand water's role in gaseous mixtures or I am a moron.

Also, thank you SO MUCH for your help so far.

 

[Borek]

OK, I see what it is about - whether ideal gas approximation is correct, or not.

To be honest with you I don't know what are limitations. A lot depends on the required accuracy. However, I don't think we have much better approximation, especially for the random mixtures of gases. I seem to remember seeing elaborate tables for some gas mixtures - note that they contain experimental data, not calculated

You can try to ask about ideal gas approximation limitations in the Engineering forum - link to this thread so that people there can check what we have already discussed here.

 

[rude man]

you don't have a liquid water here. No idea if they react in gaseous phase - but i strongly doubt.

10-4.

 

[Borek]

10-4.

Makes 6, but I fail to see your point.

 

[rude man]

That's 'Highway Patrol' talk for "OK, understood". This miserable TV show was popular back around 50-60 yrs ago.

 

[Borek]

Ah, OK.

 

[kilgorethecat]

Dang, I was hoping to come to a solid conclusion on this one today since it's due, but you know, I do appreciate your help and will certainly repost for curiosity's sake. You'll hear from me again, thanks again!