When Can the Combined Homework Statement Be Applied for Pressure Calculation?

[sgstudent]

Homework Statement

A 10cm^3 bottle was filled with 10cm3 of air at atm. Later, 50cm3 more of air at atm was pumped in. What is the new pressure taking atm to be 10^5 Pa and temperature is constant?
Their working was P1V1=P2V2 so 10^5 Pa .60cm^3 =P2. 10cm^3 and they solved P2 from there. But from this video, http://www.youtube.com/watch?feature=player_detailpage&v=GwoX_BemwHs#t=243s They said we cannot just apply that formula since n changes?

Homework Equations

P1V1/T1=P2V2/T2

The Attempt at a Solution

I think their working such that they already compile the gases together such that n is a constant as they added the initial 10cm^3 to the extra 50cm^3 when the gas was sort of in the atmosphere then when all of that was condensed into 10cm^3 the number of moles of gas remains constant so the equation is still correct. However, since there is 10cm^3 of atm air initially inside, is it safe to simply add them up as they are at atmospheric pressure so it's like assuming that the 60cm^3 of air is outside the bottle at first then later the entire bunch is condensed together to 10cm^3 later?

 

[voko]

You are quite correct. Both 10 and 50 cm^3 are at the atm pressure, so it is unimportant whether they are inside or outside, and whether they are separated.

 

[sgstudent]

You are quite correct. Both 10 and 50 cm^3 are at the atm pressure, so it is unimportant whether they are inside or outside, and whether they are separated.

oh but if the question was tweaked such that the pressure inside the bottle was not atm but instead at 115 Pa. So will the new working be 10cm3 .115 Pa +50cm3 . 105 Pa= 10cm3 . new pressure? Hence the new pressure will be 661051 Pa?

Thanks for the help voko

 

[voko]

If the pressures are different, then you need to deduce the quantity of matter in both addends, and then find out what pressure the sum will have within the given volume.

 

[BruceW]

yeah, I agree with voko. As it happens, sgstudent's answer is right. But I suspect he doesn't fully appreciate why. hint: write out the 3 ideal gas equations (one for each of the two initial gasses, and one for the final gas). What conservation law can you use to relate these equations?

 

[sgstudent]

yeah, I agree with voko. As it happens, sgstudent's answer is right. But I suspect he doesn't fully appreciate why. hint: write out the 3 ideal gas equations (one for each of the two initial gasses, and one for the final gas). What conservation law can you use to relate these equations?

Actually how can i do this P1V1+P2V2=P3V3? Shouldn't i only be able to compare P1V1=P2V2?

 

[voko]

pV/T = ?

What is on the right hand side?

 

[sgstudent]

It is nR? But if won't n change because of the different volumes of air in different pressures?

 

[voko]

If you have an amount of gas n1, and an amount of gas n2, either one at whatever pressure, volume and temperature, when you mix them together, you get total amount n = n1 + n2; obviously, n1R + n2R = nR, so p1V1/T1 + p2V2/T2 = pV/T.

 

[sgstudent]

If you have an amount of gas n1, and an amount of gas n2, either one at whatever pressure, volume and temperature, when you mix them together, you get total amount n = n1 + n2; obviously, n1R + n2R = nR, so p1V1/T1 + p2V2/T2 = pV/T.

Oh yes that makes sense, so it's something like total inital PV/T=final PV/T?

Also, what does it mean by at high pressures, gases have a smaller volume? I thought that's only true if i decrease the volume such that the pressure is very high? if i heat it up, the pressure can be high but the volume is still the same.

 

[voko]

Oh yes that makes sense, so it's something like total inital PV/T=final PV/T?

Correct.

Also, what does it mean by at high pressures, gases have a smaller volume? I thought that's only true if i decrease the volume such that the pressure is very high? if i heat it up, the pressure can be high but the volume is still the same.

Gases do not "have" any volume. They are always constrained to a volume.

 

[lurflurf]

The formula (easily derived from the ideal gas law) for the pressure when we combine (all at constant temperature) into a volume V two amounts of gas at given volumes and pressures

P=\frac{P_1 V_1+P_2 V_2}{V}

a given problem can be solved different (equivalent ways depending how
we picture it. For example we could compress the contents of the bottle into part of the bottle, and then compress more gas into the rest of the bottle; or we could compress the combined contents of the bottle and additional gas into the bottle all at once. It is easy to picture with a piston and cylinder connected to the bottle. It does not matter if some thin membrane keeps the contents apart, they still have the same pressure. Otherwise imagine cramming a balloon into the bottle.

 

[sgstudent]

The formula (easily derived from the ideal gas law) for the pressure when we combine (all at constant temperature) into a volume V two amounts of gas at given volumes and pressures

P=\frac{P_1 V_1+P_2 V_2}{V}

a given problem can be solved different (equivalent ways depending how
we picture it. For example we could compress the contents of the bottle into part of the bottle, and then compress more gas into the rest of the bottle; or we could compress the combined contents of the bottle and additional gas into the bottle all at once. It is easy to picture with a piston and cylinder connected to the bottle. It does not matter if some thin membrane keeps the contents apart, they still have the same pressure. Otherwise imagine cramming a balloon into the bottle.

oh so by saying "at high pressures" they mean to say when the air is compressed a lot?

 

[BruceW]

Also, what does it mean by at high pressures, gases have a smaller volume?

Well, they just mean that if you consider a fixed number of moles of gas, then if you also keep the temperature constant, then if you increase the pressure, what must happen to its volume? (I reckon you know the answer already, but maybe they just threw you off, because it is a quite vague statement).

Edit: or maybe you are unhappy with how the change in the pressure can 'cause' a change in the volume. Well in this case, pressure and volume are inversely related, so you can interpret either "volume change causing pressure change" or "pressure change causing volume change". There is no distinction between the two statements in this case.

 

[sgstudent]

Well, they just mean that if you consider a fixed number of moles of gas, then if you also keep the temperature constant, then if you increase the pressure, what must happen to its volume? (I reckon you know the answer already, but maybe they just threw you off, because it is a quite vague statement).

I was learning about that from YouTube and they just showed a slide which stated "the ideal gas laws doesn't apply at high pressures" so I think I got confused there :) thanks for clearing this up

 

[BruceW]

yeah, when the pressure is great enough, the ideal gas approximation will break down, so the relation between thermodynamic quantities will get more complicated. When they said "at high pressures, gases have a smaller volume", this statement does still agree with the ideal gas approximation.