TPR GC Q: Momentum

[Czarcasm]

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Q: If ten grams of each of the gases He and Ne were heated to 1000 C, which one would provide more rocket lift?

A. He
B. Ne
C. They would provide the same lift.
D. Unable to predict.

I realize that both gases have the same average kinetic energy at a given temperature and the ratio of their velocities are different (related by the square root of their masses). I also realize that both would have different momentum due to the ratio of their individual masses and velocities.

Helium has a molecular weight of 4 g/mol. Neon a molecular weight of 20 g/mol.
Helium travels ~2 times faster than Neon, but weighs 1/5 times less.

Therefore, a Helium atom has a momenta 2/5 that of a Neon atom (and Neon 5/2 that of Helium). But then because we're comparing unequal amounts of molecules for both gases, the total momenta exerted due to the individual gas molecules will be different. There are 2.5 moles of Helium and 0.5 moles Neon (in 10g) and the ratio of the two is: 5:1 (Helium:Neon).

So in this scenario, you have two variables that influence the answer to this question. We have greater moles of Helium, but greater individual momenta for Neon. So what I did to compare these variables was this:

(ratio momenta x ratio moles) = ratio total momentum

Helium: 2/5 x 5 = 2
Neon: 5/2 x 1= 2.5

With this logic, Neon would be the correct answer.

Here was their explanation though and I'm not sure I really understand it:

"We are asked to choose whether ten grams of helium or ten grams of neon will provide more rocket lift (momentum). We use the definition: momentum = mv. The total masses of both gases are the same (10g), so all we have to do is choose the gas with the greater velocity. Lighter gases at the same temperature have greater velocities, so the helium would provide more lift."

The said the correct answer was A. If anyone could provide some insight into helping me understand this, it'd be a big help. I'm not sure why they used 10g of mass for both (in p=mv) as if it was 1 giant particle, when in actuality, there's a totally different number of particles within the same amount of mass, each with their own individual velocities. If that's the case, wouldn't it be more logical to use the total velocity due to all the atoms as opposed to the ratio of their individual velocities. It just seems like an unfair comparison.

I know I'm probably losing my mind and over analyzing this question. I usually don't spend this much time on one question, but it's a fundamental concept and I've always struggled with this topic.

 

[DrknoSDN]

"We are asked to choose whether ten grams of helium or ten grams of neon will provide more rocket lift (momentum). We use the definition: momentum = mv. The total masses of both gases are the same (10g), so all we have to do is choose the gas with the greater velocity. Lighter gases at the same temperature have greater velocities, so the helium would provide more lift."

When reading the question I just felt like it would be Helium without equations, but their explanation was not the greatest.
They are basically ignoring, Flow, Volume, Time, Pressure, and anything related to Bernoulli.

Counter-Example:
If you had 10g of He and 10g of Ne at room temperature, and the volume of the container was a size so that Neon was at atmospheric pressure.
Neon would have zero driving force out of the container while Helium would. Momentum is a factor, but saying "All we have to do is choose the gas with greater velocity" is an insult to rocket engineers.

Pressure is a huge factor in rocket design, the pressure gradient cannot be ignored. Helium would still be the answer but for other factors in addition to momentum.

 

[DrknoSDN]

Ignoring pressure, Effusion and Diffusion are also factors that could be considered.

Because the answer was saying that velocity is the only significant factor because the internal energy of the gases was the same, you could relate this to an NH3 + HCl gas diffusion experiment.
The less massive gas diffuses down the tube faster in a ratio equal to the square root of the inverse of their masses at all temperatures.

So Rate Helium = Root (20/4) = 2.24 times faster than Neon.
Considering "rocket lift" must counter gravity each second, the longer it takes for the gas to exit the rocket the harder it is for the gas to overpower gravity.
The answer is still Helium but I couldn't get over how many variables the answer was ignoring. 😕

 

[getfat]

drKnoSDN gives a pretty in depth description but the best rationale i can give you is:

You are looking at the passage as a physics passage and not a General chemistry passage

If mass of Helium (10kg) =mass of Neon (10 kg),
The question doesn't suggest anything about the pressure so you have to assume mole fraction helium = mole fraction Neon.
So if Lower MW = Higher # of molecules
All gases want the as much space as possible in gaseous substance, if there is more molecules taking up space there will be a higher electrostatic force
So more # molecules (Helium) will exert more force than less # (Neon),

 

[Chrisz]

drKnoSDN gives a pretty in depth description but the best rationale i can give you is:

You are looking at the passage as a physics passage and not a General chemistry passage

If mass of Helium (10kg) =mass of Neon (10 kg),
The question doesn't suggest anything about the pressure so you have to assume mole fraction helium = mole fraction Neon.
So if Lower MW = Higher # of molecules
All gases want the as much space as possible in gaseous substance, if there is more molecules taking up space there will be a higher electrostatic force
So more # molecules (Helium) will exert more force than less # (Neon),

I agree with drKnoSDN. I am really confused where you get electrostatic force from. They both are inert gases, their orbitals are all filled already. I deeply doubt there will be any London dispersion, or electrostatic force caused by instantaneous partial charges. That is why inert gases are treated very close to ideal gases where the there is no intermolecular interactions. It is true that the propelling of a rocket is achieved due to conservation of momentum. However, the answer given by the book author is too simplified. A more complete version of conservation of momentum applied to rocket has to take into account instantaneous change in mass, instantaneous velocity of gas expelled. These all deal with calculus. However, drKnoSDN gives a legitimate approximation through the effusion rate reasoning.

 

[Czarcasm]

When reading the question I just felt like it would be Helium without equations, but their explanation was not the greatest.
They are basically ignoring, Flow, Volume, Time, Pressure, and anything related to Bernoulli.

Counter-Example:
If you had 10g of He and 10g of Ne at room temperature, and the volume of the container was a size so that Neon was at atmospheric pressure.
Neon would have zero driving force out of the container while Helium would. Momentum is a factor, but saying "All we have to do is choose the gas with greater velocity" is an insult to rocket engineers.

Pressure is a huge factor in rocket design, the pressure gradient cannot be ignored. Helium would still be the answer but for other factors in addition to momentum.

I'm sorry guys, I should have provided the relevant passage information about the rocket. (It's not in a container, instead the gas is being projected via an open compartment into the surrounding medium):

"In a chemical rocket engine, a mixture of gases are allowed to explosively react in a compartment with one open end. The hot gaseous products escape out the open end with great velocity (because they're hot) and propel the rocket in the opposite direction to conserve momentum. Hot gases escapes with momentum p, so the rocket is pushed with momentum -p."

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Intuitively, I get what their saying. For simplicity, just consider the ratio of their velocities - and the faster velocity would likely impart a greater momentum. But actually calculating everything out, you get a different answer than you would expect. Problem is, I'm not sure if my calculation is entirely correct.

 

[Czarcasm]

When reading the question I just felt like it would be Helium without equations, but their explanation was not the greatest.
They are basically ignoring, Flow, Volume, Time, Pressure, and anything related to Bernoulli.

Counter-Example:
If you had 10g of He and 10g of Ne at room temperature, and the volume of the container was a size so that Neon was at atmospheric pressure.
Neon would have zero driving force out of the container while Helium would. Momentum is a factor, but saying "All we have to do is choose the gas with greater velocity" is an insult to rocket engineers.

Pressure is a huge factor in rocket design, the pressure gradient cannot be ignored. Helium would still be the answer but for other factors in addition to momentum.

I admit I'm a bit confused why pressure is relevant here. I think flow rates, volume, pressure and anything related to Bernoulli specifically applies to fluids flowing through some fixed container and how their relative positions effect the velocity at which the fluid flows. In this instance though, we're specifically considering two gases, exiting an open container, imparting a momentum (vector), which counters that of the rocket (equal momentum in opposite direction), such that the total momentum is conserved.

 

[Chrisz]

I'm sorry guys, I should have provided the relevant passage information about the rocket. (It's not in a container, instead the gas is being projected via an open compartment into the surrounding medium):

"In a chemical rocket engine, a mixture of gases are allowed to explosively react in a compartment with one open end. The hot gaseous products escape out the open end with great velocity (because they're hot) and propel the rocket in the opposite direction to conserve momentum. Hot gases escapes with momentum p, so the rocket is pushed with momentum -p."

---

Intuitively, I get what their saying. For simplicity, just consider the ratio of their velocities - and the faster velocity would likely impart a greater momentum. But actually calculating everything out, you get a different answer than you would expect. Problem is, I'm not sure if my calculation is entirely correct.

Based on the passage information, it is very clear that the explosion takes place "instantaneously" and "completely"(suddenly, the 10 grams are completely exhausted). That means everything is exhausted. What does that mean to you? (Mrocket+Mgas)Vi=(Mrocket)(Vfrocket)+(Mgas)(Vfgas). sine 10 grams are very small compared to Mrocket, it is ignorable.
(Mrocket)Vi-(Mgas)(Vfgas)=(Mrocket)(Vfrocket)
As you can see, both gases are in the same 10 grams, the only determinant that affects the final velocity of the rocket is Vf of gas. so what is Vfgas.
Vfgas=Vi-sqroot(3RT/Molar Mass).
Why we subtract sqroot(3RT/Molar Mass) from Vi, it is because we need to know the actual velocity of gas while the -sqroot(3RT/Molar Mass) is the relative velocity of gas compared to the initial velocity of the rocket. However, Vi is the same in both cases, so the final determinant of rocket velocity still depends on the terms sqroot(3RT/Molar Mass). The bigger this term, the big the final velocity of the rocket

Edit,
I should have not taken out the term Mgas in the original derivation, it would be much easier for you to see the actual change of the momentum in rocket is actually sqroot(3RT/Molar Mass). But for calulational purpose, they provide almost the same result

 

[DrknoSDN]

I admit I'm a bit confused why pressure is relevant here. I think flow rates, volume, pressure and anything related to Bernoulli specifically applies to fluids flowing through some fixed container and how their relative positions effect the velocity at which the fluid flows. In this instance though, we're specifically considering two gases, exiting an open container, imparting a momentum (vector), which counters that of the rocket (equal momentum in opposite direction), such that the total momentum is conserved.

Conservation of momentum does apply but if you ignore everything else, it would need to be in a zero g environment in a vacuum.

For a rocket launch from earth you have to consider that "lift" must counter something like gravity, and that is going to be dependent on how much acceleration the gas expansion/effusion can produce.
Yes total momentum might be higher for one gas or another, but launch time is a far more significant factor.

Finally ran across a site had more detailed explanation of why pressure was important. Pictures below, courtesy NASA.
If Helium has more moles then at 1000 deg then the pressure difference will be higher giving more acceleration against the weight of the rocket, and more lift.

What would happen if you had 10g of Helium and 10g of Neon, and the pressure inside the open container was LOWER than atmospheric pressure? How would that produce any "lift" at all? Pressure is extremely relevant and should not be dumbed down to conservation of momentum here. Answer is still Helium. 🙂
I only make the argument for pressure because conservation of momentum won't always get you the right answer.

Pressure is in every single equation.
https://www.grc.nasa.gov/WWW/K-12/rocket/rktslaunch.html
http://www.grc.nasa.gov/WWW/k-12/airplane/rockth.html
http://www.grc.nasa.gov/WWW/k-12/airplane/mflchk.html

 

[Chrisz]

Conservation of momentum does apply but if you ignore everything else, it would need to be in a zero g environment in a vacuum.

For a rocket launch from earth you have to consider that "lift" must counter something like gravity, and that is going to be dependent on how much acceleration the gas expansion/effusion can produce.
Yes total momentum might be higher for one gas or another, but launch time is a far more significant factor.

Finally ran across a site had more detailed explanation of why pressure was important. Pictures below, courtesy NASA.
If Helium has more moles then at 1000 deg then the pressure difference will be higher giving more acceleration against the weight of the rocket, and more lift.

What would happen if you had 10g of Helium and 10g of Neon, and the pressure inside the open container was LOWER than atmospheric pressure? How would that produce any "lift" at all? Pressure is extremely relevant and should not be dumbed down to conservation of momentum here. Answer is still Helium. 🙂
I only make the argument for pressure because conservation of momentum won't always get you the right answer.

Pressure is in every single equation.
https://www.grc.nasa.gov/WWW/K-12/rocket/rktslaunch.html
http://www.grc.nasa.gov/WWW/k-12/airplane/rockth.html
http://www.grc.nasa.gov/WWW/k-12/airplane/mflchk.html

I totally agree with you that gravity and drag force definitely has to be taken into account when you build a real rocket. These are all good sources. No offense with my following reasoning. For this question we definitely can use conservation of momentum to get the correct answer with the assumption that gravity and drag forces are not present. This assumption is legitimate because no matter which gas is used, the mass of rocket and masses of two gasses are the same. Most importantly, the gravitational force and drag force are always present, whether helium or neon is used. The question is asking which gas would provide better thrust with an equivalent amount of mass. So, when a rocket is out in the space, where all the external forces are not present, conservation of momentum applies. If helium could provide higher thrust in the space, it definitely can provide a better thrust than neon near the surface of earth.

 

[Czarcasm]

Conservation of momentum does apply but if you ignore everything else, it would need to be in a zero g environment in a vacuum.

For a rocket launch from earth you have to consider that "lift" must counter something like gravity, and that is going to be dependent on how much acceleration the gas expansion/effusion can produce.
Yes total momentum might be higher for one gas or another, but launch time is a far more significant factor.

Finally ran across a site had more detailed explanation of why pressure was important. Pictures below, courtesy NASA.
If Helium has more moles then at 1000 deg then the pressure difference will be higher giving more acceleration against the weight of the rocket, and more lift.

What would happen if you had 10g of Helium and 10g of Neon, and the pressure inside the open container was LOWER than atmospheric pressure? How would that produce any "lift" at all? Pressure is extremely relevant and should not be dumbed down to conservation of momentum here. Answer is still Helium. 🙂
I only make the argument for pressure because conservation of momentum won't always get you the right answer.

Pressure is in every single equation.
https://www.grc.nasa.gov/WWW/K-12/rocket/rktslaunch.html
http://www.grc.nasa.gov/WWW/k-12/airplane/rockth.html
http://www.grc.nasa.gov/WWW/k-12/airplane/mflchk.html

Thank you for taking the time to write that. It's cleared a lot of things up! 🙂

 

[Czarcasm]

Thanks guys, I appreciate all your input.