What is the Ideal Gas Force on a Container at Temperature T?

In summary: I tried to explain both possibilities in my summary, but it is ultimately up to the person solving the problem to interpret it and choose the appropriate equation.
  • #1
Raghav Gupta
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Homework Statement


In an ideal gas at temperature T, the average force that a molecule applies on the walls of a closed container depends on T as Tq. A good estimate for q is:
A. 2
B. 1
C. 1/2
D. 1/4

Homework Equations


PV= nRT
I think, Volume of container and moles are constant.
So P ∝ T

The Attempt at a Solution


Answer should be B. 1, as force = pressure* area.
Here all would be ideal case.
But it is C. 1/2. Why?
 
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  • #2
Raghav Gupta said:
Answer should be B. 1, as force = pressure/ area.
First, that should be force = pressure × area. Second, the pressure depends not only on the force of each collision of a molecule with a wall, but also on the frequency of those collisions. Both change with temperature. You need another equation to find the force of each molecular collision.
 
  • #3
Okay edited that force= pressure * area. That was a typo.
I don't know that second equation, can you give a hint?
 
  • #4
Newton's second law.
 
  • #5
F ∝ d(mv)/dt
Now mass and velocity remains constant in ideal gas, so zero force?
 
  • #6
Raghav Gupta said:
velocity remains constant in ideal gas
?
 
  • #7
Okay on colliding it changes direction.
momentum change = 2mv ?
 
  • #8
Raghav Gupta said:
Okay on colliding it changes direction.
momentum change = 2mv ?
Yes, but only if v is the component of the velocity perpendicular to the wall.

Now, what is the relation between v and T?
 
  • #9
## v ∝ \sqrt{T} ##
Now f ∝ dv/dt
So, f ∝ √T
Got it q= 1/2.
 
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  • #10
DrClaude said:
First, that should be force = pressure × area. Second, the pressure depends not only on the force of each collision of a molecule with a wall, but also on the frequency of those collisions. Both change with temperature. You need another equation to find the force of each molecular collision.
But still pressure changes linearly with temperature,
According to ideal gas law, as p is proportional to T as V and moles are constant.
If you are saying it depends on frequency, where is the frequency part in other equation?
 
  • #11
Raghav Gupta said:
But still pressure changes linearly with temperature,
According to ideal gas law, as p is proportional to T as V and moles are constant.
If you are saying it depends on frequency, where is the frequency part in other equation?
Lets consider what happens in 1D, in a direction perpendicular to the wall. The average pressure is
$$
\langle P \rangle = \frac{\langle F \rangle}{A} = - \frac{m \langle \frac{ \Delta v }{\Delta t}\rangle}{A}
$$
When calculating the pressure, you are using the average force, which includes an average over time: you need to consider the average force per collision times number of collisions per unit time. When ##T## increases, two things change in that equation: the average force per collision increases and the average rate of collision increases. Overall, you get ##P \propto T##, but the problem only asks for the force per collision, which is ##\propto T^{1/2}##.
 
  • #12
I am not getting it correctly.
Here force ∝ pressure , as area of wall is constant.
So how if we are getting overall P∝ T, then force is not proportional to T ?
Does that word average makes a difference?
 
  • #13
Raghav Gupta said:
So how if we are getting overall P∝ T, then force is not proportional to T ?
The average total force applied to the wall is proportional to T, not the average force per collision.
 
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  • #14
Okay, got it. Thanks.
 
  • #15
@DrClaude you must see this. I think you are incorrect.

 
  • #16
Raghav Gupta said:
@DrClaude you must see this. I think you are incorrect.


But the video is considering the "time-averaged" force on the walls (over many collisions). Note that the instructor in the video, around 1:12 or so, begins calculating the time it takes to go from once side of the wall to the other, and relates that to molecule's (average) velocity. Essentially what he's doing there bringing the frequency of collisions into the overall picture.

If the problem in this thread is asking for the time-averaged force on the wall of the container, time-averaged over many collisions (however many collisions that happen within a unit amount of time [for a fixed volume, fixed number of molecules, etc.]), then yes, what's said in the video applies to this thread.

On the other hand if the problem statement is asking for the average force for per individual collision, then that's a whole different story and the video doesn't fully apply to this thread.

[Edit: In my opinion, the way the problem statement is worded it's a little ambiguous.]

[Another edit: I should point out that if the answer C. is correct, the question really is asking about the force per collision (not time-averaged over all collisions within a unit period of time), and DrClaude's interpretation is correct. The problem statement could have been worded more clearly to avoid ambiguities, whatever the case.]
 
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  • #17
Thanks collinsmark. Yeah, I think the problem statement is ambiguous.
 

Related to What is the Ideal Gas Force on a Container at Temperature T?

1. What is the ideal gas law and how does it relate to the force on a container?

The ideal gas law is a fundamental equation in thermodynamics that describes the relationship between the pressure, volume, temperature and number of moles of an ideal gas. It states that the product of pressure and volume is directly proportional to the product of the number of moles and the temperature. This law can be used to calculate the force exerted by an ideal gas on the walls of a container.

2. How is the force on a container affected by changes in temperature and volume?

According to the ideal gas law, as the temperature of an ideal gas increases, its volume also increases proportionally. This results in an increase in the force exerted on the container walls. Similarly, as the volume of the gas decreases, the force on the container also decreases. This relationship is known as Boyle's law.

3. Does the type of gas affect the force exerted on a container?

No, the ideal gas law applies to all ideal gases regardless of their composition. This means that the force exerted on a container by an ideal gas is not affected by the type of gas present.

4. Can the ideal gas law be used to calculate the exact force on a container?

No, the ideal gas law is an approximation and does not account for factors such as intermolecular forces and the volume occupied by the gas molecules. It can provide a rough estimate of the force on a container, but for more precise calculations, other equations and models may be necessary.

5. Are there any limitations to the ideal gas law in determining the force on a container?

Yes, the ideal gas law only applies to ideal gases, which do not exist in real life. Real gases deviate from ideal behavior at high pressures and low temperatures. In these cases, more complex equations, such as the van der Waals equation, must be used to accurately calculate the force on a container.

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