Constant pressure in a heated syringe

[sgstudent]

Homework Statement

When a syringe with its end sealed is heated, the piston is pushed back such that the gas pressure in the syringe remains the same. Since the force exerted is dependent on 1) frequency of collision with the walls and 2) the force of the collisions. So comparing before and after heating, is the frequency higher and is the force hitting higher?

also, is it possible to ask this question: more air is pumped into a filled incompressible tyre and it is put in a cooler. How has the pressure change?
Or this: Air is drawn out of a filled incompressible tyre and it is also heated up. How has the pressure changed?

Homework Equations

PV=PV but T is not constant here so I don't think this is applicable?

The Attempt at a Solution

The answer key gave the frequency to be the same as before but doesn't the average kinetic energy of the air particles increase? So won't the frequency be subsequently higher as well. Also, if the frequency remains constant doesn't it also mean that the force of collision is the same as well? So going with the same explanation why would the air hit with less force?

For the other 2 questions I posed, I don't think it is possible to answer them directly because they are contrasting (add air increase P cool reduce P, draw out air reduce P heat increase P). So just this empirical data will not tell us which phenomenon has a greater magnitude which doesn't allow me to say what will happen? But if eg a question says: a water bottle is filled partially with hot water and it expands slightly. Explain in terms of the kinetic model of matter this phenomenon. In this case, air is drawn out as water displaces some of it so pressure decreases, but since they already told us that the bottle expands, that means the pressure must have increased. Hence, the increase in temperature leads to the increase in pressure and this increase is more than the decrease in pressure by the removal of air?

However, this example is also weird be because water pressure should be more than air pressure? I'm quite confused with this. Thanks for the help!

 

[Infinitum]

Hi!

ut doesn't the average kinetic energy of the air particles increase? So won't the frequency be subsequently higher as well. Also, if the frequency remains constant doesn't it also mean that the force of collision is the same as well?

Remember, the pressure is constant. The speed of the molecules increases as the temperature increases. Since the pressure remains constant, that means the number of collisions with the container walls doesn’t change. If the number of collisions must stay the same, then the molecules have to travel further before they collide with the container walls. The only way this can happen is if the container volume increases, which is the experiment in question.

 

[sgstudent]

Hi!



Remember, the pressure is constant. The speed of the molecules increases as the temperature increases. Since the pressure remains constant, that means the number of collisions with the container walls doesn’t change. If the number of collisions must stay the same, then the molecules have to travel further before they collide with the container walls. The only way this can happen is if the container volume increases, which is the experiment in question.

Hi! But what about the force inflicted? Will it also remain the same? Thanks

 

[sgstudent]

Hi, so when an object is heated with a constant pressure, will the frequency of collision and force of collision the same? Thamks

 

[Infinitum]

Hi, so when an object is heated with a constant pressure, will the frequency of collision and force of collision the same? Thamks

The frequency is explained in the above post. The force of collision will increase. Can you see how? Think about the force obtained from momentum change per unit time..

 

[sgstudent]

The frequency is explained in the above post. The force of collision will increase. Can you see how? Think about the force obtained from momentum change per unit time..

But if force increases won't the pressure in the syringe be greater than before? Which is not true since the pressure remains constant in the syringe? Before and after frequency same, but before and after th force is greater so I don't quite understand this part.

Thanks infinitum!

 

[Infinitum]

Assuming the kinetic theory of gases, you have the force exerted by each collision on a cubic container wall as,

$$F = \frac{2mv}{t}$$

Where t is the time taken to transverse the length of the cube twice. Now, since frequency does not change, the time remains constant. But due to temperature rise, velocity increases. And so...

 

[sgstudent]

I know force increases, but before heated eg frequency is 1 point and force is 1 point. When heated frequency is the same so 1 point and but since force increases so it's 2 point. So now pressure isn't a constant anymore? It doesn't seem correct to me...

 

[Infinitum]

I just realized I interpreted the question wrongly(in the opposite way, actually :tongue:). The force of collision has to stay the same, because pressure is the same. But frequency of collision will not be the same, and will instead decrease, because as molecule velocity increases, collisions become lesser to maintain the same pressure.

Hope this clears it up

 

[sgstudent]

Wait, the frequency decreases? But in my ten year series the frequency remains the same..

 

[Infinitum]

Wait, the frequency decreases? But in my ten year series the frequency remains the same..

Quite sure it decreases. Say we approach this with equations,

Force on the wall = frequency * impulse of collision

Now, the impulse of collision undoubtedly increases as the temperature rises. And force also remains constant as the pressure is same( F=P/A, and nothing changed the area). So the frequency has to decrease to keep that equation true.

 

[nasu]

Wait, the frequency decreases? But in my ten year series the frequency remains the same..

Don't know what is ten year series.
Nevertheless, the frequency (f) of the collisions with the wall is proportional to the density of the gas and the average normal component of the velocity.
$$f \propto nv_{perp}$$
n is the number density of molecules and v_perp is the average velocity.
We have that
$$n \propto 1/V$$
and
$$v \propto \sqrt{T}$$
In general,
$$f \propto \sqrt{T}/V$$
For expansion at constant pressure,
$$V \propto T$$
and
$$f \propto \sqrt{T}/T=1/ \sqrt{T}$$
So as the temperature increases, the frequency of collisions decerases.
The effect of decreased density is stronger than that of increased molecular speed.

 

[sgstudent]

Oh but I don't understand why would the force remain the same. If the pressure is equal to force of collision and frequency and pressure is constant, since frequency decreases, won't force have to increase? Thanks for the help!

 

[Infinitum]

Oh but I don't understand why would the force remain the same. If the pressure is equal to force of collision and frequency

That's not correct. Collision force on the wall is equal to frequency times the impulse of collision. Pressure is the force per unit area...

and pressure is constant, since frequency decreases, won't force have to increase? Thanks for the help!

As said before, the pressure remains the same!

If the area is the same, and so is the pressure, what can you say about force, given as,

$$F = PA$$

 

[sgstudent]

The force will increase? Since area increases and and pressure is the same?

 

[Infinitum]

The force will increase? Since area increases and and pressure is the same?

Uh why will the area increase? The piston wall area remains the same!

 

[sgstudent]

The force will have to be the same? So force and pressure is constant while the impulse increases and frequency decreases?

 

[Infinitum]

The force will have to be the same? So force and pressure is constant while the impulse increases and frequency decreases?

Yes, indeed!

 

[sgstudent]

Oh, but isn't impulse similar to force?

 

[Infinitum]

Oh, but isn't impulse similar to force?

How do you define impulse?

 

[nasu]

Oh but I don't understand why would the force remain the same. If the pressure is equal to force of collision and frequency and pressure is constant, since frequency decreases, won't force have to increase? Thanks for the help!

There is some confusion between the total force on the wall and the force transferred to the wall by one individual collision.
At constant pressure, the total force is the same (F=P*A). However the average force transferred during one collision process increases due to the higher average speed of the molecules.
So there are less collisions per second and each one "applies" more force so the total force stays the same.