# Thermal expansion of air inside a bottle -- some questions...

For thermal expansion in a bottle - is there a theoretical model on the pressure exerted on the inside of the bottle due to the air inside expanding when the bottle heats up.
The air will be expanding randomly in all directions so the pressure exerted at any point on the inside of the bottle walls will be the same everywhere else inside the bottle.

Yes, it is called the perfect gas law. https://en.wikipedia.org/wiki/Ideal_gas_law
Ok I do not really understand this law and how I will use it in a practical situation with air in a bottle where I will need the molar mass to calculate the pressure.

anorlunda
Staff Emeritus
All you need to know is the change Use PV=nRT

Let's say you know P, V and T initially Then nR/V=P/T.

Now increase the temperature from ##T_1## to ##T_2##. nR remains constant. The volume of the bottle remains constant. That means

##\frac{P_1}{T_1}=\frac{P_2}{T_2}##

(be careful T is in degrees Kelvin)

Does that help with your problem?

James124900
Yes. Thank you. But how will I know nR? and just to be sure I will calculate initial pressure with PV=nRT (or is it just atmospheric pressure?).

Edit: Did you mean to calculate nR with initial P, V and T?
(I do not have P - unless it is just atmospheric pressure. However I am not sure of this since the bottle will be cooled to a low temperature before being heated up).

anorlunda
Staff Emeritus
Did you mean to calculate nR with initial P, V and T?

Yes, that's what I meant. You can't solve the problem unless you know initial P and T.

Yes, that's what I meant. You can't solve the problem unless you know initial P and T.
Ok, is initial P atmospheric pressure, or would it be different since the bottle will have been cooled?

anorlunda
Staff Emeritus
Ok, is initial P atmospheric pressure, or would it be different since the bottle will have been cooled?

I can't answer that because I don't know the circumstances. It is a good guess that at some point in history it was at atmospheric presssure. If you know the temperature and volume at that time, then that is your initial point.

russ_watters
Mentor
Ok, is initial P atmospheric pressure, or would it be different since the bottle will have been cooled?
It's your scenario, so you are going to have to provide the details.

It's your scenario, so you are going to have to provide the details.
A bottle is cooled, I know the temperature and volume of the bottle. How can I calculate the pressure in the bottle to calculate nR?

anorlunda
Staff Emeritus
Before the bottle was sealed, you need the P and T. Possibly atmospheric pressure and room temperature. Then you have P1 and T1. You can then calculate a P2 for any other T2.

If you don't know anything about the conditions when the bottle was sealed, then you're stuck.

Before the bottle was sealed, you need the P and T. Possibly atmospheric pressure and room temperature. Then you have P1 and T1. You can then calculate a P2 for any other T2.

If you don't know anything about the conditions when the bottle was sealed, then you're stuck.
Ok thanks.

Hey I got another question related to the above concepts:
I have two closed bottles one has a smaller volume that the other. (initially atmospheric pressure in both and then closed with a cap)
If I heat both bottles up, which one will increase in pressure more the larger bottle or smaller one.

anorlunda
Staff Emeritus
Hey I got another question related to the above concepts:
I have two closed bottles one has a smaller volume that the other. (initially atmospheric pressure in both and then closed with a cap)
If I heat both bottles up, which one will increase in pressure more the larger bottle or smaller one.

How? because both bottles will have the same pressure to start with (it it not like the bottle decreases in volume thus when smaller will have a larger pressure)
so I am confused which one will increase in pressure more.

davenn
Gold Member
How? because both bottles will have the same pressure to start with (it it not like the bottle decreases in volume thus when smaller will have a larger pressure)
so I am confused which one will increase in pressure more.

you have the initial P, V and T for each bottle.
you know the volume for each bottle isn't going to change so use the formula you were shown to work out change in P for changes in T for each bottle

you have the initial P, V and T for each bottle.
you know the volume for each bottle isn't going to change so use the formula you were shown to work out change in P for changes in T for each bottle
I don't understand why the smaller bottles have a larger pressure than the smaller bottles result in a larger pressure.

davenn
Gold Member
I don't understand why the smaller bottles have a larger pressure than the smaller bottles result in a larger pressure.

that didn't make sense .... did you do the working out ?

show us your working so somebody can check it

no calculations. I just don't understand why a closed smaller bottle will increase in temperature more than a larger bottle when both are heated by the same amount.

jbriggs444
Homework Helper
no calculations. I just don't understand why a closed smaller bottle will increase in temperature more than a larger bottle when both are heated by the same amount.
Do you understand why heating a swimming pool for one hour under a single heat lamp will result in less temperature rise than doing the same with your kitchen sink?

It appears that the intended query had to do with the increase in pressure when a smaller bottle has the same increase in temperature as a larger bottle. But I see no foundation for such a question. Who is trying to claim that such a difference in pressure increase will be seen?

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Do you understand why heating a swimming pool for one hour under a single heat lamp will result in less temperature rise than doing the same with your kitchen sink?
I meant both bottle's will be heated up to the same temperature.

Edit: Sorry for my other post I meant increase in pressure not temperature

jbriggs444
Homework Helper
I meant both bottle's will be heated up to the same temperature.

Edit: Sorry for my other post I meant increase in pressure not temperature
I meant both bottle's will be heated up to the same temperature.

Edit: Sorry for my other post I meant increase in pressure not temperature
Now you have me confused. What are you trying to ask and why?