Pressure and Volume Relationship in Closed Bottles

In summary, both bottles will reach the same temperature, but the pressure of the smaller bottle will increase more.
  • #1
James124900
22
0

Homework Statement


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.

Homework Equations


PV=nRT ?

The Attempt at a Solution


For this I just want to understand which one will increase in pressure more theoretically, no calculations necessary.
 
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  • #2
James124900 said:

Homework Statement


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.

Homework Equations


PV=nRT ?

The Attempt at a Solution


For this I just want to understand which one will increase in pressure more theoretically, no calculations necessary.
So what is your attempt? Your relevant equation looks promising.
 
  • #3
gneill said:
So what is your attempt? Your relevant equation looks promising.
I just don't understand which bottle (smaller or larger volume) will have a larger increase in pressure when heated. No calculations involved.
both bottles will be of same (atmospheric pressure) to start with but as heated which one increase in pressure more (end up with a larger pressure).
 
  • #4
I think you need to figure out how to determine n(no. Of moles) in your equation... Since you have a sense of all the other variables...
You need to do an attempt and show us, because this question requires one unless you have great memory
 
Last edited:
  • #5
James124900 said:
No calculations involved.
No calculations (numbers), but you do need to employ the ideal gas law at least symbolically.
 
  • #6
I don't really need to use the ideal gas law as I have two different volumes, not one volume decreasing.
Why does the smaller bottle result in a larger pressure when heated?
 
  • #7
James124900 said:

Homework Statement


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.

Homework Equations


PV=nRT ?

The Attempt at a Solution


For this I just want to understand which one will increase in pressure more theoretically, no calculations necessary.

T is average KE of the particles. A given amount Q of heat, will increase the total KE of both systems in the same amount, but given that one bottle is smaller than the other (and so it has fewer gas particles than the other, cause at the beginning both systems were at the same P and T), the average KE of the particles of the smaller bottle will be higher at the end (than the average KE of the particles of the larger bottle), so the final T of the smaller bottle will be higher than the final T of the larger bottle.

So...what happens then to the pressure at the end? (you know that volumen will not change, neither "n" (amount of matter), nor the constant "R").
 
  • #8
The answer is there is no answer unless the question is better defined. What do you mean by "heat both bottles up"? In the absence of any other indicator, I would naturally take this to mean "heat them both up to the same temperature". In which case it is easy to see from the gas equation (at constant n and V, P is proportional to T) that they will have the same final pressure.

mattt is correct that if you give each bottle the same amount of heat Q, the smaller bottle will reach a higher temperature, but I submit that this is not the most natural way to understand the phrase "heat both bottles up".
 
  • #9
mjc123 said:
The answer is there is no answer unless the question is better defined. What do you mean by "heat both bottles up"? In the absence of any other indicator, I would naturally take this to mean "heat them both up to the same temperature". In which case it is easy to see from the gas equation (at constant n and V, P is proportional to T) that they will have the same final pressure.

mattt is correct that if you give each bottle the same amount of heat Q, the smaller bottle will reach a higher temperature, but I submit that this is not the most natural way to understand the phrase "heat both bottles up".
I thought the pressure of the smaller bottle will increase more, how can temperature increase in the smaller bottle?
(also yes, each bottle is heated to the same amount - how does this affect pressure in a smaller volume compared to a larger volume?)
 
  • #10
James124900 said:
I thought the pressure of the smaller bottle will increase more, how can temperature increase in the smaller bottle?
(also yes, each bottle is heated to the same amount - how does this affect pressure in a smaller volume compared to a larger volume?)

I just spoiled it all in #7 :-), but you did not understand it, right? :-) What do you not understand of what I wrote there?
 
  • #11
ok I get it now but doesn't and increase in temperature = increase in pressure
 
  • #12
James124900 said:
ok I get it now but doesn't and increase in temperature = increase in pressure

Yes (cause in this case V, n and R are constant), so P increases proportional to the increase of T in this system, under suppossed conditions.
 
  • #13
I thought n was lower in the smaller bottle that is why it had a higher temperature?
 
  • #14
James124900 said:
I thought n was lower in the smaller bottle that is why it had a higher temperature?
Consider just one container for the moment. Using the ideal gas law, derive an expression for the pressure ##P_2## when the temperature is raised from ##T_1## to ##T_2##.
 
  • #15
P=nRT2/V
 
  • #16
James124900 said:
P=nRT2/V
I don't see ##P_1##, ##P_2##, ##T_1## and ##T_2## in there. How do you usally go about finding the new pressure when you're given starting conditions and the final temperature?
 
  • #17
gneill said:
I don't see ##P_1##, ##P_2##, ##T_1## and ##T_2## in there. How do you usally go about finding the new pressure when you're given starting conditions and the final temperature?
I thought you would just put T2 in there, nR is a constant (is it for different volumes?) and the volume for a the bottle does not change.
 
  • #18
James124900 said:
I thought you would just put T2 in there, nR is a constant (is it for different volumes?) and the volume for a the bottle does not change.
Suppose you don't know what nR or V is, only the starting pressure and temperature? You can still use the ideal gas law if you form a ratio.

Write out the ideal gas law separately for the starting and ending states. The see what ratio you can form so that the nR and V terms cancel out...
 
  • #19
ok so P1/T1 = P2/T2
 
  • #20
James124900 said:
ok so P1/T1 = P2/T2
Okay! Now notice that there's no n, R, or V in that expression. What does this tell you about the relationship between pressure and temperature for an arbitrary sample of ideal gas?
 
  • #21
If temperature increases pressure increases. I already know this. How does these relate to having two different bottles of different volume?
 
  • #22
James124900 said:
If temperature increases pressure increases. I already know this. How does these relate to having two different bottles of different volume?
In the expression that your found relating pressure and temperature, was the volume involved at all?
 
  • #23
no, oh so volume of the container is irrelevant of pressure.
 
  • #24
James124900 said:
no, oh so volume of the container is irrelevant of pressure.
Right. If the volume is constant and the contents (amount of gas particles) doesn't change, the pressure will vary with the absolute temperature according to the ideal gas law. So regardless of the size of container, the pressure will vary with temperature in the same way.

Be sure that you don't confuse temperate with heat though. Temperature and heat are two different things :wink:
 
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  • #25
James124900 said:
If temperature increases pressure increases. I already know this. How does these relate to having two different bottles of different volume?

Because of what I explained in #7, you know that the final temperature of the small bottle will be higher, and you also know that, under current conditions, for each bottle it is [tex]P = cte* T[/tex] where cte is a given constant (the SAME constant for both bottles under current conditions, why? :-) ), so...it is quite straight forward, isn't it? (What bottle has higher pressure at the end? ).
 

Related to Pressure and Volume Relationship in Closed Bottles

1. What is the relationship between pressure and volume?

The relationship between pressure and volume is an inverse one, known as Boyle's Law. This means that as the pressure of a gas increases, its volume decreases, and vice versa, as long as the temperature and amount of gas remain constant.

2. How does temperature affect pressure and volume?

According to Charles's Law, as the temperature of a gas increases, its volume also increases, as long as the pressure and amount of gas remain constant. This is because the particles of gas have more energy and move further apart, taking up more space.

3. Can pressure and volume be measured?

Yes, pressure can be measured using a device called a pressure gauge, while volume can be measured using tools such as a graduated cylinder or a volumetric flask. Both pressure and volume have standard units of measurement, such as pascals (Pa) for pressure and cubic meters (m3) for volume.

4. How do pressure and volume affect each other in a closed container?

In a closed container, pressure and volume have an inverse relationship, meaning that if one increases, the other decreases. This is because the gas particles in the container are constantly colliding with each other and the walls of the container, and as the volume decreases, the particles have less space to move, resulting in more collisions and an increase in pressure.

5. What happens to pressure and volume when a gas is compressed?

When a gas is compressed, its volume decreases, resulting in an increase in pressure. This is because the particles of gas are forced closer together, creating more collisions and a higher pressure. This is also known as Boyle's Law, which states that the pressure and volume of a gas are inversely proportional to each other, as long as the temperature and amount of gas remain constant.

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