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Temperature as a function of pressure?


Science Student
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I understand how pressure is a function of temperature, but not how temperature a function of pressure.

 

My notes have: initial volume / initial temperature = final volume / final temperature.

 

This seems to mean that adding pressure increases temperature. But how does pressure increase the average kinetic energy of the air molecules?

 

I can understand how increasing the average kinetic energy of air molecules increases the pressure on the walls of the container. But vice versa doesn't make sense to me.

 

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I'm assuming you're working with ideal gasses? The relationship between pressure, volume and temperature can be found with the ideal gas law:

 

[math]PV=mRT[/math]

 

where R is a constant and m is the mass of the gas. So if we hold V and m constant, we get:

 

[math]\frac{P}{T}=const.[/math]

 

Or, in a perhaps more useful form:

 

[math]\frac{P_1}{T_1} = \frac{P_2}{T_2}[/math]

 

This does indeed say that increasing temperature results in increased pressure. However, you seem to be confusing volume and pressure as indicated by the equation you posted. Pressure and volume are not the same thing. Also, these equations don't say what "causes" what, as you have phrased it, they simply state a correlation between different properties of a gas. In other words, they don't say that "adding pressure" will cause increase temperature, they say that there is a correlation between the pressure of a gas and its temperature. Heating a gas up causes its pressure to go up. I'm not sure what it means to "add pressure" without adding more gas or adding heat.

Edited by elfmotat
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Mechanism aside, if one were to somehow increase pressure while keeping volume constant would require the particles to hit the surfaces more often and/or with greater momentum. If the number of them is also held constant, the only way to do this is to increase their speed. You might even see that this will give both effects — they hit more often and have a larger momentum. Each effect varies with v, so the dependence on T (which depends on the average v2) should not be a surprise.

 

By what mechanism would we actually do this? I'm not sure.

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Damn, I did not explain the issue properly.

 

I should have asked how temperature can be a function of volume. In other words, how does decreasing the volume of a container increase the temperature of the gas? Or would it?

 

In terms of V1/T1 = V2/T2, I don't think my air compressor heated up when it would take in and compress the air. Although I may not have noticed.

Edited by Science Student
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Damn, I did not explain the issue properly.

 

I should have asked how temperature can be a function of volume. In other words, how does decreasing the volume of a container increase the temperature of the gas? Or would it?

 

In terms of V1/T1 = V2/T2, I don't think my air compressor heated up when it would take in and compress the air. Although I may not have noticed.

 

[math]\frac{V_1*p_1}{T}=\frac{V_2*p_2}{T}, T=const[/math]

 

When you will decrease volume (f.e. close one end syringe and push it), you will increase pressure inside.

 

In other case:

[math]\frac{V_1*p}{T_1}=\frac{V_2*p}{T_2}, p=const[/math]

 

This simplifies to:

[math]\frac{V_1}{T_1}=\frac{V_2}{T_2}[/math]

 

But both volume and temperature has to change at the same time, to have constant pressure.

 

Edited by Sensei
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But then why does the pressure change and not the temperature? Or why doesn't pressure and temperature change?

 

It can — there's an infinite number of combinations. But you can only easily solve for ones where you know that one variable is held constant.

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I was curious about this question and what happened at the molecular level so I looked it up. When you increase pressure the velocity of the enclosing moving surface(s) adds momentum to the molecules that collide with it and hence heat; rather like a swinging bat hitting a ball as opposed to a balll bouncing off a stationary wall. The heat rise will only be temporary however due to the usual heat loss pathways.

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