Volumetric Expansion: Temperature, Heat & 3 Alpha=Beta

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In summary, the conversation discusses the relationship between temperature, heat, and volume thermal expansion. The approximation \beta=3\alpha is explored and it is suggested to use definitions of linear and volumetric thermal expansion coefficients to solve the problem. Calculus is not necessary with the use of the approximation (1+\alpha)^3\approx 1+3\alpha.
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mjongem
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Hello, Could someone explain to me when referring to temperature and heat, speciffically volume thermal expansion how 3 alpha equals beta? If you could show me in a formula that would be great!
 
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  • #2
Hi mjongem, welcome to PF. The approximation [itex]\beta=3\alpha[/itex] is something you can prove to yourself.

1. Start with the definitions of linear and volumetric thermal expansion coefficiencts.
2. Considering a cube, write the volume change as a function of unit volume in terms of [itex]\alpha[/itex] and [itex]\beta[/itex].
3. Assume small expansions compared to the size of the object, and simplify power terms by a Taylor series expansion (e.g., [itex](1+a)^3\approx 1+3a[/itex] when a is small).
 
  • #3
I was wondering if you would show me the exact wat to derive my question, I have not taken calculus before, and I just wanted to see how three alpha equals beta
 
  • #4
We don't just provide solutions here, since that wouldn't help anyone learn. If you show your work in solving a problem, though, you'll usually be able to get help if you get stuck.

If you're interested in solving the problem yourself, try the steps I outlined above. Calculus isn't necessary once you have the approximation [itex](1+\alpha)^3\approx 1+3\alpha[/itex].
 

FAQ: Volumetric Expansion: Temperature, Heat & 3 Alpha=Beta

What is volumetric expansion?

Volumetric expansion is the increase in volume of a substance due to a change in temperature. This phenomenon is caused by the increase in molecular motion and the resulting increase in space between molecules.

How does temperature affect volumetric expansion?

As temperature increases, the molecules in a substance gain energy and move more rapidly, causing them to take up more space. This results in an increase in volume and is known as thermal expansion.

What is the relationship between temperature and volumetric expansion?

The relationship between temperature and volumetric expansion is directly proportional. This means that as temperature increases, the volume of a substance also increases, and vice versa. This relationship is described by the coefficient of volumetric expansion, which is a constant value for each substance.

What is the role of heat in volumetric expansion?

Heat is the energy that is transferred from a hotter object to a cooler object, causing a change in temperature. When heat is applied to a substance, its molecules gain energy and move more rapidly, resulting in an increase in volume due to volumetric expansion.

How does 3 Alpha=Beta affect volumetric expansion?

3 Alpha=Beta is a term used to describe the type of volumetric expansion that occurs in some substances, such as water, when they are cooled below a certain temperature. This results in a decrease in volume instead of the expected increase. This phenomenon is caused by the rearrangement of molecules in the substance and is also known as negative thermal expansion.

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