Volumetric Expansion: Temperature, Heat & 3 Alpha=Beta

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Homework Help Overview

The discussion revolves around the concept of volumetric thermal expansion in relation to temperature and heat, specifically addressing the relationship between the coefficients of linear expansion (alpha) and volumetric expansion (beta), expressed as 3 alpha equals beta.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants explore the definitions of linear and volumetric thermal expansion coefficients, with one participant seeking a derivation of the relationship without prior calculus knowledge. Others suggest starting from basic definitions and using approximations to understand the relationship.

Discussion Status

The conversation is ongoing, with some participants providing guidance on how to approach the problem conceptually. There is an emphasis on self-exploration rather than direct solutions, and multiple interpretations of the relationship are being discussed.

Contextual Notes

One participant mentions a lack of calculus background, which may influence their understanding of the derivation process. The forum's policy against providing direct solutions is also noted, emphasizing a learning-focused approach.

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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Hi mjongem, welcome to PF. The approximation \beta=3\alpha 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 \alpha and \beta.
3. Assume small expansions compared to the size of the object, and simplify power terms by a Taylor series expansion (e.g., (1+a)^3\approx 1+3a when a is small).
 
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
 
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 (1+\alpha)^3\approx 1+3\alpha.
 

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