Volumetric Expansion: Temperature, Heat & 3 Alpha=Beta

[mjongem]

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!

 

[Mapes]

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).

 

[mjongem]

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

 

[Mapes]

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 $(1+\alpha)^3\approx 1+3\alpha$.