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I know the relationship between change in temperature and change in length. When the ambient temperature around any substance is increased, its length increases. This is due to molecules gaining more kinetic energy due to excess heat and vibrating at a higher frequency. Due to this increased vibration, the average distance between neighboring molecules increases. Therefore, increasing the overall length of the substance.

I also know that:

$\alpha_V = \frac{1}{V}(\frac{\partial V}{\partial T})_p$

and:

$\alpha_L = \frac{1}{L}(\frac{\partial L}{\partial T})$

What I want to know is how are these two formulas derived, how did we come to know this $\frac {1}{L}$ multiplied by $\frac{\partial L}{\partial T}$ will give us the coefficient of linear expansion?

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    $\begingroup$ Actually your model (though many people use it) is wrong. Higher temperature causes more vibration and the molecules oscillate farther away from each other but also nearer to each other, so the effects cancel. Thermal expansion happens because the potential between molecules is not symmetric: they are not simple harmonic oscillators but the force rises more slowly for positive displacements but for negative ones. $\endgroup$ – RogerJBarlow Sep 16 '18 at 15:55
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We start with the approximation that the relative change in the length or volume is proportional to the change in the temperature i.e. that it is linear in the temperature. So we get an expression for the new length:

$$ L = L_0(1 + \alpha \delta T) $$

Rearranging this gives:

$$ \alpha = \frac{L - L_0}{L_0 \delta T} $$

And since $L - L_0$ is the increase in length, $\delta L$, this gives us:

$$ \alpha = \frac{1}{L_0} \frac{\delta L}{\delta T} $$

And likewise for the volume. Note however that the increase in length is only approximately linear in the temperature change. The real equation is something like:

$$ L = L_0(1 + \alpha \delta T + \beta \delta T^2 + \gamma \delta T^3 + \, ...) $$

But as long as the temperature change is small we find from experiment that the terms in $\delta T^2$, $\delta T^3$, etc are small relative to the linear term and can be neglected.

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