Why does the change in length depend on the initial length of a rod?

Question

When a rod of some length is heated then it undergoes linear expansion (also areal and volumetric) but I don't understand Why does the amount of expansion depend on the initial length of the rod ?

If we have two rods one of half the length of the other, both of same material and at same initial temperature then if both are heated to same final temperature then the one with greater length undergoes greater change in length. This is what confused me.

Shouldn't the energy per unit length be same in both the rods and so each bonds should get the same energy in both the rods and so both should expand the same amount?

Answer 1

To address the energy-per-unit-length qualm, let's assume that it is equal in both rods. We can also assume the rods have the same number of bonds-per-length, and each bond receives the same amount of energy.

One rod is longer, so it takes more energy to raise its temperature to the same temperature, because it has more mass.

Now, each bond has been given the same amount of energy and will expand by a tiny amount $\Delta{}x$. But the longer rod has more bonds, and each bond is expanding by a tiny amount. The smaller rod expands by $\frac{L}{b}\Delta{}x$ and the larger rod expands by $\frac{2L}{b}\Delta{}x$ where $b$ is the length for a single bond, so that $L/b$ is the total number of bonds along the rod's length.