the bonding-energy curve

There are two ways to describe how two atoms interact: by the force between them, or by the energy stored. The bonding-energy curve is the energy picture, a graph of the atoms' potential energy against how far apart they are. Its shape is a valley: high energy when the atoms are far apart, sloping down to a lowest point, then climbing steeply back up when they are squeezed too close. Atoms, like a ball on a hill, roll to the bottom of the valley and settle there.

The curve is the sum of two energy terms: an attractive part that lowers energy as atoms approach (roughly proportional to -1/r for ionic bonds) and a repulsive part that raises energy very steeply at short range (something like +1/r^12). The distance at the very bottom of the valley is the equilibrium spacing r0, the bond length, and the depth of the valley below the far-apart level is the bond energy E0, the energy you must pay to pull the atoms fully apart. Force and energy are two views of one thing: the force is the slope of this curve, and the atoms rest where the slope is zero, which is exactly the bottom of the valley.

Almost every mechanical and thermal property is written in the shape of this one curve. A deep valley means a strong bond, hence a high melting point and large bond energy. A sharply curved valley (steep walls) means a stiff material with a high Young's modulus, because it costs a lot of energy to move the atoms off the bottom. And the valley's asymmetry, gentler on the far side than the near side, is why materials expand when heated: as atoms jiggle harder with temperature, their average position drifts outward along the lopsided valley.