Intuition The one core ideaA covalent bond is a tug-of-war that reaches a truce : two positive nuclei push apart, a pool of shared electrons between them pulls both inward, and they settle at the gap where push exactly cancels pull. Everything in this topic — bond length, bond energy, bond order — is just a number that describes that truce point or how deep the truce is dug in .
Before you can read the parent note Covalent bonding — bond length, bond energy, bond order , you must own every symbol it throws at you. Below, each one gets: plain words → the picture → why the topic needs it. They are ordered so each rests on the one before.
Definition Nucleus, electron, and separationr
A nucleus is the tiny, heavy, positively charged centre of an atom. Picture a small red dot that repels other red dots.
An electron is a light, negatively charged speck of "glue" that is attracted to any nucleus.
r r , for radius/range ) is the distance between the two nuclei , measured centre-to-centre.
Look at the figure: two nuclei, and the double-headed arrow between them is r
r how far apart the two nuclei are . If we can plot "energy versus r r horizontal axis of everything that follows.
Units of r we use pm (picometres) and Å (ångström).
1 pm = 1 0 − 12 1 A ˚ = 100 pm 100 160
Definition Attraction and repulsion
Like charges repel (two + push apart; two − push apart).Opposite charges attract (+ and − pull together).
This is the whole of the physics we need. No equations yet — just "same signs shove, opposite signs hug."
Why the topic needs it: the entire bond is a competition between
nucleus–nucleus repulsion (both +, so they shove apart), and
nucleus–electron attraction (+ pulls the − glue, and the glue pulls back on both nuclei).
The strength of these forces changes with r r
∝ A ∝ B A B B A ∝ r 1
∝ r 1 r 1 big when r and small when r . Picture the two nuclei almost touching (r r 1 r r 1 r → 0
Why the topic needs it: this single fact — repulsion explodes at short range — is what stops the nuclei from collapsing into each other and creates a minimum in the energy curve.
E E total potential energy of the two-atom system: negative means "bound / stable / a good place to be", positive means "strained". By convention E = 0 infinitely far apart (no interaction).
Look at the well-shaped curve. The horizontal axis is r E
Far apart (r E ≈ 0 Approaching : attraction wins, E dips below zero — the system is happier closer.Too close : repulsion (∝ 1/ r E shoots up steeply.In between : E bottom of the well .
Intuition Why a "well" and not a slideA ball dropped into a valley rolls to the lowest point and rests there. The two nuclei do the same in this energy landscape: they settle at the bottom of the well. That resting spot is the bond. The two headline numbers of the topic are simply where the bottom is (its r how deep it is (its E
r e r e e means equilibrium ) is the value of r at the bottom of the well — the truce distance where push = pull. This is what we call the bond length .
In the well figure, drop a vertical line from the lowest point down to the axis: where it lands is r e
Why the topic needs it: "bond length" is not a vague size — it is precisely the r E . Naming it r e r e
D e B E D e B E bond energy ) is how far below zero the bottom of the well sits — the vertical drop from E = 0 pay to climb back out , i.e. to break the bond and separate the atoms to infinity.
In the figure, the vertical black bar marks this depth. Since the bottom is at E = − D e E = 0 B E = 0 − ( − D e ) = D e > 0.
Common mistake "Bond energy should be negative because forming a bond releases energy."Why it feels right: falling into the well does release energy (the drop is downward).
Fix: By definition B E climb out , not the fall in. Climbing costs energy → B E positive . Forming releases the same size with a minus sign.
Units of energy: kJ mol− 1 = kilojoules per mole . A mole is just a fixed huge count of bonds (6.022 × 1 0 23 − 1
Picture stacking more blobs of glue between the two magnets. More glue = stronger pull inward = the nuclei sit closer and the well gets deeper.
Intuition Why bond order is the master dialTurn BO up and two things move together : r e D e narrower/left-shifted (shorter r e and deeper (bigger D e .
The parent also computes BO for diatomic molecules from Molecular Orbital Theory :
Intuition Why subtract, then halveBonding electrons build the pull, antibonding electrons cancel it — so the net helpful count is N b − N a 2 bond is made of a pair , and dividing a headcount of electrons by 2 converts "electrons" into "pairs". That is why an odd leftover (like one lone antibonding electron) gives a half , e.g. BO = 2.5
∑ Δ H
∑ add up all of these ". ∑ B E ( bonds broken ) Δ H delta H ) means "change in heat energy " of a reaction. Δ Δ H
Why the topic needs them: the payoff formula
Δ H rxn = ∑ B E ( broken ) − ∑ B E ( formed ) cost to break minus payback from forming . These belong to Hess's Law and Enthalpy ; here they are only the bookkeeping symbols. (Related: Sigma and Pi Bonds explains why the first bond is strongest, and Electronegativity and Bond Polarity plus VSEPR and Molecular Geometry and Resonance and Delocalisation refine the picture later.)
Charge: like repel, opposite attract
Distance r between nuclei
Proportional 1 over r repulsion
Energy E vs r curve, the well
Bottom position gives bond length r_e
Well depth gives bond energy D_e
Shared electron pairs bond order BO
MO electron counts N_b and N_a
Master trend and enthalpy
Cover the right side and test yourself.
What does the symbol r The distance between the two nuclei; the horizontal axis of the energy curve.
Convert: 1 A ˚ 100
What does A ∝ r 1 r → 0 It blows up (becomes huge) — this is the exploding nucleus–nucleus repulsion.
Where is E When the atoms are infinitely far apart (no interaction).
r e The horizontal position (r
D e B E The vertical depth of the well = energy to break the bond; always positive.
Why is bond energy reported positive, not negative? It is defined as the energy to break (climb out of) the well, and climbing costs energy.
What does bond order count? The number of shared electron pairs between the two atoms.
In BO = 2 N b − N a N b N a Electrons in bonding orbitals and in antibonding orbitals.
Why divide by 2 in the MO formula? To convert an electron headcount into a count of pairs (a bond = one pair).
What do ∑ Δ H "Add all of these" and "change in heat energy of the reaction".
