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How do we experimentally measure the strengths of covalent versus ionic bonds? Also, for a given molecule, how can we know if two pairs of atoms is covalently or ionically bound?

Related question: Textbooks say that in solution of water, two neighboring water molecules (call them A and B) will be 'attracted' because the O in molecule A will attract the H of the 'nearby' molecule B. It is said that H will alternate states between A and B. How do we know this is true experimentally and how can it be measured?

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You measure the energy of any bond by carrying out reactions and seeing how much energy (usually in the form of heat) has to be invested. For example, if you react one mole of $\ce{C}$ with one mole of $\ce{O2}$, you get $\ce{CO2}$ and heat. Measure the energy with a calorimeter and divide by how many atoms and you get the energy you invested to produce one bond. Not coincidentally, bonds are generally measured in units of energy/mol (e.g. kJ/mol).

Whether a bond is ionic or covalent has to do with whether the bonding electron resides on one atom or both. A perfectly covalent bond finds the electron half on one atom and half on the other. A perfectly ionic bond has the electron 100% on one atom and not at all on the other. Real systems are a mix of both. When we say a bond is ionic (such as $\ce{NaCl}$), we are really referring to the fact that we have measured a circumstance where the electron truly favors one atom. There are various ways to measure this, one of the most common is to melt the solid and see if it conducts electricity. In the case of $\ce{NaCl}$, the bond is so ionic that a small voltage will cause the ions to wander from their neighbor.

With a covalent bond such as you find in $\ce{SiO2}$, atoms can't wander without breaking the bond, so a small voltage doesn't cause a current. A large voltage may break the covalent bonds and cause a current, but it also changes the substance (in this case releasing oxygen gas and producing silicon metal).

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