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There are three somewhat different types of interactions that can hold atoms together, all called bonds: there are ionic bonds, covalent bonds, and hydrogen bonds.
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Ionic Bonds.
Covalent Bonds.
Hydrogen Bonds.
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IONIC BONDS exist between, not too surprisingly, ions of opposite charges: the attraction between positive and negative ions, usually from atoms in Columns 1 and 2 in the Periodic Table (which have positive charges from discarding electrons in the outer shell) and atoms from Column 7 and sometimes 6 (negative ions from stealing electrons to fill their shells). This bond can hold ions together extremely tightly, like two powerful magnets, and is commonly the bond found in salts, including sodium (+1) chloride (-1), NaCl, table salt. Ionic bonds are often disrupted by water, which itself is somewhat charged (see the next section on water properties) and which tends to push apart and surround the ions; this makes ionic compounds somewhat rare in watery biological systems, while dissolved ions are common (when you "taste salt," you are not tasting the compound sodium chloride, but detecting the ions separated in your saliva). The ion form, with a full outer electron shell, is the stable, natural form here, which makes the sharing of electrons we'll see later in covalent bonds below not work. When compounds separate into ions (or when radicals become ions), the act is called ionization (water can spontaneously separate into a positive Hydrogen ion, H+, and a negative Hydroxide ion, OH-). Larger molecules can ionize when parts are lost that either take electrons with them or leave them behind, and small parts of very large molecules might be ionized.
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Ionic Bonding - Works even after ions have already formed. (Video)
Image of dry ionic compound atomic structure.
Video on ionization of water.
...And it all might work differently under non-Earth-surface conditions...
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COVALENT BONDS are produced when atoms can fill their outer electron shells more easily by borrowing electrons from another atom, while sharing some of theirs. Atoms from the middle Columns of the Periodic Table, from 3 to 6, often form these kinds of bonds, even when coming together as ions. The number of bonds an atom can form is the number of electrons it needs to fill its outer shell. The vast majority of atoms in biological molecules come from just a set of four: Hydrogen from Column 1 (but whose tiny outer shell only needs one more to be full), Oxygen from Column 6 (needing 2 electrons to fill its bigger shell to its capacity of 8), Nitrogen from Column 5 (needs 3 electrons), and Carbon from Column 4 (needs 4 electrons). You might remember them as HONC, which puts their bonding properties in order, 1-2-3-4.
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HONC molecules, showing bonds as lines.
Much more on bond types.
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Although bonds often hold together atoms on a one-bond: one-atom pairing, it is possible for more than one electron to be shared between two atoms. These are multiple bonds. There are double bonds, as is found in the common atmospheric form of oxygen, O2, or triple bonds, found in the atmospheric form of nitrogen, N2. Atmospheric nitrogen is held so tightly together by those bonds that it is very stable and difficult to break; getting nitrogen for use in biological molecules, where it is sometimes a critical ingredient, can only be done by a handful of organisms on which the rest all depend. Quadruple bonds probably do not appear much in nature, since the ones made in labs tend to explode. Some bonds resonate, shifting between single and double bonds in complex molecules, especially rings of mostly carbon.
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Imaging multiple bonds.
More on multiple bonds than you really want to know.
Why there aren't quadruple bonds.
Resonating bonds.
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Life is often talked about as Carbon-based because, with its ability to share electrons and bond to 4 other atoms simultaneously, carbon makes possible the complicated atomic arrangements needed in large molecules. Science fiction has toyed with the idea of silicon-based life: silicon is also in Column 4 and is fairly common on rocky planets like ours. There are reasons to accept that this is theoretically possible but reasons to believe it is also extremely unlikely. The term organic originally was applied to any materials with carbon in them, on the assumption that any such compounds were somehow related to living things. The term has been updated for molecules, and now requires carbon and hydrogen, a combination that is rare outside of living systems. The term should require oxygen in the compounds as well, but it was never updated for that.
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Many carbon-based molecules.
Silicon-based Life?
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Nitrogen, with its important ability to hold three atoms together, is also important in such molecules as proteins and DNA (breakdown products from these are called nitrogenous wastes). Oxygen is commonly a "bridge" inside molecules and sometimes, held on with a double bond, a potentially reactive outer atom. Hydrogen can be thought of as a "capper," attaching to those bond areas that face outward in molecules and must be attached to something.
Bonds exist in three-dimensional space: depending upon the connections, the angles at which atoms are arranged vary. This is partly why strings of amino acids in proteins curl and bend and kink and can produce molecules in a vast array of shapes, including small rings and large specimens with very complex outer "surfaces."
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The Nitrogen Cycle.
Molecular geometry.
Bonding angles. (Video)
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Covalent bonds may result from atoms sharing electrons, but the sharing may not be completely equal: atoms with more protons may be able to hold their own and other atom's borrowed electrons more strongly than their partner atoms can, so that electrons spend more time around the larger atom in a covalent bond. This can create slight charge differences across whole molecules or parts of molecules - areas that electrons spend more time in are slightly more negative, while areas where they spend less time are slightly positive. Molecules with such partial charges are called polar molecules. Water, with a big oxygen toward one side and little hydrogens toward the other, tends to be somewhat negative on the oxygen end and somewhat positive on the hydrogen end. This makes it bipolar, but not every polar arrangement is so neatly 2-ended. In water, however, this is perhaps the most important feature of the molecule, responsible for much of makes water a unique substance.
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More on polarity.
Polarity. (Video)
Water, showing bipolar nature.
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Like full ionic charges, these partial charges can attract one another, which is the third type of bond: HYDROGEN BONDS, called this because little, weak hydrogen is a common participant. These can hold separate molecules loosely together, as happens in water, or can hold different parts of one molecule together, creating the spiral pattern of DNA or the complex forms of proteins. Hydrogen bonds, based on a wide variety of partial charges, also can be widely different in strength, approaching ionic bond strength in rare cases. In molecules whose activity depends upon their shapes, factors that disrupt hydrogen bonds, such as heat or loose ions, can seriously denature them (shut down their activity).
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Animations showing water and hydrogen bonds.
How hydrogen bonds hold proteins in particular shapes.
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There is another class of bond, metallic bonds, solids where the electrons are shared among all of the atoms, but they have no real role in biological systems. They do, apparently, produce transitory examples of things like quadruple, even sextuple bonds.
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Metallic bonds.
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