Lattice Enthalpy and the Born–Haber Cycle

Transcript

0:00Introduction

Hello everyone! Sodium chloride and magnesium oxide: their ions are of similar size. Sodium chloride has charges plus one and minus one; magnesium oxide, plus two and minus two. Yet pulling magnesium oxide apart into its ions takes almost five times the energy! Doubled charges, five times the energy? Strange, right? Today we crack this puzzle and learn the Born Haber cycle, which JEE papers do ask about. So, let's begin.

0:32Lattice enthalpy: pulling a crystal apart

So, what is lattice enthalpy? It is the enthalpy change for separating one mole of an ionic solid into its gaseous ions. One mole, gaseous ions: both matter. Okay? Now think. The positive and negative ions attract each other strongly. So to pull them apart, do we put energy in, or does energy come out? Energy goes in, correct! Sodium chloride, N a C l: the solid, s in brackets, goes along the arrow to the ions N a plus and C l minus, gases, g in brackets. The symbol: a Greek capital delta, lattice written small below, then H, and its value is plus seven hundred and eighty eight kilojoules per mole, written k J mol to the minus one. Plus, as energy goes in. Right? One warning. Some books define it for the reverse process, gaseous ions forming the solid. That gives out energy, it is exothermic, so their value is negative. In this course we mean separation, with a plus. Got it?

1:39What makes lattice enthalpy large?

Alright, what makes a lattice enthalpy large? Picture two pairs of ions: one with charges plus one and minus one, the other with plus two and minus two. Which pair pulls together harder? Plus two and minus two, exactly! So lattice enthalpy grows with the product of the ionic charges, z plus times z minus: z with a small plus below for the cation, z with a small minus below for the anion, numbers only, no signs. With me? Second, size. r plus and r minus are the radii of the cation and the anion, so r plus, plus r minus, is the interionic distance. Bigger ions, centres farther apart, weaker pull, so lattice enthalpy falls. Clear? Together: lattice enthalpy is roughly proportional to z plus times z minus, over r plus, plus r minus; the sign in the middle means proportional to. It is a guide for comparing, not an exact formula. Now our puzzle: for magnesium oxide, M g O, what is the charge product? Two times two, four! Against one times one for sodium chloride, so the charges alone give four times, not two. That was the trap! The magnesium and oxide ions are also somewhat smaller than sodium and chloride ions, so a little more: almost five times. Puzzle solved, okay?

3:05Topper's trick: charges first, then size

Now a topper's trick for ordering questions. Step one: charges first. The biggest charge product wins, by far. Clear? Step two: only when the charges are the same, compare sizes. The smaller ions give the larger lattice enthalpy. Right? The greater than sign opens towards the bigger value. Magnesium oxide and calcium oxide, both two plus, two minus, come first; the calcium ion is bigger, so calcium oxide is second. Then the one plus, one minus salts: sodium fluoride, sodium chloride, potassium chloride, potassium iodide, with bigger and bigger ions. Now you try: the lithium halides, L i F, L i C l, L i B r and L i I, in which order? Same charges, so only size matters. Fluoride is the smallest halide ion, iodide the biggest. So, lithium fluoride, greater than lithium chloride, greater than lithium bromide, greater than lithium iodide. Very good! Got it?

4:07Born–Haber cycle: two routes, same total

Now, to calculate a lattice enthalpy, we use the Born Haber cycle. From the elements to the solid M X: one direct route, or small steps through gaseous ions. Same start, same end, same total. M is the metal, X the non-metal. Metal first: M solid to M gas is sublimation, delta sub H of M; then M gas loses an electron to give M plus, ionisation, delta i H of M, with sub and i written small. With me? Now the non-metal. The salt needs just one X, so we take half of X two, the fraction one half on screen, and break it: half delta diss H of X two, diss for dissociation. Then X gains an electron to give X minus: the electron gain enthalpy, delta e g H of X, okay? Last step: the gaseous ions come together to form the solid. We defined lattice enthalpy for pulling apart, with a plus. So for this step, the other way round, plus or minus? Minus, exactly! It is the reverse of separation, so it is minus delta lattice H, and here energy comes out. Clear? Route two must equal route one, delta f H, the enthalpy of formation, with a small f. So, delta f H equals delta sub H, plus delta i H, plus half delta diss H, plus delta e g H, minus delta lattice H, and remember that minus! One condition: this form is for a salt M X, with ions M plus and X minus. Got it?

5:49Example 1: order KCl, NaCl, MgO, KBr by lattice enthalpy

Let's try one. Arrange in increasing order of lattice enthalpy: potassium chloride, sodium chloride, magnesium oxide and potassium bromide. Use the topper's trick. Pause the video, and try it first. Done? Okay, let's check. Charges first. Magnesium oxide is two plus, two minus; the other three are one plus, one minus. So magnesium oxide is far the largest. Right? Now the one plus, one minus salts: same charges, so size decides. Bromide is bigger than chloride, so potassium bromide is lowest, then potassium chloride. The sodium ion is smaller than the potassium ion, so sodium chloride is above both. The less than sign opens towards the bigger value. Okay? Now, careful. The question says increasing. So which salt do we write first? The smallest, yes! Potassium bromide, less than potassium chloride, less than sodium chloride, less than magnesium oxide. Writing the decreasing order out of habit is a common slip. Read the question twice!

6:57Example 2: lattice enthalpy of NaF by Born-Haber cycle

Now a harder one. Sodium fluoride, N a F, in kilojoules per mole: delta sub H of sodium is one hundred and eight, delta i H of sodium is four hundred and ninety six, half delta diss H of F two is seventy nine, delta e g H of fluorine is minus three hundred and twenty eight, and delta f H of sodium fluoride is minus five hundred and seventy four. Find the lattice enthalpy. Pause and try. Hint: watch the signs. Done? Okay, let's check. Into the Born Haber equation: minus five seventy four equals one hundred and eight, plus four ninety six, plus seventy nine, minus three twenty eight, minus delta lattice H, with me? Now add the four known terms. What do you get? Three hundred and fifty five, correct! One hundred and eight plus four ninety six is six hundred and four; plus seventy nine, six eighty three; minus three twenty eight, three fifty five. Got one thousand and eleven? Oh no, careful! The electron gain enthalpy is minus three twenty eight, so adding it means taking away three twenty eight. Right? So, minus five seventy four equals three fifty five, minus delta lattice H, clear? Move it across: delta lattice H equals three fifty five plus five seventy four, that is, nine hundred and twenty nine kilojoules per mole. Positive, as it must be. And in brackets, sodium chloride's seven eighty eight: sodium fluoride is larger, because fluoride is smaller than chloride. Perfect!

8:42Quick revision: charges first, then size, and the cycle

So, quick revision. One: lattice enthalpy separates one mole of solid into gaseous ions: positive here, but check your book's sign. Two: charges first, then size, roughly z plus times z minus, over r plus, plus r minus. Three: in the Born Haber cycle, the steps add up to delta f H, with minus delta lattice H at the end. Now try the practice problems on this page. All the best!