Gibbs Energy and Spontaneity

Transcript

0:00Introduction

Hello everyone! Okay, a small puzzle first. Leave an ice cube on the table: it melts, all by itself, right? But melting takes in heat: the enthalpy goes up, not down! And in the freezer, water does the opposite, and gives out heat. So who decides the direction: the heat, or the disorder? Today we meet the quantity that settles this fight: Gibbs energy. It comes up often in JEE, so let's begin.

0:30Gibbs energy: judge from the system alone

So, by the second law, a process is spontaneous when the total entropy of system plus surroundings goes up. But tracking the surroundings is painful, right? So we define a quantity for the system alone: G equals H minus T S, okay? G is the Gibbs energy, H the enthalpy, T the temperature in kelvin, and S the entropy. Clear? Now, at constant temperature, delta G equals delta H minus T delta S: the heart of today's lesson. The little triangle is the capital Greek letter delta, meaning change in. Okay? Now think. The second law uses delta S total, for the whole universe; delta G looks only at the system. How are the two linked? Like this: at constant temperature and pressure, the T and P on screen, delta G equals minus T delta S total; the small words below, total and surr, mean the universe and the surroundings. It uses delta S surr equals minus delta H over T, true only at constant temperature and pressure. So when the total entropy goes up, delta G goes down. With me?

1:42The sign of ΔG decides

So, at constant temperature and pressure, the sign of delta G decides. Delta G less than zero, the less than sign on screen: spontaneous. Delta G equal to zero: equilibrium. Right? And delta G greater than zero: non-spontaneous. But careful: then the reverse process is spontaneous. Got it? Now tell me. If a reaction has a large negative delta G, it must also be fast, right? No, no, careful! Delta G tells you whether a reaction can go, not how fast. The rate depends on the activation energy. That is a common trap. Okay?

2:21Four cases: when does temperature decide?

Now, delta G equals delta H minus T delta S has two parts, and their signs can fight. Remember, T in kelvin is always positive. First case: delta H negative, delta S positive. Can delta G ever become positive? Never, exactly! Delta H is negative, and minus T delta S is negative too. So delta G stays negative: spontaneous at all temperatures. Okay? Now the exact opposite: delta H positive, delta S negative. Both terms push delta G up, so it is never spontaneous, at any temperature. Clear? Now the interesting ones: delta H and delta S both negative. Delta H helps, but minus T delta S is positive, growing as temperature rises. So, spontaneous at low temperature, or high? At low temperature, yes! When T is small, the delta H term wins. So it is spontaneous only below T equals delta H over delta S, the temperature where delta G becomes zero. Got it? And the last case: both positive. Now a higher temperature makes minus T delta S more negative, pulling delta G down. So it is spontaneous only above T equals delta H over delta S, the same switch temperature. With me?

3:47The four cases on one graph

Now let's see all four cases on one graph. Delta G is on the vertical axis, temperature T on the horizontal one. Each coloured line is delta G equals delta H minus T delta S: its intercept, where it starts, is delta H, and its slope is minus delta S, as written below the graph. Okay? When delta H and delta S have opposite signs, the line never crosses zero. The green line stays below zero: always spontaneous. The orange line stays above zero: never spontaneous. Right? When the signs are the same, the line crosses zero at T equals delta H over delta S, the black dot. The dark blue line, both negative, is below zero only before that dot: spontaneous at low temperature. The purple line, both positive, goes below zero only after it: spontaneous at high temperature. Got it?

4:44Back to the ice puzzle

Now, back to our ice. Melting takes in heat, so delta H is positive; liquid water is more disordered, so delta S is positive too. Both positive: spontaneous only above zero degrees Celsius; the small circle means degrees. Okay? Freezing is the reverse, both negative: spontaneous only below zero degrees Celsius, in your freezer. Heat and disorder fight, and the temperature decides who wins. Puzzle solved! Now, a trick toppers use. At the melting point, solid and liquid are at equilibrium, so delta G is zero. So the melting point is simply delta H over delta S, one division! Clear?

5:27Minus ΔG: the maximum useful work

One more meaning. Minus delta G is the maximum useful work a process can do at constant temperature and pressure; useful means non-expansion work. Right? For example, the electrical work of a cell. In a hydrogen fuel cell at two hundred and ninety eight kelvin, delta G is minus two hundred and thirty seven point one kilojoules per mole of water; kJ means kilojoules. So it gives at most that much electrical work. Got it?

5:56Example 1: ΔG at 298 K and 500 K, the switch at 400 K

Now, Example one. Delta H is plus twenty kilojoules, and delta S is plus fifty joules per kelvin: capital J is joules, and capital K to the power minus one means per kelvin. Find delta G at two hundred and ninety eight kelvin, and at five hundred kelvin. Pause the video, and try it. Done? Okay, let's check. Delta G equals delta H minus T delta S, so twenty, minus two hundred and ninety eight times fifty. That's it, right? No, no, careful! That is the trap. Delta H is in kilojoules, but delta S is in joules per kelvin. So first, fifty joules per kelvin becomes zero point zero five zero kilojoules per kelvin. Clear? At two hundred and ninety eight kelvin: twenty minus two hundred and ninety eight times zero point zero five zero, which is plus five point one kilojoules. Positive, so not spontaneous. Okay? At five hundred kelvin: twenty minus five hundred times zero point zero five zero, that is, twenty minus twenty five: minus five point zero kilojoules. Negative, so now it is spontaneous! Right? This is our both-positive case: spontaneous only at high temperature. The switch is at delta H over delta S: twenty thousand joules over fifty joules per kelvin, that is, four hundred kelvin. Perfect!

7:31Example 2: ΔG for ammonia, not spontaneous above 464 K

Now a harder one: making ammonia. N two plus three H two gives two N H three; the small numbers are subscripts, and the arrow means gives. Delta H is minus ninety two point two kilojoules; delta S, minus one hundred and ninety eight point seven joules per kelvin. Find delta G at two hundred and ninety eight kelvin, and the temperature above which it is not spontaneous. Pause and try. Watch the units and the signs. Done? Okay, let's check. In kilojoules per kelvin, delta S is minus zero point one nine eight seven. So delta G equals minus ninety two point two, minus two hundred and ninety eight times minus zero point one nine eight seven, in brackets. With me? Now, two hundred and ninety eight times zero point one nine eight seven is about fifty nine point two. So, minus ninety two point two, minus fifty nine point two, right? No, no, wait! Minus times minus is plus. So it is minus ninety two point two, plus fifty nine point two: minus thirty three point zero kilojoules. Negative, so spontaneous at this temperature. Clear? Now the second part. Delta H and delta S are both negative. From our four cases: spontaneous below some temperature, or above it? Below, exactly! Both negative: the delta H term wins only at low temperature. So it is spontaneous only below T equals delta H over delta S, okay? The minus signs cancel: ninety two thousand two hundred joules, over one hundred and ninety eight point seven joules per kelvin. That's about four hundred and sixty four kelvin; the wavy equals sign means approximately. Above that, delta G turns positive: not spontaneous. Perfect!

9:32Quick revision: ΔG = ΔH - TΔS and the four cases

So, quick revision. One: delta G equals delta H minus T delta S; at constant temperature and pressure, a negative delta G means spontaneous. Two: when delta H and delta S have the same sign, temperature decides, and the switch comes at delta H over delta S, okay? Three: change delta S to kilojoules per kelvin before you subtract. Now go and try the practice problems on this page. All the best!