Equilibrium Constant, Gibbs Energy, Enthalpy and Entropy

Transcript

0:00Introduction

Hello everyone! Two small puzzles first. One: a fully discharged battery gives zero volts, yet its chemicals are still inside. Why does it stop? Two: the Daniell cell gives just one point one zero volts. Small, right? But hidden inside that voltage is a number of about ten to the power thirty seven! Today: cell potential, the equilibrium constant, Gibbs energy, entropy and enthalpy, all connected. This comes up often in JEE, so let's begin.

0:34Equilibrium: the cell potential falls to zero

So, start from the Nernst equation: E cell equals E naught cell, minus zero point zero five nine one over n, times log Q, at two hundred and ninety eight kelvin. The little circle, read naught, means standard: one molar, one bar, usually at two hundred and ninety eight kelvin. Q is the reaction quotient, n the number of electrons, and log is to base ten. Okay? Let the cell run: reactants get used up, products pile up. So what happens to Q, and what happens to E cell? Q goes up, yes! That is the up arrow. Log Q grows, and because of the minus sign, E cell falls: the down arrow. The double arrow means implies. With me? And where does it stop? When E cell reaches zero: equilibrium, where Q has become capital K, the equilibrium constant. That is our discharged battery, a cell at equilibrium. Puzzle one, solved! Clear? So, set E cell to zero and Q to K: zero equals E naught cell, minus zero point zero five nine one over n, times log K, okay? Rearrange: log K equals n E naught cell, over zero point zero five nine one. But remember, zero point zero five nine one is for two hundred and ninety eight kelvin only. Got it?

1:51How big is K? The Daniell cell

Now, puzzle two. In the Daniell cell, zinc gives two electrons to copper two plus, so n is two, and E naught cell is one point one zero volts. Using our new formula, what is log K? Two times one point one zero, over zero point zero five nine one. About thirty seven point two three. The wavy equals sign means approximately. Okay? So K is about one point seven times ten to the power thirty seven! There is our giant number. With K that huge, the reaction goes essentially to completion. Right? Topper's trick: for a one electron reaction, every zero point zero five nine one volts of E naught multiplies K by ten. So zero point five nine one volts, ten such steps, gives K equals ten to the power ten. Neat, isn't it?

2:39Gibbs energy ties it all together

Delta G naught, the standard Gibbs energy change, equals minus n F E naught cell; the triangle, capital delta, means change. Capital F is the Faraday constant, about ninety six thousand five hundred coulombs per mole. Okay? And delta G naught also equals minus R T l n K, the link to equilibrium. R is the gas constant, capital T the temperature in kelvin, and l n the natural log. This one holds at any temperature. With me? Set the two equal, using l n K equals two point three zero three log K, okay? And two point three zero three R T over F, at two hundred and ninety eight kelvin, is about zero point zero five nine one volts. There is our number! Now tell me. If E naught cell is positive, what is the sign of delta G naught, and is K bigger than one? Negative delta G naught, and K greater than one. Correct! The double headed arrows mean each statement implies the other: products are favoured. Clear?

3:41Temperature coefficient: entropy and enthalpy

Now, a cell's potential changes a little with temperature. Its temperature coefficient is partial E by partial T, okay? The curly d means a partial derivative, and the small p outside the bracket means constant pressure. The unit is volts per kelvin. Now the best part. Delta S, the entropy change, equals n F times this coefficient. Simple! So tell me. If the cell potential rises when you warm the cell, what is the sign of delta S? Positive, exactly! Since n and F are positive, delta S takes the sign of the coefficient. Got it? Now enthalpy. Rearrange the Gibbs equation: delta H equals delta G plus T delta S, right? Put in minus n F E for delta G, and for delta S, n F times the coefficient. So delta H equals minus n F E, plus n F T times partial E by partial T, okay? From E and its change with temperature, we get all three quantities! With me?

4:40Example 1: K for Cu + 2Ag+ with E0 = 0.46 V, the n = 2 trap

Example one. Copper, C u, reacts with two silver ions, A g plus, giving copper two plus and two silver atoms; the two half arrows mean equilibrium. E naught is zero point four six volts. Find K, okay? Pause the video and try it. Done? Okay, let's check. First, n. Each silver ion takes just one electron. So n is one, right? No, no, careful! Look at copper. Copper becomes copper two plus, that is the arrow, and gives two electrons, written e minus; the two silver ions take one each. So n is two, not one. Always count the electrons in the balanced equation. Clear? Now, log K equals two times zero point four six, over zero point zero five nine one: zero point nine two over zero point zero five nine one, about fifteen point five seven. With me? So K is ten to the power fifteen point five seven, about three point seven times ten to the power fifteen.

5:40Example 2: Delta G, Delta S, Delta H from E0 and dE/dT

Right? Now a harder one, JEE level. Here n is two, E naught is one point one zero volts at two hundred and ninety eight kelvin, and the temperature coefficient is minus four point zero times ten to the power minus four volts per kelvin. Find delta G naught, delta S naught and delta H naught. Pause and try. Hint: watch the units. Done? Okay, let's check. Delta G naught is minus two, times ninety six thousand five hundred, times one point one zero: minus two hundred and twelve thousand three hundred joules, or minus two hundred and twelve point three kilojoules. Okay? Now entropy. The temperature coefficient here is negative. So what is the sign of delta S naught? Negative, yes! Two, times ninety six thousand five hundred, times minus four point zero times ten to the power minus four. That gives minus seventy seven point two joules per kelvin. Right? Now enthalpy: delta H naught equals delta G naught, plus T delta S naught. So, minus two hundred and twelve point three, plus two hundred and ninety eight times minus seventy seven point two? No, no, careful! Delta G naught is in kilojoules, but delta S naught is in joules per kelvin. So first, T delta S naught: two hundred and ninety eight times minus seventy seven point two, about minus twenty three thousand and six joules, or minus twenty three point zero kilojoules. Got it? Now add, both in kilojoules. Minus two hundred and twelve point three, minus twenty three point zero: minus two hundred and thirty five point three kilojoules. That's it!

7:22Quick revision: K from E0, Delta G, Delta S and Delta H

So, quick revision. One: at equilibrium, E cell is zero and Q equals K, so log K equals n E naught cell over zero point zero five nine one, at two hundred and ninety eight kelvin. Two: delta G naught equals minus n F E naught cell, and also minus R T l n K, at any temperature. Three: delta S is n F times the temperature coefficient, and delta H is delta G plus T delta S, in the same units. Now try the practice problems on this page. All the best!