Chap 7: The 2nd and 3rd Laws

Entropy and Gibbs Free Energy

• Why do things fall apart?

• Why doe some things happen spontaneously?

• Why does anything worthwhile take work?– I’m not sure that our discussions will completely answer this, but we’ll give it a go.

7.1 Spontaneous Change

• Last chapter, we learned about work, heat and enthalpy (at least I hope we did, it will be on the final…)

• We learned that all energy in the universe is conserved and constant– But that’s not really very useful– We want to predict the behaviour of the

universe around us.

Spontaneous ChangesWe know from our own

experiences that some

things will happen

spontaneously

Metal cools Gas expands

We can make the reverse happen (heat a metal cube up, partition a gas), but that takes work (energy)

Spontaneity in Processes

• A process is spontaneous if it has the tendency to occur without being driven by an external influence– Don’t confuse ‘spontaneous’ with ‘speedy’

or ‘rapid’!

7.2: Entropy and Disorder

• If we think about the universe around us and look at all of the spontaneous changes we have observed over our lifetimes, one thing is certain:

Energy and Matter tend to disperse in a disorderly fashion

• Gas molecules don’t pile up on one side of a flask• Buildings fall apart• Thing left untended will get worse

The 2nd Law of Thermodynamics

The entropy of an isolated system

increases in the course of any

spontaneous change

• We can summarize this law mathematically as:

€

ΔS =q

T

The 2nd Law

• What does this mean?

– If a lot of heat energy is transferred, there is a large increase in entropy

– This change in entropy is more noticeable at lower temperatures than higher temperatures (relatively)

• Temperature must be in Kelvin and heat must be in Joules

€

ΔS =q

T

Entropy

• Entropy is a measure of disorder, according to the second law of thermodynamics

• The entropy of an isolated system increases in any spontaneous reaction

• Entropy is a state function

Changes in Entropy

• The equation we obtained from the second law

is valid for isothermal situations (change of state, gas expansion) but we frequently want to be able to determine the entropy as temperature changes…

€

ΔS =q

T

Entropy Change as a Function of Temperature at Constant Volume

• If T2 > T1, then the logarithm is ‘+’ and entropy increases– Makes sense since we are raising the temperature

and thermal motion will increase

• The greater (higher) the heat capacity, the higher the entropy change

€

ΔS = C lnT2

T1

⎛

⎝ ⎜

⎞

⎠ ⎟

Entropy Change as a Function of Changing Volume

• We can use a similar logic to derive the change in entropy when the volume changes:

• When V2 > V1, the entropy increases

• Note: Units are still J/K€

ΔS = nR lnV2

V1

⎛

⎝ ⎜

⎞

⎠ ⎟

Entropy Change as a Function of Pressure

• Remember Boyle’s Law?

• We can substitute this relationship into the equation for entropy change as a function of volume to get:

€

ΔS = nR lnP1

P2

⎛

⎝ ⎜

⎞

⎠ ⎟

Entropy decreases for a sample that has been compressed isothermally (P1>P2)

Entropy Changes Occurring as a Function of Physical State Changes

• What happens to the entropy of a system as we change state?

• Remember: Melting point temp = Tf

Boiling Point temp = Tb

• Temperature doesn’t change as we heat a sample to cause a phase change

• Let’s look at liquid water --> water vapor

Boiling Water and Entropy

Let’s get 3 facts straight:

1. At a transition temperature (Tf or Tb), the temperature remains constant until the phase change is complete

2. At the transition temperature, the transfer of heat is reversible

3. Because we are at constant pressure, the heat supplied is equal to the enthalpy

Water Boiling and Entropy

• We use the ‘ º ’ superscript to denote the standard entropy (the entropy at 1 bar of pressure)€

ΔS =ΔHvap

Tb

(at the boiling temperature)

€

ΔSo =ΔHvap

o

Tb

Ice Melting and Entropy

• We use the same logic to determine the entropy of fusion, ΔSfus

€

ΔSfus =ΔH fus

Tb

€

ΔSfuso =

ΔH fuso

Tb

Trouton’s Rule

• Because the entropy increases so much in going from liquid to a gas, many liquids have similar ΔS°vap values

Trouton’s Rule

• Why is this?– Positional disorder of a gas versus a solid

or liquid (which are pretty close to the same thing)

• Exceptions: Molecules with very weak or very strong intermolecular forces– Helium, Water and Methanol

7.5: A Molecular Interpretation of Entropy

• We’ve looked at the changes to the entropy of a system, but now let’s look at the absolute entropy of the system itself

• If we had a perfect crystal, the positional disorder would be ____

• If the temperature was 0K, the would be ___ thermal disorder, so the entropy would be ___

The 3rd Law

The entropies of all perfect crystals approach zero as the absolute temperature approaches zero.

The Boltzmann Formula

• W is a reflection of the ensemble, the collection of molecules in the system

• This entropy value is called the statistical entropy

S = k lnW

Where:

k = Boltzmann’s constant

= 1.381 x10-23 J/K

W=# of ways atoms or molecules in the system can be arranged and still give the same total energy

The Boltzmann Formula

• Let’s think about W (Wahrscheinlichkeit) for a moment– Word translates as “probability” or “likelihood”

• If we could only arrange the atoms/molecules in the system one way, there would be ___ entropy since ln(1) = __

• As we start to increase the population of the system, we can arrange the members of the system in different ways and still have the same total energy, so W increases.

The Boltzmann Formula

Each molecule can be oriented 2 ways

W = 2 x 2 x 2 x 2 = 16

Example 7.7

Residual Entropy• We know Boltzmann’s concept of the ensemble

is correct from observations of molecules at low temperature (and using logic with some cynicism)

• As we get near absolute zero, the entropy within the crystal becomes increasingly a function of the positional entropy within the ensemble cause by the packing of the components

• Let’s think about this for a minute…

Residual EntropyThe entropy of 1 mole of FClO3 at T = 0 K is 10.1 J/K. Suggest and interpretation.

•How many molecules do we have?

•How many ways can each molecule be arranged?

•What is the residual entropy?

•Pretty close!