The Story of Spontaneity and Energy Dispersal

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Spontaneity

Spontaneous process are those that occur naturally.

Hot body cools

A gas expands to fill the available volume

A spontaneous direction of change is where the direction of change does not require work to bring it about.

Spontaneity

The reverse of a spontaneous process is a nonspontaneous process

Confining a gas in a smaller volume

Cooling an already cool object

Nonspontaneous processes require energy in order to realize them.

Spontaneity

Note:

Spontaneity is often interpreted as a natural tendency of a process to take place, but it does not necessarily mean that it can be realized in practice.

Some spontaneous processes have rates sooo slow that the tendency is never realized in practice, while some are painfully obvious.

Spontaneity

The conversion of diamond to graphite is spontaneous, but it is joyfully slow.

The expansion of gas into a vacuum is spontaneous and also instantaneous.

2ND LAW OF THERMODYNAMICS

Physical Statements of the 2nd Law of Thermodynamics

Kelvin Statements

―No process is possible in which the sole result is the absorption of heat from a reservoir and its complete conversion into work‖

―It is impossible for a system to undergo a cyclic process whose sole effects are the flow of an amount of heat from the surroundings to the system and the performance of an equal amount of work on the surroundings.‖

―It is impossible for a system to undergo a cyclic process that turns heat completely into work done on the surroundings.‖

Clausius statement

It is impossible for a process to occur that has the sole effect of removing a quantity of heat from an object at a lower temperature and transferring this quantity of heat to an object at a higher temperature.

Heat cannot flow spontaneously from a cooler to a hotter object if nothing else happens

The 2nd Law of Thermodynamics

The 2nd Law of Thermodynamics recognizes the two classes of processes, the spontaneous and nonspontaneous processes.

Implications of the 2nd Law

No heat engine can have an efficiency as great as unity

No macroscopic process can decrease the entropy of the universe

Are you kidding me?

Thermodynamic Cat

Hot

Reservoir

Heat Engine Work

Heat Cold

Reservoir

I approve!

What determines the direction of spontaneous change?

The total internal energy of a system does NOT determine whether a process is spontaneous or not.

Per the First Law, energy is conserved in any process involving an isolated system.

What determines the direction of spontaneous change?

Instead, it is important to note that the direction of change is related to the distribution of energy.

Spontaneous changes are always accompanied by a dispersal of energy.

Energy Dispersal

Superheroes with energy blasts and similar powers as well as the Super Saiyans are impossible characters.

They seem to violate the Second Law of Thermodynamics!

Power

Genki dama

Energy Dispersal

A ball on a warm floor can never be observed to spontaneously bounce as a result of the energy from the warm floor

Energy Dispersal

In order for this to happen, the thermal energy represented by the random motion and vibrations of the floor atoms would have to be spontaneously diverted to accumulate into the ball.

Energy Dispersal

It will also require the random thermal motion to be redirected to move in a single direction in order for the ball to jump upwards.

This redirection or localization of random, disorderly thermal motion into a concerted, ordered motion is so unlikely as to be virtually impossible.

Energy Dispersal and Spontaneity

Spontaneous change can now be interpreted as the direction of change that leads to the dispersal of the total energy of an isolated system!

INDEED!

Entropy

A state function, denoted by S.

While the First Law can be associated with U, the Second Law may be expressed in terms of the S

Entropy and the Second Law

The Second Law can be expressed in terms of the entropy:

The entropy of an isolated system increases over the course of a spontaneous change:

ΔStot > 0

Where Stot is the total entropy of the system and its surroundings.

Entropy

A simple definition of entropy is that it is a measure of the energy dispersed in a process.

For the thermodynamic definition, it is based on the expression:

Entropy as a State Function

To prove entropy is a state function we must show that ∫dS is path independent

Sufficient to show that the integral around a cycle is zero or

Sadi Carnot (1824) devised cycle to represent idealized engine

dSdq

T 0

Hot Reservoir

Cold Reservoir

Engine

-w2

-w1 w3

w4

qh

qc

Th

Tc

Step 1: Isothermal reversible expansion @ Th

Step 2:Adiabatic expansion Th to Tc

Step 3:Isothermal reversible compression @ Tc (sign of q negative)

Step 4: Adiabatic compression Tc to Th

Carnot Engine

How is that

possible?

Carnot Cycle

Carnot Cycle

Step 1: ΔU=0

Step 2: ΔU=w

Step 3: ΔU=0

Step 4: ΔU=-w

Efficiency of Heat Engines

Efficiency is the ratio of the work done by an engine in comparison to the energy invested in the form of heat for all reversible engines

e 𝑜𝑟 η =𝑤

𝑞ℎ=𝑞ℎ − 𝑞𝑐

𝑞ℎ=𝑇ℎ − 𝑇𝑐

𝑇ℎ= 1 −

𝑇𝑐𝑇ℎ

All reversible engines have the same efficiency irrespective of their construction.

Refrigeration/ Heat pump

Refrigeration

Coefficient of performance (COP or β or c)

𝐶𝑂𝑃 =𝑞𝑐𝑤=

𝑞𝑐𝑞ℎ − 𝑞𝑐

=𝑇𝑐

𝑇ℎ − 𝑇𝑐

COP describes the qc in this case as the minimum energy to be supplied to a refrigeration-like system in order to generate the required entropy to make the system work.

SUPERENGINE?

Carnot Cycle - Thermodynamic Temperature Scale

The efficiency of a heat engine is the ratio of the work performed to the heat of the hot reservoir

e=|w|/qh

The greater the work the greater the efficiency

Work is the difference between the heat supplied to the engine and the heat returned to the cold reservoir

w = qh -(-qc) = qh + qc

Therefore, e = |w|/qh = ( qh + qc)/qh = 1 + (qc/qh )

Hot Reservoir

Heat Engine Work

Heat Cold

Reservoir

qh

-qc

w

Carnot Cycle - Thermodynamic Temperature Scale

Hot Reservoir

Heat Engine Work

Heat Cold

Reservoir

qh

-qc

w

William Thomson (Lord Kelvin) defined a substance-independent temperature scale based on the heat transferred between two Carnot cycles sharing an isotherm

He defined a temperature scale such that qc/-qh = Tc/Th

e = 1 - (Tc/Th )

Zero point on the scale is that temperature where e = 1

Or as Tc approaches 0 e approaches 1

Efficiency can be used as a measure of temperature regardless of the working fluid

Applies directly to the power required to maintain a low temperature in refrigerators

Efficiency is maximized

Greater temperature difference between reservoirs

The lower Tc, the greater the efficiency

Entropy

For a measurable change between two states,

In order to calculate the difference in entropy between two states, we find a reversible pathway between them and integrate the energy supplied as heat at each stage, divided by the temperature.

Example

Practice

Reversible temperature changes

∆𝑆 = 𝐶𝑝𝑑𝑇

𝑇

𝑇2

𝑇1

The specific heat of water is 4.184 J/gK.

Change in entropy of the surroundings: ΔSsur If we consider a transfer of heat dqsur to the surroundings, which

can be assumed to be a reservoir of constant volume.

The energy transferred can be identified with the change in internal energy

dUsur is independent of how change brought about (U is state function

Can assume process is reversible, dUsur= dUsur,rev

Since dUsur = dqsur and dUsur= dUsur,rev,

∴ dqsur must equal dqsur,rev

That is, regardless of how the change is brought about in the system, reversibly or irreversibly, we can calculate the change of entropy of the surroundings by dividing the heat transferred by the temperature at which the transfer takes place.

Change in entropy of the surroundings: ΔSsur

For adiabatic change, qsur = 0, so DSsur = 0

Entropy changes: Expansion

Entropy changes in a system are independent of the path taken by the process

ΔS = 𝑛𝑅 𝑙𝑛𝑉2

𝑉1

Total change in entropy however depends on the path:

Reversible process: ΔStot = 0

Irreversible process: ΔStot > 0

Irreversible processes

Entropy changes: Phase Transitions

ΔtransS =ΔtransHTtrans

Trouton’s rule:

An empirical observation about a wide range of liquids providing approximately the same standard entropy of vaporization, around 85/88/90 J/mol K.

ΔvapS = 10.5 R

Entropy of gas mixing

Exercise

w = 0

ΔU = q = 312 J

ΔS = 1.00 J/K

Third Law of Thermodynamics

At T = 0, all energy of thermal motion has been quenched, and in a perfect crystal all the atoms or ions are in a regular, uniform array.

The localization of matter and the absence of thermal motion suggest that such materials also have zero entropy.

This conclusion is consistent with the molecular interpretation of entropy, because S = 0 if there is only one way of arranging the molecules and only one microstate is accessible (the ground state).

Third Law of Thermodynamics

The entropy of all perfect crystalline substances is zero at T = 0.

Nernst heat theorem

The entropy change accompanying any physical or chemical transformation approaches zero as the temperature approaches zero: ΔS 0 as T 0

provided all the substances involved are perfectly crystalline.

Lewis statement

If the entropy of each element in some crystalline state be taken as zero at the absolute zero of temperature, every substance has a finite positive entropy—but at the absolute zero of temperature the entropy may become zero, and does so become in the case of perfect crystalline substances.

Unattainable absolute zero

Giauque’s adiabatic demagnetization has led to temperatures of less than 0.000001 K (1 μK) in the nuclear spins of a magnetizable system.

Opposing laser beams that effectively stop the translational motion of atoms have acheved an effective temperature of 3 × 10−9 K (3 nK) (Saubamea and friends)

William Francis Giauque,

1895–1982, was an

American chemist who

discovered that ordinary

oxygen consists of three

isotopes. He received

the 1949 Nobel Prize in

chemistry for

pioneering the process of

adiabatic

demagnetization to attain

low

temperatures.

Third-Law entropies or Absolute Entropies

These are entropies reported on the basis that S(0) = 0.

Measurement of Entropy (or molar entropy) for heating

Measurement of Entropy (or molar entropy)

The terms in the previous equation can be calculated or determined experimentally

The difficult part is assessing heat capacities near T = 0.

Such heat capacities can be evaluated via the Debye extrapolation

Measurement of Entropy (or molar entropy)

In the Debye extrapolation, the expression below is assumed to be valid down to T=0.

𝐶𝑝, 𝑚 = 𝑎𝑇3

𝐶𝑣, 𝑚 = 𝑎𝑇

3 + 𝑏𝑇

Exercises

Statistical Entropy: A molecular look

Boltzmann formula:

𝑆 = 𝑘 ln𝑊

𝑆𝑠𝑡 = 𝑘𝐵 ln

/W

thermodynamic probability

Reflects the number of microstates, or the ways in which the molecules of the system can be arranged.

Entropy is a reflection of the microstates, the ways in which the molecules of a system can be arranged while keeping the total energy constant.

Statistical entropy is a measure of the lack of information about the mechanical state of a system.

Example: statistical entropy of a deck of cards

General equations for entropy during a heating process

S as a function of T and V, at constant P

Δ𝑆 = 𝑛𝐶𝑣 𝑙𝑛𝑇𝑓𝑇𝑖+ 𝑛𝑅 ln

𝑉𝑓𝑉𝑖

S as a function of T and P, at constant V

Δ𝑆 = 𝑛𝐶𝑝 𝑙𝑛𝑇𝑓𝑇𝑖− 𝑛𝑅 ln

𝑃𝑓𝑃𝑖

HELMHOLTZ AND GIBBS ENERGIES

Clausius inequality

𝑑𝑆 ≥ 𝑑𝑞

𝑇

The Clausius inequality implies that dS ≥ 0.

―In an isolated system, the entropy cannot decrease when a spontaneous change takes place.‖

Criteria for spontaneity

𝑑𝑆 −𝑑𝑞

𝑇≥ 0

In a system in thermal equilibrium with its surroundings at a temperature T, there is a transfer of energy as heat when a change in the system occurs and the Clausius inequality will read as above:

Criteria for spontaneity

When energy is transferred as heat at constant volume:

𝑑𝑆 −𝑑𝑞

𝑇≥ 0

*dq = dU

𝑇𝑑𝑆 ≥ 𝑑𝑈

At either constant U or constant S:

𝑑𝑆𝑈, 𝑉 ≥ 0 𝑑𝑈𝑆, 𝑉 ≤ 0

Which leads to 𝑑𝑈 − 𝑇𝑑𝑆 ≤ 0

Criteria for spontaneity

When energy is transferred as heat at constant pressure, the work done is only expansion work and we can obtain

𝑇𝑑𝑆 ≥ 𝑑𝐻

At either constant H or constant S:

𝑑𝑆𝐻, 𝑝 ≥ 0 𝑑𝐻𝑆, 𝑝 ≤ 0

Which leads to 𝑑𝐻 − 𝑇𝑑𝑆 ≤ 0

Criteria for spontaneity

We can introduce new thermodynamic quantities in order to more simply express 𝑑𝑈 − 𝑇𝑑𝑆 ≤ 0 and 𝑑𝐻 − 𝑇𝑑𝑆 ≤ 0

Helmholtz and Gibbs energy

Helmholtz energy, A:

A = U - TS

dA = dU – TdS

dAT,V ≤ 0

Gibbs energy, G:

G = H - TS

dG = dH – TdS

dGT,p ≤ 0

Helmholtz energy

A change in a system at constant temperature and volume is spontaneous if it corresponds to a decrease in the Helmholtz energy.

Aside from an indicator of spontaneity, the change in the Helmholtz function is equal to the maximum work accompanying a process.

Helmholtz energy

, useful

Variation of the Gibbs free energy with temperature

Variation of the Gibbs free energy with pressure

Variation of the Gibbs free energy with pressure

Homework

1. When 1.000 mol C6H12O6 (glucose) is oxidized to carbon dioxide and water at 25°C according to the equation C6H12O6(s) + 6 O2(g) 6

CO2(g) + 6 H2O(l), calorimetric measurements give ΔrHθ= -2808 kJ

mol-1 and ΔrSθ = +182.4 J K-1 mol-1 at 25°C. How much of this energy

change can be extracted as (a) heat at constant pressure, (b) work?

2. How much energy is available for sustaining muscular and nervous activity from the combustion of 1.00 mol of glucose molecules under standard conditions at 37°C (blood temperature)? The standard entropy of reaction is +182.4 J K-1 mol-1.

3. Calculate the standard reaction Gibbs energies of the following reactions given the Gibbs energies of formation of their components

a) Zn(s) + Cu2+(aq) Zn2+(aq) + Cu(s)

b) C12H22O11(s) + 12 O2(g) 12 CO2(s) + 11 H2O(l)

One for the road

Life requires the assembly of a large number of simple molecules into more complex but very ordered macromolecules. Does life violate the Second Law of Thermodynamics? Why or why not?