Stuff

---Tonight: Lecture 2 July 16

---Assignment 1 has been posted along with all the reading material that will support doing the homework.

---Presentation Assignment by Tuesday? We shall see.

--Thermo and more thermo. Problems only next Tuesday.

Thermodynamics Three Laws and a little calculus.

Dr. Richard Gross Chem 247.53Room 115

"A theory is the more impressive the greater the simplicity of its premises, the more varied the kinds of things that it relates and the more extended the area of its applicability. Therefore classical thermodynamics has made a deep impression on me. It is the only physical theory of universal content which I am convinced, within the areas of the applicability of its basic concepts, will never be overthrown." -- Einstein (1949)

My goal for thermodynamics is to review basic concepts, tough to understand concepts, but most importantly to apply concepts in the lab.

1. Isothermal titration calorimetry--applicable to nearly all biomolecule “binding studies”

--great for determining !U, !H, !G, Kbeq , !S in two hours or less--automatically.

2. Differential Scanning Calorimetry--great for heat capacity, Cp and enthalpy of conformational transitions in biomolecules, protein folding, DNA.

Thermodynamics is the study of energy interchanges between two or more equilibrium states.

Organic Compounds

Biosynthesis

Surroundings

Transport of Ions and

Molecules

Motility and Motion

Photosynthesis

Redox Reactions

HeatATPIon

Gradients

Reduced NADH

Solar Energy

The exchange of energy is necessary for life as we know it.

Thermodynamics considers the energetics between two equilibrium states, but says nothing about the rate of a chemical reaction (kinetics).

Time

Pote

ntia

l Ene

rgy

James Joule showed that mechanical work and heat are manifestations of the same thing, energy.

Joule found 1 cal heat = 4.1868 J work

amount of heat required to raise the temperature of 1g of water by 1˚C

W = Force x distance

The "Oth law" states that if two systems A and B are in thermal equilibrium with a third system C, they are in thermal equilibrium with each other (i.e. same Temperature).

The 0th law defines temperature.

One of the biggest pains in physical chemistry is keeping track of units. Blame the chemists for violating the SI unit convention. Watch pressure and energy terms.

system

energy out

energy in

Thermodynamic Formalism: A system exchanges matter and energy with its surroundings across a boundary.

+

-

surroundings

How Many Forms of Energy can you think of?

• Radiant energy comes from photons from the sun and is earth’s primary energy source.

• Electrical energy comes from charges moving through a potential difference

• Thermal energy is the energy associated with the translation of atoms and molecules.

• Chemical energy is the energy stored within the chemical bonds of substances.

• Nuclear energy is the energy stored within the mass (neutrons and protons) in the atom (E = mc2)

• Kinetic energy: energy associated with moving mass 1/2mv2

• Potential energy: energy available by virtue of an object’s position or height above a reference height.

• Mechanical energy: stored in spings, F x d

How Physicist Richard Feynman Described Energy

“There is a fact, or if you wish, a law, governing natural phenomena that are known to date. There is no known exception to this law — it is exact so far as we know. The law is called conservation of energy. It states that there is a certain quantity, which we call energy that does not change in manifold changes which nature undergoes. That is a most abstract idea, because it is a mathematical principle; it says that there is a numerical quantity, which does not change when something happens. It is not a description of a mechanism, or anything concrete; it is just a strange fact that we can calculate some number, and when we finish watching nature go through her tricks and calculate the number

again, it is the same.”—The Feynman Lectures on Physics

• Translational kinetic energy.• Molecular rotation.• Bond vibrations.• Intermolecular attractions.• Electrons.• Nuclear • Electrochemical

• Translational kinetic energy.

• Rotational energy

• Vibrational energy

• Electrostatic energy

Thermodynamic Postulate: A system has a bulk property that we call “internal energy”.

On a microscopic scale the internal energy is:

Surroundings

Surroundings

+q = heat added

+w = work done on the system

-q = heat lost from the system

-w = work done by the system

System!U

energy out

energy in + -

The 1st Law of Thermodynamics says that total energy of a system + surroundings remains constant—i.e. energy is neither created nor destroyed just transformed between the two boundaries.

!Usystem + !Usurroundings = 0

The internal energy, U can be changed only by exchanging heat (q) or work (w) with its surroundings.

It’s a state function!

Surroundings

Surroundings

+q = heat added

+w = work done on the system

-q = heat lost from the system

-w = work done by the system

System!U

energy out

energy in + -

Mechanical work

HeatInternal energy

Otherwork

∆U = q + w + w�

A process is an event in which a physical or chemical change causes a state property of the system to change. When a process occurs matter, heat or work must cross a boundary between the system and surroundings.

A system is defined when its variable of states (independent variables) are defined.

A path is a specific way to carry out a process from initial state through intermediate states and final state. Work and heat are both path functions.

A system is that part of the universe that exchanges matter and energy with its surroundings across a boundary.

Thermodynamic Vernacular & Definitions

The surrounding that part of the universe which is not the system.

There are three-classes of “systems” depending on how each exchanges matter and energy with its surroundings.

Closed System Isolated System

No matter exchangedNo heat or work exchanged

Matter not exchangedHeat or work exchanged

Matter

Diathermal Diathermal Adibatic

q q

Open SystemMatter exchangedHeat or work exchanged

There are 4- types of restricted paths typically imposed on a system---often with an experimental simplification in mind.

• Adiabatic Change: system is thermally insulated, zero heat exchange between system and surroundings. (q = 0)

• Isobaric Change: no change in pressure ( P = 0) from the initial state to the final state.

• Isometric Change: no change in volume ( V = 0) in the system from the initial state to the final state.

• Isothermal Change: no change in temperature ( T = 0) in system.

Ideal Gas Law!

“Equations of state” are experimentally-derived functions that exactly describe the bulk properties or “state” of a system. Here are two examples:

PV = nRT

P + (V – nb) = nRTan2

V2( )}

Pobserved

}

correction due to actual volume of

molecules

correctionfor

intermolecularattraction

forces

Vcontainer

Van der Waals

• Van der Waals corrects the ideal gas law for:

1. a accounts for intermolecular forces of attraction2. b accounts for volume of gas molecules

A state function is a function that has one unique value between some initial and a defined final state. The value of a state function does not depend on history or path taken between the two states.

!U = Ufinal - Uinitial

!P = Pfinal - Pinitial

!V = Vfinal - Vinitial

!T = Tfinal - Tinitial

!H = Hfinal - Hinitial

All the functions below are state functions or in math speak “exact differentials”

PE = mghh

0

Final State

Initial StateThe relevance and utility of a state function is that it has an infinite number of paths between two equilibrium states--but only one value characterizing it.

It’s a path integralno !!

We can write the first law of thermodynamics for a closed system in differential form. The symbols

This equation conveys that a small change in the internal energy can be caused by a small change in heat or a small change in work or both together.

The notation also says that U is an exact differential (i.e. a state function) and path independent while the partial signs indicate that q and w are not state functions and path dependent.

dU = ∂q + ∂w Infinitesimal changes

Large changes

�dU = ∆U = U2 − U1 = q + w

If we integrate dU we get the delta form. Get used to thinking this way.

• Heat q. Is the transfer of energy across the sys-surr boundary which manifests as a consequence of a temperature difference between system and surroundings. Heat “flows” from higher T to lower T.

• heat flows from a body at high T to a body at low T• heat arises from random motion of molecules• heat will cease to “flow” if there is no difference in Ts

• heat is an extensive quantity• heat manifests by an effect in the surroundings.

• we must look at the surroundings not at the system to identify what is happening with heat or work flow.

• Work, w. Energy which manifests as a consequence of a force x distance. It is not a state function (i.e. an exaxt differential).

• a change of state must occur to observe work• there are many forms of work that can be considered• both work and heat have convention of + if it results in a net

increase in internal energy of the system.

• work appears only at the boundry of a system• heat manifests by an effect in the surroundings.

There are many important types of work that are useful--but they r often omitted from the initial analysis.

All work is a directed force x displacement. If the F is in the direction of the displacement then we can forego keep it simple (no vectors)

Force is constant throughout.

Work = Force×Distance

= F∆x

Work = Force×Distance

=�

Area

Area

�F ×Distance

=�

F

A

�×A Distance

Pext is constant throughout. Note that the opposing pressure!

Mechanical PV Work: When external pressure, Pext is constant.

= −Pext

� V2

V1

dV = Pext∆V

y(x) is dependent on x (not constant) so we have to have the functional form of P(V) to integrate it. If not show stops!

W =� 10

0ax2dx

TYPES OF WORK: When work is not a constant or is a function over the path, we need the functional form.

=� V2

V1

−P (V )dV =

Work is a path function. The magnitude of work done from some initial state to a final state depends on the path.

work = w =� b

a−P dV

Pf

PiPiPi

PfPf

Vf Vf VfVi Vi Vi

f ff

i i i

The area underneath these curves = work is different for each path that is taken.

It is helpful to realize that for an ideal gas plotting P vs V at constant T and n gives a hyperbola for a ideal gas. Keep that curve in mind.

Pres

sure

, P

Volume, V

Common Restricted Paths In Thermodynamics

1. Free Expansion ==> P = Pext = 0 pressure

2. Constant Volume (Isochoric) ==> Vf - Vi = 0 = dV = 0

3. Expansion Against Constant P (Isobaric) => P = Pext = Constant

4. Reversible Expansion/Compression => Pext = Pin (ideal gas)

5. Ideal Gas use PV = nRT

6. Non-Ideal Gas use van der waals equation

7. Irreversible Expansion P = Pfinal = Constant

1. Free Expansion: P = Pext = 0 pressure

2. Constant Volume (Isochoric) ==> Vf - Vi = 0 = dV = 0

W =�−P (V )dV =

�−(0)dV = 0

W =�−P (V )dV =

�−P (V )(0) = 0

3. Expansion Constant P (Isobaric) => P = Pext = Constant

W =�−P (V )dV

=�−PextdV

= −Pext

� Vf

Vi

dV = −Pext(Vf − Vi)

5. Isothermal reversible Expansion/Compression => Pext = Pin with dn and dT = constant

This case represents an infinitesimal change that is conceptual only. Pext = Pint.

It does not mean Pext = constant--instead it will follow an equation of state like the idea gas

Same mechanics as before: Start with:

W =�−P (V )dV P =

nRT

V

=�−PextdV =

� Vf

Vi

nRT

VdV = nRT

� Vf

Vi

1V

dV = −nRT lnVf

Vi

use ideal gas, T constant

6. Non-Ideal Gas use van der waals equation isothermal

�P +

an2

V 2

�(V − bn) = nRT

P =nRT

(V − bn)− an2

V 2

This is a homework problem

W =�−P (V )dV =

�−PintdV =

rearranging

same basic set-up

using ideal gas lawconstant n and T

same basic set-up

7. Ideal Isothermal Irreversible Expansion P = Pfinal = ConstantThis case represents Pext = constant Same mechanics as before: Start with:

W =�−P (V )dV

=�−PextdV

= −Pext

� Vf

Vi

dV = −Pext(Vf − Vi)

= −nRT

Vf(Vf − Vi)

Isothermal Compression of An Ideal Gas

Reversible Limit

Irreversible Limit

Closer to Reversible

Notice how the area under the curve (work) is larger in irreversible processes and reaches a

minimum when Pext = Pin = Ideal Gas

=�−PextdV =

� Vf

Vi

nRT

VdV = nRT

� Vf

Vi

1V

dV = −nRT lnVf

Vi

Reversible Isothermal Expansion of An Ideal Gas

Reversible Limit

Irreversible Limit

Closer to Reversible

Notice how the area under the curve (work) is smaller in the irreversible processes and reaches

a maximum when Pext = Pin = Ideal Gas

=�−PextdV =

� Vf

Vi

nRT

VdV = nRT

� Vf

Vi

1V

dV = −nRT lnVf

Vi

Reversible processes get the most out for the least in (Maximum Work for expansion minimum work from compression.)

--If the surroundings does work on the system, it takes the least work to compress it.

--If a system expands it does the most work on the surroundings

To get the big change integrate the itty bitty.

We can write the the first law in terms of differentials.

for itty bitty changes

� f

i∂w = w �= wf − wi heat and work are path

functions so we have to integrate over the specific path. These paths must be known or given!!

� f

i∂q = q �= qf − qi

∆U = q + w = q − P∆V for large changes

dU = ∂q − ∂w

�dU = ∆U = Uf − Ui

Let’s Play with the first law closed system and calculus

Calculus tells us we can also write dU like this:dU = ∂q − PdV

dU =�

∂U

∂T

�

V,ni

dT +�

∂U

∂V

�

T,ni

dV

This term is important! It’s is the heat capacity at constant volume! Let’s review heat capcity.

This is the total differential

Observation: The amount of heat, q, transferred from an object at higher temperature to an object at lower temperature is proportional to the difference in temperature of the two objects. In math terms we write:

T1 T2

Object 1 Object 2Heat q

ThermometerThermomeTh

Thq α ∆T

q = C∆T

C =q

∆T

The heat capacity (C) of a substance is the amount of heat (q) required to raise the temperature of a quantity of the substance by one degree Celsius or Kelvin (units of J/˚C or J/K or cal/˚C). It is an extensive quantity.

Heat CapacityC =

q

∆T

Don’t forget our notation: big changes are represented by q and small changes as "q. Also, the partial derivative means that heat is path-dependent--we need to have knowledge of the process variables or a function describing q. We write C(T) just in case C is not a constant. It might be and if it is between T1 and T2 we can pull it outside the integral sign.

C =∂q

dTq =

� T2

T1

C(T )dT

Molar Heat Capacity C =

C

n= Cn

where n is the number of moles of substance. We can normally look these up in a table.

Units: J/KHeat capacity of gases depends on P and V we define:

Heat Capacity Constant Volume

from large change ===> small change

Cp =�

∂qp

∂T

�

P,n

Cp =qp

∆T=

∆H

∆T

Cv =�

∂qv

∂T

�

V,n

Cv =qv

∆T=

∆U

∆T

Heat Capacity Constant Pressure

qp =� T2

T1

CpdTqv =� T2

T1

CvdT

KE gives this

KET and the equipartition of energy links internal energy and heat capacity, C of ideal gases.

Cv =�

∂qv

∂T

�

V

=

∂(

32nRT )

∂T

V

=32nR

U =32nRT

IMPORTANT: This equation says that the internal energy of a gas only depends on T and nothing else!

We can take the derivative and link the heat capacity to bulk properties

U =�

∂qv

∂T

�

V

n moles of monoatomic gas

U =32RT

U =32RT +

22RT + (3N − 5)RT

U =32RT +

22RT + (3N − 6)RT

(linear molecule)

(non-linear molecule)

(monoatomic)

Cv =32R +

22R + (3N − 6)R

Cv =32R +

22R + (3N − 5)R

Cv =32R

(linear molecule)

(non-linear molecule)

(monoatomic)

U

Cv

We can include the equipartition of energy to include:

Show that the heat capacity at constant volume, Cv are the given values on the previous slides.

Key Linkage to the First Law of Thermodynamics

dU = ∂q − PdV

dU =�

∂U

∂T

�

V,ni

dT +�

∂U

∂V

�

T,ni

dV

∂qv = CvdT =�

∂U

∂T

�

V,n

First Law Differential Form

Total Differential

dU = CvdT +�

∂U

∂V

�

T,ni

dV Total Differential

The power of thermodynamics is in general expressions combined with the calculus. Let’s review as this is where the comfort zone normally begins and ends.

--Derivatives and integrals of single variables.

--Derivatives of functions with 2 or more variables.

--The differential of a multi-variable function.

--Practice using these tools like you would a screwdriver, socket wrench and a chain saw. Only then will it be your tool to use.

A Derivative Example: Reduction of Br2 to Bromide

Br2(aq) + HCOOH (aq) 2Br- (aq) + 2H+ (aq) + CO2 (g)

time

393

Br2(aq)

t = 0

393 nm lightDetector

Br2(aq) 2Br-

t = ti

We can easily monitor the change in [Br2] with an lab instrument.

Time (s) [Br2] (mM)

0.0 12.0 mM50.0 10.0 mM100.0 8.46 mM150.0 7.10 mM200.0 5.96 mM250.0 5.00 mM300.0 4.20 mM350.0 3.55 mM400.0 2.96 mM

Br2(aq) + HCOOH (aq) 2Br- (aq) + 2H+ (aq) + CO2 (g)

Suppose we monitor the color change and we plot [Br2] vs Time (s) as shown below:

2.0

4.5

7.0

9.5

12.0

0 50 100150200250300350400

Time (seconds)

[Br 2

] (m

illiM

olar

)

average rate of disappearance

![Br2]!t

= -[Br2]final – [Br2]initial

tfinal - tinitial= -

Time (s) [Br2] (mM) ![Br2]

0.0 12.0 mM50.0 10.0 mM100.0 8.46 mM150.0 7.10 mM200.0 5.96 mM250.0 5.00 mM300.0 4.20 mM350.0 3.55 mM400.0 2.96 mM

2.00

1.54

1.36

1.14

0.96

0.80

0.65

0.59

AvgRate

0.04

0.031

0.027

0.023

0.019

0.016

0.013

0.01

Just like driving in a car and sometimes going fast, then slow, then fast again---the average rate of a chemical reaction also varies over time!

2.0

4.5

7.0

9.5

12.0

0 50 100 150 200 250 300 350 400

Time (seconds)

[Br 2

] (m

illiM

olar

)

![Br2]

!t

The average rate of disappearance of Br2 is the slope of the line between any two points on the curve (plot of [] vs time). There are an infinite number of averages.

[Br2]final – [Br2]initial

tfinal - tinitial

rate = -

2.0

4.5

7.0

9.5

12.0

0 50 100 150 200 250 300 350 400

Time (seconds)

[Br 2

] (m

illiM

olar

)

!t

![Br2]

The instantaneous rate of disappearance of Br2 is the slope of the line tangent at any point along the curve.

= -[Br2]t+#t – [Br2]t

!t

Instantaneous Rate of ChangeSlope of the tangent line at a point

A derivative of a function, f(x) is just another function that when evaluated at a point gives the slope of the line at that point.

Plot of f(x) = y(x) = x2

f�(x) =d

dx(x2) = 2x

instantaneous rate of change at a point P(x0, y0)

average rate of change between two points Q and P

P(x,f(x))

Q(z,f(z))

x z

z - x = h

h

slope =rise

run=

f(z)− f(x)z − x

slope =rise

run=

f(x + h)− f(x)h

f(x,y)

x

y

f �(x) = limh→0

f(x + h)− f(x)h

f �(x) = limz→x

f(z)− f(x)z − x

f(x,f(x))

f(z,f(z))

x zh

There are an infinite number of “average rates”. The derivative gives the “instaneous rate” at a particular point, x. It is line tangent to the point where we want to take the rate.

f(x,y)

x

y

A function

f’(x)

f(x)

It’s derivative

We can map the slopes to a different graph.

f�(x) =d

dx(3x2) = 6x

Graph of f(x) = y(x) = 3x2

Graph of f(x) = y(x) = x2

f�(x) =d

dx(x2) = 2x

y = 3x2

y’(x) = Slope = 6x

y’(1) = 6(1) = 6

y = x2

y’(x) = Slope = 2x

y’(1) = 2(1) = 2

Rules of Derivatives: Basically Mechanical Process

d

dx(c) = 0

d

dx(ax) = a

d

dx(xn) = (nxn−1)

d

dx[f(x) + g(x)] =

d

dx[f(x)] +

d

dx[g(x)]

the derivative of a constant is 0.

the derivative of x times a constant is the constant.

the derivative of a sum is the sum of derivatives

bring down the power and subtract 1. The power rule.

d

dx[f(x)g(x)] = g(x)

d

dx[f(x)] + f(x)

d

dx[g(x)] the derivative of a product is the first

times the derivative of the second plus the second times the derivative of the first.

d

dx[f(x)/g(x)] =

g(x)d

dx[f(x)]− f(x)

d

dx[g(x)]

[g(x)]2the derivative of a quotient is the first times the derivative of the second plus the second times the derivative of the first over the denominator squared.

Rules of Derivatives: A Few More That You Will See

d

dx(ex) = ex

d

dx(lnx) =

1x

d

dx(ax) = ax ln a

d

dx(loga x) =

1x ln a

Some Derivative Practice

y(x) = x3 − 13x5

+ 2√

x− 3x

+1− 2x

x3

y(x) = 6x4 − 7x3 + 2x + 8

y(x) =�

3x− 1x2 + 3

�2

y(x) =(2x− 3)1/3

(√

4x− 9)

Some Common Integral Properties

Common Integrals to Have At Hand We can also have functions of two variables: z(x,y). This function, z(x,y) is a rule that assigns to each ordered pair (x,y) of real numbers some unique set of real numbers f(x,y) for its domain.

xy

z

z(x, y) = 2x2 + 2y2 − 4

It a 3 d surface.

The partial derivative, fx or fy, is the slope of a tangent line at a point f(x,y). We view this by fixing either x or y (infinite number of ways to do this) and looking at the rate of change of the other variable.

We fix either x or y and looking at the rate of change of the other variable. Look at the two red tangent lines showing fx and fy.

Tangent line fx(x0, y0)

y

x

z = f(x,y)

y = y0(x0, y0)

Plane at y0

Plane at x0

Tangent line fy(x0, y0)

Curve z = f(x0, y)

Some Partial Derivative Practice Some Partial Derivative Practice

Some Partial Derivative Practice The total differential of a function gives the total infinitesimal change in a function through the combined change of each of its independent variables.

Suppose we are given f(x,y) which is well behaved.

We can write the total differential of the function f(x,y)

df(x, y) =�

∂f

∂x

�

y

dx +�

∂f

∂y

�

x

dy

When !x is small the approximation distance CT (= dy) better approximates CB (!y). Line AT represents the tangent to the curve at point A.

The total differential dy of the function y = f(x) is used as an approximation for a change !x. Here’s the geometric interpretation.

!y = CB = f(x + !x) - f(x)

dy = CT = f '(x)!x .

x

C

B

A

∆x

T

In words if we know the derivative of a function, f’(x) and we change x by some amount !x then we can use dy as a good guesstimate of what actually happens to !y (compare distance CT to distance CB).

x

!y = CB = f(z) - f(x)

z

dy = f�(x)∆x =dy

dx∆x

y = f(x)

y = f(x)y’ = f’(x)

Write the total differential of for the function P(n,V,T)?

Write the total differential of for the function:z = x2 - 3xy2.

df(x, y) =�

∂f

∂x

�

y

dx +�

∂f

∂y

�

x

dy

df(x, y) =�

∂f

∂x

�

y

dx +�

∂f

∂y

�

x

dy

Write the total differential of for the equation of state of an ideal gas P(n,V,T)?

Write the total differential of for the function:z = x2 - 3xy2.

It is convenient from an experimental point of view to define a new state function: enthalpy, H.

Enthalpy = !H = qP = !U + P!V

Path 2

!U = qP - P!V

At constant pressure: !P=0

qP = !U + P!V

!U = qV - P!V

At constant volume: !V=0

!U = qV

Path 1

Let’s write the enthalpy in terms of its total differential using T and P as the natural variables.

!H = qP = !U + P!V

dH =�

∂H

∂T

�

P

dT +�

∂H

∂P

�

T

dP Total Differential

Large changes

At dP = 0dH = ∂qp =�

∂H

∂T

�

P

dT

By definitionCp =�

∂qp

∂T

�

p

=�

∂H

∂T

�

P

dH = CpdT +�

∂H

∂P

�

T,ni

dP Recasting

Chemists measure the enthalpies many chemical reactions, give them names and tabulate these values in Handbooks. They are useful in the real-world.