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# Classical Physical Chemistry Sets the Stage

Shaun Williams, PhD

## The Classical Properties of Matter

### What do we measure?

• We measure few things directly
• We can directly measure time and distance
• How do we measure mass?
• By how far a spring in a scale is compressed
• How do we measure temperature?
• By how far the mercury rises

### Which is ice and which is diamond?

• By observation they appear the same

### Which is ice and which is diamond? Can we tell by mass?

• For equal volumes of each, the two substances will compress the spring different amounts

### Which is ice and which is diamond? Can we tell by heating the samples?

• The ice will melt at a lower temperature.

### Classical Parameters - Volume

• Everything occupies three dimensions in space
• We can measure the distance occupied along each coordinate axis to determine the volume
• SI unit of volume is m3
• Good for buildings and bulldozers
• $$1\,\mathrm{L}\equiv 10^{-3}\,\mathrm{m}^3$$ and $$1\,\mathrm{mL} = 1\,\mathrm{cm}^3 = 10^{-6}\,\mathrm{m}^3$$

### Classical Parameters - Mass and Moles

• Classical chemistry established that atoms of the same element share the same (or nearly the same) mass
• Relation of mass of a substance to number of particles is the mole (defined by Avogadro's number) $$1\,\mathrm{mol} = 6.022\times 10^{23}$$
• Number of moles of particles of mass $$M$$ is $$n=\frac{M}{\mathcal{M}}$$
• $$\mathcal{M}$$ is the molar mass ($$\mathrm{g}\,\mathrm{mol}^{-1}$$)

### Classical Parameters - Pressure

• Explanation of how matter changes requires force
• Force is proportional to the application area
• Measuring a system's pressure will provide the force by $$P=\frac{F}{A}$$
• SI unit: Pascal (Pa)
• $$1\,\mathrm{bar} = 10^5\,\mathrm{Pa}$$
• $$1\,\mathrm{bar}$$ is about 2% of typical atmoshperic pressure at sea level

### Classical Parameters - Temperature

• Usually related to the average speed of particles
• If ice is surrounded by air, the air molecules are traveling faster than the ice molecules
• Collision between the faster air molecules with the ice molecules transfers energy to the ice, thereby warming it up

### Measuring Temperature

• Temperature was first measured by its ability to cause mercury to expand
• Or to alter the voltage in of certain circuit components
• Typically we use Celsius scale in the lab
• In P Chem we nearly always use Kelvin where $$T(\mathrm{K}) = T(^\circ \mathrm{C}) + 273.15$$
• This is also called the absolute temperature scale

## Thermodynamics

### What is thermodynamics?

• Thermodynamics is the study of the conversion of energy among different forms and the transfer of energy between different systems
• Chemists are interested in molecular structure and how to change it
• They merge in chemical thermodynamics
• Probe the flow of energy during the transformation of reactants to products

### Energy

• Energy is divided into three classes
• Kinetic
• Potential
• Energy is the capacity to apply a force or (equivalently) to oppose an existing force
• In this course we will neglect the conversion of mass into energy

### Zero Energy

• The minimum possible energy at which all the particles in the system are
1. Fully assembled
2. Present in whatever container we have constructed for them
• This energy is called internal energy of the system
• Experimental properties typically only measure changes in energy, $$\Delta E = E_f - E_i$$

### The First Law

• Divides the change of energy into two contributions
• Work, $$w$$, is the energy change that drives a net shift in the distribution of mass in the system
• Heat, $$q$$, is any other energy change – in particular the energy associated with changes in the random motions of the individual particles in the system
• Mathematically: $$\Delta E \equiv E_f-E_i =q+w$$

### Signs

• For work, $$w$$
• Positive work means work is done to the system, increasing the samples energy
• Negative work means work is done by the system
• For heat, $$q$$
• Positive heat means heat is entering the system
• Negative heat means heat is leaving the system

### A Steam Example

• Let’s say we want to get work out of the steam in a pipe might want to raise the temperature of the steam by heating the pipe
• Some of that energy would instead by diverted into expanding the volume of the gas a piston at one end of the pipe
• If we calculate $$\Delta E$$, we need to include the heating of the gas and the work done by the expansion

### A Steam Example, Signs

• The gas is heated so $$q$$ is positive (heat is entering the system)
• The gas does work so $$w$$ is negative
• Therefore, the total change in energy, $$\Delta E$$, is less than the energy needed to heat the steam $$\Delta E = q+w < q$$
• Because of this difference, we calculate the change in a new variable called enthalpy, $$H$$

### Enthalpy

• At constant pressure, $$\Delta H=q$$
• If the pressure changes then the energy needed to do the work changes and then enthalpy isn’t valid

### Back to the steam pipe

• Some of the input energy goes into warming the piston rather than pushing it forward
• Josiah Willard Gibbs invented a way of looking at systems, leading to the Gibbs free energy, $$G$$
• Gibbs and others found a way to define the maximum amount of work obtainable other than by expansion
• Defined at constant temperature and pressure

### Blocks of diamond and ice

• Add energy - the diamond’s temperature increases
• Add energy – the ice’s temperature stays the same as long as there is both water and ice
• Clearly, energy and temperature are not the same phenomenon
• There is clearly some other variable that connects energy and temperature

### Entropy and the Second Law

• Rudolf Clausius formulated a relationship between heat and temperature
• Based on a new parameter he called entropy
• Clausius realized that the heat used to drive an engine is always greater than the useful work the engine could do
• The difference between input heat and the work done as a heat transfer to the surroundings, $$q_{surr}$$

### Defining Entropy

• Lost energy tends to grow in proportion to the operating temperature
• Clausius defined entropy, $$S$$, as $$\Delta S_{surr} = \int \frac{q_{surr}}{T}$$
• $$\Delta S_{surr}$$ is the change in entropy of the surroundings
• After a complete cycle, an engine would be in its initial state but the surroundings would be warmer

### Alternative Entropy Definition

• Entropy is central to thermodynamics
• It explains how temperature and energy are different
• Clausius defined entropy in terms of the independent variable temperature
• Alternatively we can let entropy be independent $$T \equiv \left( \frac{\partial E}{\partial S} \right)_{V,n}$$

### Explanation

• Understanding $$T \equiv \left( \frac{\partial E}{\partial S} \right)_{V,n}$$
• $$\partial$$ is like a tiny $$\Delta$$
• So we have a tiny (infinitesimal) change in energy on top
• On the bottom is a tiny change in entropy
• Ratio these two tiny changes
• This ratio is temperature when the volume and number of moles are held constant

### Melting Ice

$$T\equiv \left( \frac{\partial E}{\partial S} \right)_{V,n}$$

• As we begin heating the ice, the energy rises much faster than the entropy
• The equation tells us that this means $$T$$ is rising
• At the melting point, suddenly the energy and entropy are rising at the same rate
• The equation tells us that this means $$T$$ is not changing

### Another Clausius realization

• From numerous measurements, Clausius realized that $$\Delta S_T \ge 0$$
• The change in the total (overall) entropy can never increase
• $$\Delta S_T=0$$ is reserved for an ideal case called a reversible process
• Basically this means that there is always some inefficiency in real systems

## The Kinetic Theory of Ideal Gases

### Towards our modern molecular theory of matter

• The kinetic theory of gases imagined that gases are
• Independent particles
• Randomly moving
• Ideal gas is a system of point masses that interact with only the walls of the container
• Interactions with the walls are purely elastic collisions

### The Maxwell-Boltzmann Distribution

• James Clerk Maxwell began with the idea that molecular motions are random
• Work by botanist Robert Brown – pollen particles in water
• Results of random events tend to cluster around an average

### Example: Flipping 50 coins

• About 11% of the time we actually get the average
• Less probability away from the average
• This is holds for all random events
• The distribution of random events is a Gaussian function, a function of the form $$e^{-x^2}$$
• Our coin flipping curve is fit by $$\sqrt{\frac{2}{\pi N}} e^{-\frac{2\left(h-\expect{h}\right)^2}{N}}$$
• $$h$$ is the number of time we get a head
• $$\expect{h}$$ is the mean value of $$h$$
• $$N$$ is the number of coin flips

### Derivation of Maxwell-Boltzmann Distribution

• Chapter 1 contains the original derivation of the distribution equation
• This is not a short derivation.
• It is covered on pages 50 – 53 of the textbook
• Rather than going through it all, I will summarize it

### Summary of Maxwell-Boltzmann Distribution Derivation

• Maxwell assumed a Gaussian distribution for the particle velocities, $$v_x$$, $$v_y$$, and $$v_z$$
• To eliminate the dependence of direction, we integrated the distribution over all angles
• We solved for a constant $$a$$
• Normalized the resulting function $$\mathcal{P}_v(v)=4\pi \left( \frac{\mathcal{M}}{2\pi RT}\right)^{\frac{3}{2}} v^2 e^{-\frac{\mathcal{M}v^2}{2RT}}$$

### The Ideal Gas Law

• Reflections are totally elastic
• These reflection allow us to define the forces acting in the box
• The time-averaged force $$F_1$$ exerted on wall 1 by a single particle with mass $$m$$ traveling along the $$x$$ axis at speed $$v_x$$ is balanced by an equal and opposite force on the particle of $$–F_1$$
• This yields: $$-F_1=ma_x=m\frac{dv_x}{dt} \approx m\frac{\Delta v_x}{\Delta t} = 2m\frac{-v_x}{\Delta t}$$
• $$\Delta t$$ is the time of a single round trip between the walls

### Continuing with the force

• Remembering that the box is $$a$$ long in the $$x$$ axis $$\Delta t = \frac{2a}{v_x}$$
• Plugging that in we find $$F_1 = 2m\frac{v_x}{\Delta t} = 2m\frac{v_x}{\bfrac{2a}{v_x}} = \frac{mv_x^2}{a}$$
• To extend this to the entire gas $$\expect{F} = N\expect{F_1} = N \expect{\frac{mv_x^2}{a}} = \frac{Nm\expect{v_x^2}}{a}$$

### Working with the velocity part of the expression

• The overall mean squared speed, $$\expect{v_x^2}$$, can be separated $$\expect{v^2} = \expect{v_x^2+v_y^2+v_z^2} = \expect{v_x^2} + \expect{v_y^2} + \expect{v_z^2}$$
• The motion on each of the axes should be exactly the same so all these means are equal, thus $$\expect{v^2} = 3\expect{v_x^2}$$
• It turns out that for ideal gases $$\expect{v^2} = \frac{3RT}{\mathcal{M}}$$

### Continuing on...

• Combining out equations $$\expect{F} = \frac{Nm\expect{v_x^2}}{a} = \frac{NmRT}{\mathcal{M}a} = \frac{n\mathcal{M}RT}{\mathcal{M}} = \frac{nRT}{a}$$
• Now we can solve for pressure $$P=\frac{\expect{F}}{A} = \frac{nRT}{aA} = \frac{nRT}{V}$$
• This yields our well known ideal gas law $$PV = nRT$$

### Another Maxwell-Boltzmann Distribution

• We can write the Maxwell-Boltzmann equation in terms of kinetic energy of the molecules rather than molecular speeds $$\mathcal{P}_v(v) = 4\pi \left( \frac{\mathcal{M}}{2\pi RT} \right)^\bfrac{1}{2} \left(\frac{E_m}{\pi RT}\right) e^{-\bfrac{E_m}{RT}}$$
• $$E_m$$ is the molar energy

### Chemical Equilibrium

• At equilibrium, the concentrations of products and reactants does not change
• The balance between the reactant and product concentrations is determined by the rate of motion of the atoms $$\frac{\conc{products}}{\conc{reaectants}} \approx \frac{e^{-\bfrac{E_{prod}}{RT}}}{e^{-\bfrac{E_{reac}}{RT}}} = e^{-\bfrac{\left( E_{prod} - E_{reac} \right)}{RT}} = e^{-\bfrac{\Delta_{rxn}E^\ominus}{RT}}$$
• $$\Delta_{rxn} E^\ominus$$ - change in energy from react. to prod.

### Other Conditions

• If the reaction is allowed to exchange heat and work with its surroundings then it is the change in the Gibbs free energy, $$\Delta_{rxn}G^\ominus$$, that determines the balance between reactants and products
• For the equilibrium $$\chem{A\rightleftharpoons B}$$, we express the equilibrium constant, $$K_{eq}$$, as $$K_{eq}=e^{-\bfrac{\Delta_{rxn}G^\ominus}{RT}}\approx \frac{\conc{B}}{\conc{A}}$$

### Time-Dependent Chemical Reactions

• Without waiting for equilibrium we can study a reaction
• The rate a reaction occurs is given by the rate law $$\chem{A\rightarrow B}\text{ yields }Rate=-\frac{d\conc{A}}{dt}=k\conc{A}$$
• $$k$$ is the rate constant
• The rate constant obeys the Arrhenius equation $$k=Ae^{-\bfrac{E_a}{RT}}$$
• $$E_a$$ is the activation energy

### Final Thoughts

• In equation after equation we find these exponential expression.
• The expressions pop up when studying different situations
• Why? How are they all connected?
• It is the job of physical chemistry to discover the how they are linked and use the links to improve our understanding of the nature of matter

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