Reviewer Physics II

8/13/2019 Reviewer Physics II

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Kinetic Theory of Gases

Atomic hypothesis

All things are made of atoms little particles that move around in perpetual motion, attractingeach

other when they are a little distance apart, but repellingupon being squeezed into one another.

What happens if there were no attraction and repulsion?

If there was no attraction, there would be no objects in the first place because particles would

not come together and form something. If there was no repulsion, everything would just mix up - there

would be no identity. This is the reason for the obvious rule that no two objects can occupy the same

space at the same time.

What causes attraction and repulsion?

All particles have mass and charges. The effect of mass is to attract other masses. The effect of

charge is to attract the opposite charge and repel the same charge. The effect of gravity is much weaker

than the electromagnetic forces. Gravity only dominates for very large aggregation of particles (e.g.sun). At the smallest scales, electromagnetic force dominates. Imagine how varied the properties of

elements are when, in fact, their only difference is in the number and arrangement of electrons and

protons.

A lot of physical phenomena can be explained by atomic hypothesis

The fragrance of flowers are knocked out flower particles by the ever-moving air particles, and

which are brought to our noses.

The mercury in a thermometer rises because the mercury atoms jiggle more when the

thermometer is in contact with a hot object.

Laws of Boyle, Gay-Lussac and Charles

Ideal Gas Law

:= pressure := volume := temperature := particles per unit volume := Boltzmann constant

Kinetic Theory: Deriving the ideal gas law using Newtons Laws

Collection of particles in constant motion Collisions are perfectly elastic (e.g. collision of billiard balls) Spaces between particles are much greater than the size of the particles (Why?)


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Can the kinetic theory be used to explain why the mercury in a thermometer rises with temperature?

No, because the mercury in a thermometer is a liquidthe particles are still close enough with each

other and have enough interaction.

Pressure of a Gas

Gas exerts pressure because of its (perpetual) motion.

What does positive work mean?

Positive work means that the work is done BY the system.

Quantifying the Force F

Force is the change in momentum per second of the particles Two steps: change in momentum of ONE particle times the number of collisions per second.

Why 2?

Apply conservation of momentum to get the total momentum gained by the piston upon collision with

one particle.

Those within the distance = vx t will hit the piston in time t. The volume occupied by those atoms will be

A vx t. The number of atoms that will actually hit IN TIME t is n vx t A (n is the number of particles PER

VOLUME). So the number of atoms that will hit the piston per 1 sec is n vx A.

Does the equation make sense?

Yes, because the larger the n, more particles will exert force, and with higher velocities, they will impart

greater momentum. The larger the area, more particles will be able to collide at once.

Quantifying the Pressure P

Not all particles have the same . Half of the particles have , half have . Therefore, .


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Why?

The equation shows that we are able to relate a MACROSCOPIC QUANTITY (such as pressure)

with a microscopic quantity such as kinetic energy of a particle. Note also that certainty is

abandoned because we do not assign to each particle a specific kinetic energy; we just need

their average.

Nothing special about the x direction: so that

Pressure can be known given the speeds of the gas.

Pressure and Internal Energy

where is the total number of gas particles

where is the internal energy (energy contained in the system)Is the equation true for particles capable of vibration and/or rotation?

If the gas is compressed, what happens to the internal energy?

No, because the average kinetic energy pertains to TRANSLATIONAL kinetic energy only. The particles

are therfore assumed to be point objects (e.g. monoatomic gases)

The internal energy increases is the gas is compressed because the particles hit the walls of the piston

more often.

Adiabatic compression

All the work done in compressing the system goes into changing the internal energy of thesystem (translational kinetic energy for point particles)

for monoatomic gasses Non-adiabatic: because when compression is done fast, some of the heat is lost to the

environment?


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Temperature and kinetic energy

What happens when we leave the figure above on its own?

In its current state, the atoms on the right are contained in a smaller volume, there are more collisions

on the wall so that the kinetic energies of the atoms are higher on average than those of the atoms on

the left. When left alone, the piston will then move to the left. If in case it moves too much to the left,

then the opposite will happen: the atoms on the left will have higher kinetic energies on average so that

the piston moves to the right. After some time, the piston will reach an equilibrium, where the

average kinetic energies of the particles on both sides are the same.

Temperature is defined as something that is proportional to the kinetic energy that two bodies in

contact reached. When you dip your hand in a basin of cold water, initially you will feel cold, but after

sometime your hand adjusts to it because the energetic particles of your hand have collided with the

less energetic particles of water. At this point, the water and the hand have reached the same

temperature. Therefore, temperature is not a property of one object. Its a property of the objects in

contact which reached equilibrium.

Where did the Boltzmann constant come from?

Heat versus Temperature

Temperature is a property that two systems in contact share as they reach equilibrium (i.e. they have

the same average kinetic energy).

Heat is the energy that one system with an initially higher average kinetic energy imparts on the other

system with an initially lower average kinetic energy as their particles collide.

What does a thermometer measure?

Is a temperature difference necessary for heat to flow?

A thermometer measures the temperature of both the person and the thermometer. So the measured

temperature of the body is actually less than the actual temperature of the body before being in contact

with the thermometer, because somebody particles have already collided and imparted a little of their

energy to the mercury atoms (i.e. a little heat has transferred from the body to the thermometer).

Yes, if there was no temperature difference, the collisions between the systems would not change the

kinetic energy of the particles on average.

Deriving the Ideal Gas Law


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Laws of Thermodynamics

Steam engines: using heat to produce motion

Why is a temperature difference needed?

A temperature difference is needed because if the temperature were the same during expansion and

compression, the NET work done would be zero; machine is useless.

Other engines are basically steam engines: in automobiles, combustion of hydrocarbons; in nuclear

power plants, fission of radioactive materials.

Thermodynamics began with the attempt of Sadi Carnot to investigate the maximum efficiency of

engines.

Two ways to change the energy of the system:

1. Heat (due to temperature difference)2. (Mechanical) Work

What does a positive/negative heat mean? What does a positive/negative work mean?

Positive heat/work means that energy is ADDED to the system. Negative heat/work means that energy is

TAKEN AWAY from the system.

First Law of Thermodynamics

Conservation of Energy: if one has a system and puts heat into it, and does work on it, then its internalenergy is increased by the heat put in and the work done:

Why does the First Law of Thermodynamics imply that the energy of the world is constant?

What is if the system is isolated?The First Law of Thermodynamics imply that the energy of the world is a constant because whatever

energy that is gained/lost by the system is lost/gained by the environment.

If the system is isolated, no exchange of energy with the environment is allowed. Therefore, the change

in internal energy is zero.

Many ways of stating the Second Law of Thermodynamics

Heat cannot flow spontaneously from cold to hot (e.g. when a cold hand touches a hot mug, themug does not get hotter and the hand does not get colder)


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Heat cannot be taken in at a certain temperature and converted into work with no other changein the system or the surroundings. (e.g. heat from car engines propels the wheels, but in the

process dumps heat and sound to the environment)

The entropy of the world is always increasingSecond Law of Thermodynamics

Not all processes are allowed by Nature.

When an apple hits the ground, the heat from the ground cannot be harnessed to propel the apple

again upward.

A room during a hot summer day does not get colder without an air-conditioner.

A broken mug does not get whole without exerting effort to put the pieces back together.

Carnots POSTULATE:Heat cannot be all converted to work.

If heat is to be used to do work, as in steam engines, it must dump waste heat to the environment.

Work can be converted easily to heat (e.g. rubbing of hands) but heat cannot all be converted to work.

By experience, all machines heat up as they operate. Hence, some input heat is wasted.

If not all heat can be converted to work, then what could be the maximum efficiency of an engine?

Reversible engines: the ideal engine

Reverse heat flow is impossible if the difference in temperature is finite, but if one makes sure that heat

flows always between two things at essentially the same temperature, with just an infinitesimal

difference to make it flow in the desired direction, the flow is said to be reversible.

Can reversible engines exist in reality?

Analogy: imagine a ball on a plane with friction. If we push the ball to the left, the planes surface is

agitated and heat is expelled. If we return the ball to its initial position, the total work done on the ball is

zero, but the planes surface is even more agitated. The system (ball) is the same but there is change in

the environment. But if the plane is friction less, the process of pushing the ball around can be reversed

so that there will be no change in both the system and the environment.

No, because the concept of infinitesimal is only an idealization. But, nonetheless, the concept of

reversibility is still useful as it is the best all engines could ever be. Analogy: Getting a 100 is (almost) an

idealization. But it is nonetheless useful to know that, at best, one gets a 100 and nothing more. Thus,

no unnecessary effort would be wasted in trying to get a 101 because it is impossible.

Carnot cycle

Is the Carnot engine a reversible engine?

Heat reservoir is an object wherein heat can be taken in or out without affecting the temperature.


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All expansion and compression processes are done infinitesimally slow so that the reversibility is

preserved.

Step 1 Isothermal expansion at T1: The chamber is brought in contact with a heat reservoir at T1; heat

Q1 flows in and the chamber expandswork is done by the system.

Step 2 Adiabatic expansion from T1 to T2: The chamber is isolated and expanded; temperature drops toT2.

Step 3 Isothermal expansion at T2: The chamber is brought in contact with a heat reservoir at T2; heat

Q2 flows out and the chamber compresseswork is done by the environment.

Step 4 Adiabatic compression from T2 to T1: The chamber is isolated and compressed; temperature rises

to T1.

NET work is derived from machine and because machine goes back to initial state, the process can

indefinitely be repeated.

No other engine is more efficient than a Carnot engine.

Suppose that given T1 and T2, a reversible engine A does work W and an arbitrary engine B does work

W. The implication of W>W is that we can extract heat from heat bath 2 and convert that to work with

no change in the system or surrounding (no heat expelled). Hence W>W is a violation of Carnots

postulate!

Now suppose that engine B is also reversible. Then W should be greater than W. The amount of work

obtained to deliver heat from T1 to T2 is not dependent on the design of the engine. This is a property of

the world and not of the engine! The net result, the total amount of work done, is that universal

function, that great function which is independent of substance. So we see that a substances properties

must be limited in a certain way; one cannot make up anything he wants, or he would be able to invent

a substance which he could use to produce more than the maximum allowable workTHIS PRINCIPLE,

THIS LIMITATION IS THE ONLY REAL RULE THAT COMES OUT OF THERMODYNAMICS.

Standard temperature and efficiency of a Carnot engine

Forget everyday notion of temperature in terms of degrees Celsius or Kelvins. Just keep in mind that

temperature is related to the average kinetic energy of the molecules and we are free to choose any

scale we want. Therefore, for a given system with a certain average kinetic energy, let us SET or DEFINE

its temperature to be 1deg. We can make a standard amount of heat Qs to flow to this given system by

connecting it to a heat engine, right? The heat engine will take in Q at T, do work W and expel Qs at

1deg. We dont know what exactly Q and T are, we just know that Q must increase as T increases. So if

an engine takes Q=x*Qs to deliver Qs at 1deg, then let us DEFINE T to be xdeg. Theres no harm in doingthis because we followed the restriction that Q increases as T increases.

Efficiency of an Carnot engine

Can ever be less than ? What does a negative efficiency mean?


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When heat goes in a system, it should obviously disturb the system and the environment. But in system

such as a Carnot engine, which operates under a reversible process, something in BOTH the system and

the environment does not change as heat goes in at T1 and goes out at T2. That something is what we

call entropy. If the process is then irreversible, then the total entropy must change. This is what happens

in real life when engines operatethey heat up as the undergo (irreversible) processes and the system

and environment are not the same as before.

For a machine to do POSITIVE work, T1 should never be less than T2. Negative efficiency means that the

machine does NEGATIVE work. For example, instead of a car running on itself, negative efficiency means

that a person has to push the car just to move.

Entropy

Measure of disorder in a system.

A system which has a given macroscopic property is more disordered if there are more ways of

arranging its particles which preserves the macroscopic property.

Why is a broken glass more disordered than when it is whole?

Which has more entropy, a gold bar or molten gold?

A broken glass is more disordered because the bonds between particles have been broken, thus giving

more freedom to the particles and more ways of rearranging the system.

Molten gold has more entropy because the particles are loose and have more ways of rearranging

themselves.

Third Law of Thermodynamics

The entropy of the system at absolute zero is zero.

Absolute zero cannot be attained in a finite number of thermodynamic steps.

What is the physical significance of the third law of thermodynamics?

Classical versus quantum mechanical definition of temperature: negative temperatures, possible

connections to cosmology

Reviewer Physics II

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