Olber’s Paradox.

“The comedian George Carlin used to do a sketch where he was a kind of hippy weather man. “And the weather tonight is … dark, man.”

But he didn’t ask the question, why is it dark? Once you get past the flippant answer “duh, because it’s night-time”, you see why it is an interesting question. If there are an infinite number of stars, shouldn’t the sky be bright?”

Since the sky isn’t very bright at night – a lot of dark plus a few pinpoints of light – this is called Olber’s Paradox. Put simply, if the universe is infinite then wherever you look you should see so many stars that the night would be brighter than the day.

Johannes Kepler considered this question but he argued that the universe must be finite. Otherwise the total flux from all the stars would make the night sky “as luminous as the sun.”

Suppose we gaze out in any direction from Earth, imagining a thin sphere of radius R around us. Unlike Kepler, Newton’s model assumed a uniform, infinite (and static, or not expanding) universe, the number of stars in the shell is proportional to R² and the intensity of radiation from the shell reaching Earth is proportional to 1/R². So, according to Newton’s model such shells stretch to infinity so the sky can never be dark.

Of course, now we know that the Universe is expanding – it has a beginning – and stars and galaxies aren’t tastefully arranged in neat little spheres.

Lots of bits are dark and the combined effect is to make the night sky dark.

Furthermore, there’s more ‘dark’ than ‘light’ and the Universe expansion probably isn’t uniform. Best estimates suggest that if we calculate the energy needed to overcome gravity, dark energy ( the stuff that accelerates the expansion of the Universe) makes up roughly 68% of it. Dark matter makes up another 27%, leaving the “normal” or baryonic matter that we are familiar with to make up less than 5% of the cosmos.

 

Very Basic Thermodynamics

Thermodynamics is the study of energy. (IB Core: Section 3.2, Option B.2 part)

A MOLE is an amount of stuff ~6×10²³ particles’ worth. (SI unit)

This number is the Avogadro constant, N_A (mol^-1) – the number of constituent particles in 1 mole of substance. In 12g of C¹² or 18g of water, there are 6×10²³ carbon atoms or water molecules respectively – one mole. One mole of electrons contains 6×10²³ electrons, and so on.

We should remember that for a fixed mass of an ideal gas, the ideal gas equation (below) applies The equation is considered most accurate for monatomic gases at high temperatures and low pressures. Check the link so you understand the assumptions of the kinetic theory of gases.

so k=R/N_A

Temperature is a measure of the degree of hotness of a body, as compared to a fixed scale. Normally we calculate in kelvins (K) – a base unit – where a difference of 1K corresponds to a difference of 1⁰C. For now, from the Ideal Gas Laws,

Energy exists in many forms, such as heat, light, chemical energy, and electrical energy. Energy is the ability to bring about change or to do work.

Laws of Thermodynamics                               Zeroth Law

If A is in thermal equilibrium with B and B is in thermal equilibrium with C then A and C are also in thermal equilibrium.

All thermal equilibrium means is that the rate of transfer of heat from A to B is the same as that from B to A. If B is hotter than A, B transfers heat more rapidly to A than A does to B. But – the transfer is still two-way. Temperature difference between two bodies determines the net flow rate of energy between them.

First Law

The First Law of Thermodynamics or Law of Conservation states that the total energy in the universe is always conserved; it cannot be created or destroyed.  Energy can only be converted from one form into another. For a fixed mass of an ideal gas, the gas can either do work on its surroundings, delta W, gain heat from its surroundings, delta Q or its internal energy increases. delta U change in internal energy is a function of temperature: U is a large scale concept. We cannot talk about the “thermal energy” of something – it has no meaning. Instead we refer to the internal energy of a body which is the total potential energy (arising from intermolecular forces) + random kinetic energy (translational and rotational) of the molecules in a sample of material and clearly we can only measure change in (delta) U not U directly. Easier to see with symbols.

Let’s imagine a frictionless piston, containing a fixed mass of an ideal gas (very low density, pressure and high temperature). Now, let’s make some changes to it. Isothermal changes are changes without change in temperature, thus the internal energy of the gas is unchanged. So, all the heat supplied = all the work done by the gas. Take note of which processes are slow (so heat gets in or out) and fast (no heat in or out)

Adiabatic Process. No heat in or out. Any work done is done fast on or by the gas and is reflected in a change in internal energy. Pure adiabatics are quite rare.

The nearest thing to a fast adiabatic process is a bursting tyre. The rubber of the tyre is an insulator, so no heat enters or leaves the gas and the work done by the gas on escaping through the hole is at the expense of a fall in temperature of the remaining gas inside the tyre. On the graph below, this would be the thick green line from high T to low T

NB: the AREA UNDER the pV graph is the work done on or by the interaction. (fave exam question where you have to count squares)

You might now ask yourselves what the pV graph might look like if the volume was NOT allowed to change – the “sticky piston” problem called an ISOVOLUMETRIC change

and, yes, it’d be a vertical line at some constant volume between two isothermals.

An ISOBARIC (constant pressure) line would be a horizontal line between two isothermals. It might be helpful to sketch both on the graph above.

A CYCLE of events means that we make changes to the gas to get back to our starting point. The (shaded) area swept out by these changes is a measure of the work done during one cycle. T1 is greater than T2, clearly. Here’s an idealised diagram. In reality heat enters (BC) and leaves  DA) the gas so BC and DA aren’t perfect adiabatics in real systems.

This looks complicated, but it’s not, really. Please make sure you look at the problem below from IB 2008. As you can see it consists of two isobaric and two isovolumetric events.

Now calculate the overall energy transferred in one cycle and explain whether, after one complete cycle, the internal energy of the gas goes up, down or stays the same.

One-Way Processes.

The Second law of thermodynamics states that that the entropy or measure of disorder of an isolated system always increases, because isolated systems spontaneously evolve towards thermodynamic equilibrium – the state of maximum entropy (or minimum potential energy – a ball spontaneously rolls down a hill and not vice-versa.) A cup of tea tends to cool as energy is dissipated to the surroundings and not vice-versa. An increase in entropy is the Universe doing the most likely thing – the probable is what usually happens. (when you blow up a building it tends not to spontaneously reassemble as if the film ran backwards)

A mechanical watch will run until the potential energy in the spring is converted, and not again until energy is reapplied to the spring to rewind it. A car that has run out of gas will not run again until you refuel the car. In the process of energy transfer, some energy will dissipate as heat. Entropy always increases and is a measure of the disorder of the Universe – put another way, the more energy is transferred from one body to another the greater are the number of ways in which that energy dissipation can take place. For example, a waterfall turns a paddle wheel which drives a turbine which turns an alternator which produces electricity, dissipating energy into many different forms along the way.