Apollo 11: Does my Grade 5 Teacher Still Have My Poster? :)

When I was in grade 5, you could write to NASA, and they would send you whatever information or pictures you asked for and a lot more. On one occasion, I got a large envelope in our mailbox with a beautiful poster of the Apollo 11 mission. I brought it to school to share it with my classmates. My teacher was impressed. She put it on the classroom wall, but at the end of the year, she liked it so much that she refused to give it back to me!

Yesterday I came across the following from an old NASA poster, that may or may not have been from the one I once owned:

The first thing that caught my attention was the ratio of liquid hydrogen to that of liquid oxygen in Stage 2 of the rocket, which as the poster explains, takes the rocket from an altitude of 36 miles to 108 miles where it attains its orbital velocity. I guess not to scare 10 year-olds, they express the figures in gallons and not in moles, which would have been more meaningful. Moles are bundles of molecules and proportional to them so that from a chemical equation you know, for example, that it takes two moles of hydrogen to react with every one mole of oxygen.

Why? Since atoms are not destroyed in chemical reactions it takes to two molecules of diatomic hydrogen to bond to one molecule of diatomic oxygen to produce two molecules of water. That way, 4 atoms of hydrogen and a pair of oxygens go into the reaction, and the same number of atoms of hydrogen and oxygen are part of the water being produced.

But did NASA pack the second stage with such a ratio of 2:1? It turns out, no. At first I thought it was because an exact stoichiometric ratio of 2 parts hydrogen to 1 part oxygen is too explosive (stoichiometric just means the ratio from the balanced chemical equation). If for example, you fill a balloon with hydrogen and just rely on the oxygen from the air to react with it after you ignite it, the explosion is a lot more tame than if you ignite two parts of hydrogen and 1 part of oxygen in a balloon. You’ll even remark no color in the latter case. But when there is a shortage of oxygen gas, the excess hydrogen glows orange by the same mechanism that the unburnt particles incandesce in a candle flame. Film footage of the fuel burning from the second stage of the Saturn V rocket confirms that there is excess hydrogen. So why include more? A small symmetric molecule like hydrogen stores less vibrational and rotational energy for a given amount of heat from combustion. By having excess (unburnt) hydrogen, more of the translational energy it picks up from the combustion can be converted to kinetic energy. That results in a higher exhaust velocity, implying that the rocket gets more of a push forward.

So how do you go about calculating the mole-ratio from the chart’s 267 700 US gallons of liquid hydrogen and 87 400 gallons of liquid oxygen? We should note that the problem would be trivial if the gallons were just volumes of gases. You see thanks to Avogadro, we’ve known for more than two centuries that equal volumes of different gases under the same conditions of temperature and pressure have the same number of molecules. That’s because in the gas state, the volume is not determined by the size of individual molecules. They are so small compared to the distance between them that each atom’s size is irrelevant. But in the liquid or solid state, an atom’s radius does matter. So after converting US gallons to liters, you need to convert the volume of each liquid into mass by multiplying by its density (1.14 kg/L for liquid oxygen and 0.071 kg/L for liquid hydrogen) and then converting to moles by dividing each by its molar mass ( 0.0320 kg/mole for oxygen and 0.0020 kg/mole for hydrogen). When that’s done we obtain a mole-ratio of 3.05 parts of hydrogen for every 1 part of oxygen instead of the stoichiometric ratio of 2 : 1. The fuel mixture in the rocket’s second stage was fuel-rich indeed.

But what you really want to know is: does my grade 5 teacher still have my poster? Like many people of their age, they have no online presence and are impossible to reach. 🙂

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More On Mersenne Primes

This is a second post about Mersenne primes(M_p), which are prime numbers of the form 2^p-1, where p itself is prime. They’ve been dubbed “the prime jewels of numbers”, and all of the Mersenne primes discovered from 1952 onwards ,39 of the known 51 are bigger than the number of atoms in the universe (10⁸⁰). There is a cooperative project that borrows the collective power of volunteer personal computers called GIMPS (the Great Internet Mersenne Prime Search). But throughout the two-plus years of COVID19, it has not found a new M_p . The latest news from the venture comes from December 2021, when they announced that all p values for 2^p-1 below 107 million have been tested, so the December 2018-discovery of 2^82,589,933-1 remains supreme for the moment.

Why it’s taking a little longer is not a total surprise. The bigger the numbers get, the more daunting a task it becomes to test them. Moreover, we know from previous discoveries that Mersenne primes become even rarer than prime numbers become as numbers get larger. To get a clearer idea of what we mean by this, consider the data.

Of all the numbers between 1 to 1000, there are only 168 prime numbers (16.8%). Of these only 14 or 8.3% turned out to generate Mersenne primes. Scale up the search to numbers up to 10 000, and the percentage of prime numbers shrinks to 12.2%. But that was still 1229 primes and an extra 1000 primes to test to see if applying 2^p-1 to each one will generate a prime number. And only 8 more passed the test, meaning that for the smallest 1229 prime numbers only 1.8 % gave rise to M_ps. So Mersenne primes rarer than primes themselves.

During their they search for the 13 most recent prime M_ps, they had to test an extra 4.8 million primes! More precisely the prime number used to generate the 51st M_p, 82 589 933, corresponding to M_p= 2^82,589,933-1 = M51(not the Messier object M51, the Whirlpool Galaxy, but the 51st Mersenne prime 🙂 ), is the 5 428 681 st prime. Out of that total, only 0.00086% have turned out to be M_ps. Amazing, considering that 100% of the first 4 primes and 40% of the first 25 primes pass the test.

Unfortunately there doesn’t seem to be the formula-equivalent of n/ln(n) which would gives how many M_ps to expect out of the first n numbers or out of the first primes. Recall that all perfect numbers, numbers that are the sum of its factors, except itself) can be expressed as (2^p-1)( 2^p-1). Thus each time a new M_p is found we also find the next perfect number. And these always end in either 6 or 8. But the last six M_ps all end in 6.

Why Don’t the Nucleus’ Protons Repel?

Don’t you wish students asked that question when learning about the atom? Most know that like-charges repel, so why aren’t they troubled by the idea of an extremely small nucleus holding tightly packed positively charged and neutral particles? (An equally important question is: why aren’t quarks and gluons part of the high school curriculum when the concepts have been established for half a century!)

In my final year of teaching, one student (Paul) thought it was neutrons that held protons together. That’s kind of true but how?

The repulsive force is stronger than we imagine at distances like the 3.8 × 10^-15 meters between the pair of protons in a helium nucleus. The force is  9 × 10⁹ Nm²/C² ( 1.6 × 10^-19C)²/(3.8 ×10^-15m)² = 16 N or about 3.5 pounds for a single atom!  So if you imagine trying to bond two magnets while aligning the North pole of 1 magnet with the North of the other, you need the equivalent of a “Velcro” between the the two bar magnets. Beyond a very short distance, the Velcro would not help bind the magnets, but when the magnets are close enough, the Velcro’s little loops and hooks could overcome the repulsive force of two like-poles. The force required to hold protons and neutrons together has to be something stronger than electrostatics, and this strong force has to be acting only over short distances.

The strong force has a dual role. It binds the inner parts of a proton , and it also holds separate protons and neutrons together.

(1) For starters, we need to reveal the inner guts of a proton and those of a neutron. Each neutron and proton consists of 3 quarks. The proton has two up quarks, each with a charge of +2/3, and one down quark of charge -1/3 for a sum-charge of +1. The recipe for the neutron is the reverse: two down quarks one up quark for a sum-charge of 0. But what’s more important in keeping them together is another property of quarks called color (unrelated to the everyday sense of the word). In any nucleon (proton or neutron) each quark must be of a different color (red, blue and green).

The real Velcro is in the form of gluons emitted by each quark. These change the color-properties of quarks without affecting their charge. Equally important is that they keep the overall color balance within each quark. This is made possible by the fact that a gluon carries both a color and an anti-color, designated by a bar above the letter of the color. The exchange is what keeps the quarks bound together in each neutron and proton. 

HOW QUARKS REMAIN BOUND TO EACH OTHER IN A NUCLEON (Example: in a proton) When a “red” up quark emits a gluon with anti-green and red, that gluon is absorbed by a green up quark in the proton. Anti-green and green cancel, leaving us with a red up quark. That green up quark’s release of a green-antiblue gluon in turn changes a blue down quark into a green one.

(2) But that in itself doesn’t explain why a proton would overcome the repulsive Coulombic force. So what is it that prevents the protons of an atomic nucleus from flying apart? Within a proton, an up quark can release a gluon which becomes a down quark and an anti-down quark. The down quark replaces the up quark in the proton converting it into a neutron. Meanwhile a neutron will do something similar, but the gluon that the neutron releases (and whose up quark converts it into a proton) turns into an  up quark and an anti-up quark. The anti- up quark will annihilate the up quark from the proton and give back a gluon. At the same time, the proton-emitted gluon’s anti-down quark annihilates a down quark from the neutron to give back the other gluon. And then the whole cycle begins again. If particles are busy bartering they can’t be apart. It’s shown in the film https://www.youtube.com/watch?v=FoR3hq5b5yE, but it became clearer (at least in my mind) when I summarized it with this diagram: 

HOW NUCLEONS ARE ATTRACTED THROUGH THE STRONG FORCE (EXAMPLE PROTON and NEUTRON)