Tag: conservation of mass

The missing mass of the electron

Annihilating mass in communicating science

Keith S. Taber

An episode of 'In Our Time' about the electron

The BBC radio programme 'In Our Time' today tackled the electron. As part of the exploration there was the introduction of the positron, and the notion of matter-antimatter annihilation. These are quite brave topics to introduce in a programme with a diverse general audience (last week Melvyn Bragg and his guests discussed Plato's Atlantis and next week the programme theme is the Knights Templar).

Prof. Victoria Martin of the School of Physics and Astronomy at the University of Edinburgh explained:

If we take a pair of matter and antimatter, so, since we are talking about the electron today, if we take an electron and the positron, and you put them together, they would annihilate.

And they would annihilate not into nothingness, because they both had mass, so they both had energy from E=mc2 that tells us if you have mass you have energy. So, they would annihilate into energy, but it would not just be any kind of energy: the particular kind of energy you get when you annihilate an electron and a positron is a photon, a particle of light. And it will have a very specific amount of energy. Its energy will be equal to the sum of the energy of electron and the positron that they had initially when they collided together.

Prof. Victoria Martin on 'In Our Time'
"An electron and the positron, and you put them together, they would annihilate…they would annihilate into energy" – but this could be misleading.

Now, I am sure that is somewhat different from how Prof. Martin would treat this topic with university physics students – of course, science in the media has to be pitched at the largely non-specialist audience.

Read about science in the media

It struck me that this presentation had the potential to reinforce a common alternative conception ('misconception') that mass is converted into energy in certain processes. Although I am aware now that this is an alternative conception, I seem to recall that is pretty much what I had once understood from things I had read and heard.

It was only when I came to prepare to teach the topic that I realised that I had a misunderstanding. That, I think, is quite common for teachers – when we have to prepare a topic well enough to explain it to others, we may spot flaws in our own understanding (Taber, 2009)

So, for example, I had thought that in nuclear processes, such as in a fission reactor or fusion in stars, the mass defect (the apparent loss of mass as the resulting nuclear fragments have less mass than those present before the process) was due to that amount of mass being converted to energy. This is sometimes said to explain why nuclear explosions are so much more violent than chemical explosions, as (given E=mc2): a tiny amount of mass can be changed into a great deal of energy.

Prof. Martin's explanation seemed to support this way of thinking: "they would annihilate into energy".

An alternative conception of particle annihilation: This scheme seems to be implied by Prof. Martin's comments

What is conserved?

It is sometimes suggested that, classically, mass and energy were considered to be separately conserved in processes, but since Einstein's theories of relativity have been adopted, now it is considered that mass can be considered as if a form of energy such that what is conserved is a kind of hybrid conglomerate. That is, energy is still considered conserved, but only when we account for mass that may have been inter-converted with energy. (Please note, this is not quite right – see below.)

So, according to this (mis)conception: in the case of an electron-positron annihilation, the mass of the two particles is converted to an equivalent energy – the mass of the electron and the mass of the positron disappear from the universe and an equivalent quantity of energy is created. Although energy is created, energy is still conserved if we allow for the mass that was converted into this new energy. Each time an electron and positron annihilate, their masses of about 2 ✕ 10-30 kg disappear from the universe and in its place something like 2 ✕ 10-13 J appears instead – but that's okay as we can consider 2 ✕ 10-30 kg as a potential form of energy worth 2 ✕ 10-13 J.

However, this is contrary to what Einstein (1917/2004) actually suggested.

Einstein did not suggest that matter could be changed to energy

Equivalence, not interconversion

What Einstein actually suggested was not that mass could be considered as if another kind/form of energy (alongside kinetic energy and gravitational potential, etc.) that needed to be taken into account in considering energy conservation, but rather that inertial mass can be considered as an (independent) measure of energy.

That is, we think energy is always conserved. And we think that mass is always conserved. And in a sense they are two measures of the same thing. We might see these two statements as having redundancy:

  • In a isolated system we will always have the same total quantity of energy before and after any process.
  • In a isolated system we will always have the same total quantity of mass before and after any process.

As mass is always associated with energy, and so vice versa, either of these statements implies the other. 1

Two conceptions of the shift from a Newtonian to a relativistic view of the conservation of energy (move the slider to change the image)
No interconversion?

So, mass cannot be changed into energy, nor vice versa. The sense in which we can 'interconvert' is that we can always calculate the energy equivalence of a certain mass (E=mc2) or mass equivalence of some quantity of energy (m=E/c2).

So, the 'interconversion' is more like a change of units than a change of entity.

Although we might think of kinetic energy being converted to potential energy reflects a natural process (something changes), we know that changing joules to electron-volts is merely use of a different unit (nothing changes).

If we think of a simple pendulum under ideal conditions 2 it could oscillate for ever, with the total energy unchanged, but with the kinetic energy being converted to potential energy – which is then converted back to kinetic energy – and so on, ad infinitum. The total energy would be fixed although the amount of kinetic energy and the amount of potential energy would be constantly changing. We could calculate the energy in joules or some other unit such as eV or ergs (or calories or kWh or…). We could convert from one unit to another, but this would not change anything about the physical system. (So, this is less like converting pounds to dollars, and more like converting an amount reported in pounds {e.g., £24.83} into an amount reported in pence {e.g., 2483p}.)

Using this analogy, the electron and positron being converted to a photon is somewhat like kinetic energy changing to potential energy in a swinging pendulum (something changes), but it is not the case that mass is changed into energy. Rather we can do our calculations in terms of energy or mass and will get (effectively, given E=mc2) the same answer (just as we can add up a shopping list in pounds or pence, and get the same outcome given the conversion factor, 1.00£ = 100p).

So, where does the mass go?

If mass is conserved, then where does the mass defect – the amount by which the sum of masses of daughter particles is less than the mass of the parent particle(s) – in nuclear processes go? And, more pertinent to the present example, what happens to the mass of the electron and positron when they mutually annihilate?

To understand this, it might help to bear in mind that in principle these process are like any other natural processes – such as the swinging pendulum, or a weight being lifted with pulley, or methane being combusted in a Bunsen burner, or heating water in a kettle, or photosynthesis, or a braking cycle coming to a halt with the aid of friction.

In any natural process (we currently believe)

  • the total mass of the universe is unchanged…
    • but mass may be reconfigured
  • the total energy of the universe is unchanged…
    • but energy may be reconfigured; and
  • as mass and energy are associated, any reconfigurations of mass and energy are directly correlated.

So, in any change that involves energy transfers, there is an associated mass transfer (albeit usually one too small to notice or easily measure). We can, for example, calculate the (tiny) increase in mass due to water being heated in a kettle – and know just as the energy involved in heating the water came from somewhere else, there is an equivalent (tiny) decrease of mass somewhere else in the wider system (perhaps due to falling of water powering a hydroelectric power station). If we are boiling water to make a cup of tea, we may well be talking about a change in mass of the order of only 0.000 000 001 g according to my calculations for another posting.

Read 'How much damage can eight neutrons do? Scientific literacy and desk accessories in science fiction.'

The annihilation of the electron and positron is no different: there may be reconfigurations in the arrangement of mass and energy in the universe, but mass (and so energy) is conserved.

So, the question is, if the electron and positron, both massive particles (in the physics sense, that they have some mass) are annihilated, then where does their mass go if it is conserved? The answer is reflected in Prof. Martin's statement that "the particular kind of energy you get when you annihilate an electron and a positron is a photon, a particle of light". The mass is carried away by the photon.

The mass of a massless particle?

This may seem odd to those who have learnt that, unlike the electron and positron, the photon is massless. Strictly the photon has no rest mass, whereas the electron and positron do have rest mass – that is, they have inertial mass even when judged by an observer at rest in relation to them.

So, the photon only has 'no mass' when it is observed to be stationary – which nicely brings us back to Einstein who noted that electromagnetic radiation such as light could never appear to be at rest compared to the observer, as its very nature as a progressive electromagnetic wave would cease if one could travel alongside it at the same velocity. This led Einstein to conclude that the speed of light in any given medium was invariant (always the same for any observer), leading to his theory of special relativity.

So, a photon (despite having no 'rest' mass) not only carries energy, but also the associated mass.

Although we might think in terms of two particles being converted to a certain amount of energy as Prof. Martin suggests, this is slightly distorted thinking: the particles are converted to a different particle which now 'has' the mass from both, and so will also 'have' the energy associated with that amount of mass.

Mass is conserved during the electron-positron annihilation

A slight complication is that the electron and position are in relative motion when they annihilate, so there is some kinetic energy involved as well as the energy associated with their rest masses. But this does not change the logic of the general scheme. Just as there is an energy associated with the particles' rest masses, there is a mass component associated with their kinetic energy.

The total mass-energy equivalence before the annihilation has to include both the particle rest masses and their kinetic energy. The mass-energy equivalence afterwards (being conserved in any process) also reflects this. The energy of the photon (and the frequency of the radiation) reflects both the particle masses and their kinetic energies at the moment of the annihilation. The mass (being perfectly correlated with energy) carried away by the photon also reflects both the particle masses and their kinetic energies.

How could 'In Our Time' have improved the presentation?

It is easy to be critical of people doing their best to simplify complex topics. Any teacher knows that well-planned explanations can fail to get across key ideas as one is always reliant on what the audience already understands and thinks. Learners interpret what they hear and read in terms of their current 'interpretive resources' and habits of thinking.

Read about constructivism

A physicist or physics student hearing the episode would likely interpret Prof. Martin's statement within a canonical conceptual framework. However, someone holding the 'misconception' that mass is converted to energy in nuclear processes would likely interpret "they would annihilate into energy" as fitting, and reinforcing, that alternative conception.

I think a key issue here is a slippage that apparently refers to energy being formed in the annihilation, rather than radiation: (i.e., Prof. Martin could have said "they would annihilate into [radiation]"). When the positron and electron 'become' a photon, matter is changed to radiation – but it is not changed to energy, as matter has mass, and (as mass and energy have an equivalence) the energy is already there in the system.

Energy is reconfigured, but is not formed, in the annihilation process.

So, this whole essay is simply suggesting that a change of one word – from energy to radiation – could potentially avoid the formation of, or the reinforcing of, the alternative conception that mass is changed into energy in processes studied in particle physics. As experienced science teachers will know, sometimes such small shifts can make a good deal of difference to how we are interpreted and, so, what comes to be understood.

Addenda:

Reply from Prof. Victoria Martin on twitter (@MamaPhysikerin), September 30:

"E2 = p2c2 + m2c4 is a better way to relate energy, mass and momentum. Works for both massive and massless states."

@MamaPhysikerin
Work cited:

Notes

1 In what is often called a closed system there is no mass entering or leaving the system. However, energy can transfer to, or from, the system from its surroundings. Classically it might be assumed that the mass of a closed system is constant as the amount of matter is fixed, but Einstein realised that if there is a net energy influx to, or outflow from, the system, than some mass would also be transferred – even though no matter enters or leaves.

2 Perhaps in a uniform gravitational field, not subject to to any frictional forces, with an inextensible string supporting the bob, and in thermal equilibrium with its environment.

How much damage can eight neutrons do?

Scientific literacy and desk accessories in science fiction

Keith S. Taber

Is the principle of conservation of mass that is taught in school science falsified all the time?

I am not really a serious sci-fi buff, but I liked Star Trek (perhaps in part because it was the first television programme I got to see in colour 1) and I did enjoy Blakes7 when it was broadcast by the BBC (from 1978-1981).

Logo of the BBC television series 'Blakes7' with (a) home desk equipment or (b) manual control lever for the most advanced spaceship in the galaxy (Image by Francis Ray from Pixabay).

Blakes7 was made with the same kind of low budget production values of Dr Who of the time. Given that space scenes in early episodes involved what seemed to be a flat image of a spacecraft moving across a star field with no sense of depth or perspective (for later series someone had built a model), and in one early episode the crew were clearly given angle-poise lamps to control the craft, it was certainly not a case of 'no expense spared'. So, it was never quite clear if the BBC budget had also fallen short of a possessive apostrophe in the show title credits or Blakes7 was to be read in some other way.

After all, it was not made explicit who was part of Blake's 7 if that was what the title meant, and no one referred to "Blake's 7" in the script (perhaps reflecting how the doctor in Dr Who was not actually called Dr Who?).

The Blakes7 team on the flight desk of the Liberator – which was the most advanced spaceship in the galaxy (and was, for plot purposes, conveniently found drifting in space without a crew) – at least until they forgot to clean the hull once too often and it corroded away while they were on an away mission.

Blake's group was formed from a kind of prison break and so Blake was something of a 'rough-hero' – but not as much as his sometime unofficial lieutenant, sometime friend, sometime apparent rival, Avon, who seemed to be ruled by self-interest (at least until the script regularly required some act of selfless heroism from him). 'Rough-heroes' are fictional characters presented in the hero role but who have some traits that the audience are likely to find morally questionable if not repugnant.

As well as Blake (a rebel condemned as a traitor, having 'recovered' from brainwashing-supported rehabilitation to rebel again) and Avon (a hacker convicted of a massive computer fraud intended to make himself extremely rich) the rest of the original team were a smuggler, a murderer and a petty thief, to which was added a terrorist (or freedom fighter if you prefer) picked up on an early mission. That aside, they seemed an entirely reasonable and decent bunch, and they set out to rid the galaxy of 'The Federation's tyrannical oppression. At least, that was Blake's aspiration even if most of his companions seemed to see this as a stop-gap activity till they had decided on something with more of a long-term future.

At the end of one season, where the fight with the Federation was temporarily put aside to deal with an intergalactic incursion, Blake went AWOL (well, intergalactic wars can be very disruptive) and was assumed dead/injured/lost/captured/?… for much of the remaining run without affecting the nature of the stories too much.

Among its positive aspects for its time were strong (if not exactly model) roles for women. The main villain, Servalan, was a woman – Supreme Commander of the Federation security forces (and later Federation president).

As the ruthless Supreme Commander of the Federation security forces, Servalan got to wear whatever she liked (a Kid Creole, or Mel and Kim, look comes to mind here) and could insist her staff wore hats that would not upstage hers

In Blake's original team (i.e., 7?), his pilot is a woman. (Reflecting other SciFi series, the spacecraft used by Blakes7 require n crew members to operate effectively, where n is an integer that varies between 0 and 6 depending on the specific plot requirements of an episode.) In a later series, after Avon has taken over the role of 'ipso facto leader-among-equals', the group recruits a female advanced weapons designer/technologist and a female sharpshooter.

The Blakes7 team later in the run. (Presumably they are checking the monitor and having a quick recount.) Was Soolin (played by Glynis Barber, far right) styled as a subtle reference to the 'Seven Samurai'?

When I saw Blakes7 was getting a rerun recently I re-watched the series I had not seen since it was first aired. Despite very silly special effects, dodgy story-lines, and morally questionable choices (the series would make a great focus for a philosophy class) the interactions between the main characters made it an enjoyable watch.

But, it is not science

Of course, the problem with science fiction is that it is fiction, not science. Star Trek may have prided itself on seeking to at least make the science sound feasible, but that is something of an outlier in the genre.

Egrorian and his young assistant Pinder (unfortunately prematurely aged somewhat by a laboratory mishap) show Avon and Vila around their lab.

This is clear, for example, in an episode called 'Orbit' where Avon discuses the tachyon funnel, an 'ultimate weapon', with Egrorian, a renegade scientist. Tachyons are hypothetical particles that travel faster than the speed of light. The theory of special relatively suggests the speed of light is the theoretical maximum speed anything can have, but some other theories suggest tachyons may exist in some circumstances. As always in science, theories that are widely accepted as our current best understanding of some aspect of nature (e.g., relativity) are still open to modification or replacement if new evidence is found that suggests this is indicated.

In the Blakes7 universe, there seemed to be a surprisingly high frequency of genius scientists/engineers who had successfully absconded from the tyrannical and paranoid Federation with sufficient resources to build private research facilities on various obscure deserted planets. Although these bases are secret and hidden away, and the scientists concerned have normally been missing for years or even decades, it usually transpires that the Blakes7 crew and the Federation manage to locate any particular renegade scientist during the same episode.

This is part of the exchange between this particular flawed genius scientist and our flawed and reluctant 'rough hero', Kerr Avon:

Egrorian: You've heard of Hoffal's radiation?

Avon: No.

Ah… Hoffal had a unique mind. Over a century ago he predicted most of the properties that would be found in neutron material.

Neutron material?

Material from a neutron star. That is a… a giant sun which has collapsed and become so tightly compressed that its electrons and protons combine, making neutrons.

I don't need a lecture in astrophysics. [But presumably the scriptwriter felt the audience would need to be told this.]

When neutrons are subjected to intense magnetic force, they form Hoffal's radiation. Poor Pinder [Egrorian's lab. assistant] was subjected for less than a millionth of a second. He aged 50 years in as many seconds. …

So neutrons are part of the tachyon funnel.

Um, eight of them … form the core of the accelerator. 

From the script of 'Orbit' (c) 1981 by the British Broadcasting Corporation – made available 'for research purposes'

Now, for anyone with any kind of science background such dialogue stretches credibility. Chadwick discovered the neutron in everyday matter in 1932, so the neutron's properties could be explored without having to obtain samples from a neutron star – which would certainly be challenging. When bound in nuclei, neutrons (which are electrically neutral, thus the name, and so not usually affected by magnetic fields) are stable.

Thinking at the scale of a neutron

However, any suspension of disbelief (which fiction demands, of course) was stretched past breaking point at the end of this exchange. Not only were the generally inert neutrons the basis of a weapon that could destroy whole worlds – but the core of the accelerator was formed of, not a neutron star, nor a tonne of 'neutron matter', but eight neutrons (i.e., one for each member of Blake's 7 with just a few left over?)

That is, the intensely destructive beam of radiation that could destroy a planet from a distant solar system was generated by subjecting to a magnetic field: a core equivalent to (the arguably less interesting) half of a single oxygen atomic nucleus.

Warning – keep this away from strong magnetic fields if you value your planet! (Image by Gerd Altmann from Pixabay )

Now free neutrons (that is, outside of an atomic nuclei – or neutron star) are unstable, and decay on a timescale of around a quarter of an hour (that is, the half-life is of this order – following the exponential decay familiar with other kinds of radioactivity), to give a proton, an electron and a neutrino. The energy 'released' in this process is significant on the scale of a subatomic particle: 782 343 eV or nearly eight hundred thousand eV.

Eight hundred thousand seems a very large number, but the unit here is electron volt, a unit used for processes at this submicroscopic scale. (An eV is the amount of work that is done when one single electron is moved though a potential difference of 1v – this is about 1.6 x10-19 J). In the more familiar units of joules, this is about 1.25 x 10-13 J. That is,

0.000 000 000 000 125 J

To boil enough water at room temperature to make a single cup of tea would require about 67 200 J. 2 So, if the energy from decaying neutrons were used to boil the water, it would require the decay of about

538 000 000 000 000 000 neutrons.3

That is just to make one cup of tea, so imagine how many more neutrons would have to decay to provide the means to destroy a planet. Certainly, one would imagine,

more than 8.

E=mc2

Now since Einstein (special relativity, again), mass and energy have been considered to have an equivalence. It is commonly thought that mass can be converted to energy and the equation E=mc2 tells you how much of one would be converted to the other: how many J per kg or kg per J. (Spoiler alert – this is not quite right.)

In that way of thinking, the energy released by a free neutron when it decays is due to a tiny part of the neutrino's mass being converted to energy.

The neutron's mass defect

The mass (or so called 'rest mass') of a neutrino is about 1.67 x 10-27 kg. In the usual mode of decay the neutrino gives rise to a proton (which is nearly, but not quite, as heavy as a neutron), an electron (which is much lighter), and a neutrino (which is considered to have zero rest mass.)

Before decayRest mass / 10-31 kgAfter decayRest mass / 10-31 kg
neutron16 749.3proton16 726.2
electron9.1
neutrino–
total16 749.316 735.3
[rest] mass defect in neutrino decay

So, it seems like some mass has disappeared. (And this is the mass sometimes said to have been converted into the released energy.) This might lead us to ask the question of whether Hoffal's discovery was a way to completely annihilate neutrons, so that instead of a tiny proportion of their mass being converted to energy as in neutron decay – all of it was.

Mass as latent energy?

However, when considered from the perspective of special relativity, it is not that mass is being converted to energy in processes such as neutron decay, but rather that mass and energy are considered as being different aspects of something more unified -'mass-energy' if you like. Energy in a sense carries mass, and mass in a sense is a manifestation of energy. The table above may mislead because it only refers to 'rest mass' and that does not tell us all we need to know.

When the neutron decays, the products move apart, so have kinetic energy. According to the principle of mass-energy equivalence there is always a mass equivalence of any energy. So, in relativity, a moving object has more mass than when it is at rest. That is, the 'mass defect' table shows what the mass would be if we compared a motionless neutron with motionless products, not the actual products.

The theory of special relativity boldly asserts that mass and energy are not the independent quantities they were once thought to be. Rather, they are two measures of a single quantity. Since that single quantity does not have its own name, it is called mass-energy, and the relationship between its two measures is known as mass-energy equivalence. We may regard c2 as a conversion factor that enables us to calculate one measurement from the other. Every mass has an energy-equivalent and every energy has a mass-equivalent. If a body emits energy to its surroundings it also emits a quantity of mass equivalent to that energy. The surroundings acquire both the energy and mass in the process.

Treptow, 2005, p.1636

So, rather than thinking mass has been converted to energy, it may be more appropriate to think that the mass of a neutron has a certain (latent) energy associated with it, and that, after decay, most of this energy is divided between products (according to their rest masses), but a small proportion has been converted to kinetic energy (which can be considered to have a mass equivalence).

So, whenever any process involves some kind of energy change, there is an associated change in the equivalent masses. Every time you boil the kettle, or go up in an elevator, there is a tiny increase of mass involved – the hot water is heavier than when it was cold; you are heavier than when you were at a lower level. When you lie down or burn some natural gas, there is a tiny reduction in mass (you weigh less lying down; the products of the chemical reaction weigh less than the reactants).

How much heavier is hot water?

Only in nuclear processes does the energy change involved become large enough for any change in mass to be considered significant. In other processes, the changes are so small, they are insignificant. The water we boiled earlier to make a cup of tea required 67 200J of energy, and at the end of the process the water would not just be hotter, but also heavier by about

0.000 000 000 000 747 kg

0r about 0.000 000 000 75 g. That is easy to calculate 4, but not so easy to notice.

Is mass conserved in chemical reactions?

On this basis, we might suggest that the principle of conservation of mass that is taught in school science is falsified all the time – or at least needs to be understood differently from how it is usually presented.

Type of reactionMass change
endothermicmass of products > mass of reactants
exothermicmass of products < mass of reactants
If we just consider the masses of the substances then mass is not conserved in chemical change

Yet, the discrepancies really are tiny – so tiny that in school examinations candidates are expected to pretend there is no difference. But, strictly, when (as an example) copper carbonate is heated in a crucible and decomposes to give copper oxide and carbon dioxide there is a mass decrease even if you could capture all the CO2. But it would not be measurable with our usual laboratory equipment – so, as far as chemistry is concerned, mass is conserved. 'To all intense and purposes' (even if not absolutely true) mass is always conserved in chemical reactions.

Mass is conserved overall

But actually, according to current scientific thinking, mass is always conserved (not just very nearly conserved), as long as we make sure we consider all relevant factors. The energy that allowed us to boil the kettle or be lifted in an elevator must have been provided from some source (which has lost mass by the same extent). In an exothermic chemical reaction there is an extremely slight difference of mass between the reactants and products, but the surroundings have been warmed and so have got (ever so slightly) heavier.

Type of reactionMass change
endothermicenergy (and equivalent mass) from the surroundings
exothermicenergy (and equivalent mass) to the surroundings
If we just consider the masses of the substances then mass does not seem to be conserved in chemical change

As Einstein himself expressed it,

"The inertial mass of a system of bodies can even be regarded as a measure of its energy. The law of the conservation of the mass of a system becomes identical with the law of the conservation of energy, and is only valid provided that the system neither takes up nor sends out energy."

Einstein, 1917/2015, p.59

Annihilate the neutrons!

So, if we read about how in particle accelerators, particles are accelerated to immense speeds, and collided, and so converted to pure energy we should be suspicious. The particles may well have been destroyed – but something else has now acquired the mass (and not just the rest mass of the annihilated particles, but also the mass associated with their high kinetic energy).

So, we cannot convert all of the mass of a neutron into energy – only reconfigure and redistribute its mass-energy. But we can still ask: what if all the mass of the neutron were to be converted into some kind of radiation that carried away all of its mass as high energy rays (perhaps Hoffal's radiation?)

Perhaps the genius scientist Hoffal, with his "unique mind", had found a way to do this (hm, with a magnetic field?) Even if that does not seem very feasible, it does give us a theoretical limit to the energy that could be produced by a process that converted a neutron into radiation.6 Each neutron has a rest mass of about

1.67 x 10-27 kg

now the conversion factor is c2 (where c is the speed of light, which is near enough 3 x 108 ms-1, so c2 =(3×108ms-1)2 , i.e., about 1017m2s-2), so that mass is equivalent to about 1.50 x 10-10 J 5 or,

0.000 000 000 150 J

Now that is a lot more energy than the 1.25 x 10-13 J released in the decay of a neutron,

0.000 000 000 150 000 J

>

0.000 000 000 000 125 J

and now we could in theory boil the water to make our cup of tea with many fewer neutrons. Indeed, we could do this by annihilating 'only' about 7

448 000 000 000 000 neutrons

This is a lot less neutrons than before, i.e.,

448 000 000 000 000 neutrons

< 538 000 000 000 000 000 neutrons

but it seems fair to say that it remains the case that the number of neutrons needed (now 'only' about 448 million million) is still a good deal more than 8.

448 000 000 000 000 neutrons

> 8 neutrons

So, if over 400 million million neutrons would need to be completely annihilated to make a single cup of tea, how much damage can 8 neutrons do to a distant planet?

A common learning difficulty

In any reasonable scenario we might imagine 8 neutrons would not be significant. This is worth emphasising as it reflates to a common learning difficulty. Quanticles such as atoms, atomic nuclei, neutrons and the like are tiny. Not tiny like specs of dust or grains of salt, but tiny on a scale where specs of dust and grains of salt themselves seem gigantic. The scales involved in considering electronic charge (i.e., 10-19C) or neutron mass (10-27 kg) can reasonably said to be unimaginatively small – no one can readily visualise the shift in scale going from the familiar scale of objects we normally think of as 'small', to the scale of individual molecules or subatomic particles.

Students therefore commonly form alternative conceptions of these types of entities (atoms, electrons, etc.) being too small to see, but yet not being so far beyond reach. And it is not just learners who struggle here. I have even heard someone on a national news programme put forward as an 'expert' make a very similar suggestion to Egrorian, in this case that a "couple of molecules" could be a serious threat to public health after the use of chemical nerve agent. This is a preposterous suggestion to a chemist, but was, I am sure, made in good faith by the international chemical weapons expert.

It is this type of conceptual difficulty which allows scriptwriters to refer to 8 neutrons as being of some significance without expecting the audience to simply laugh at the suggestion (even if some of us do).

It also explains how science fiction writers get away with such plot devices given that many in their audiences will readily accept that a few especially malicious molecules or naughty neutrons is a genuine threat to life.8 But that still does not justify using angle-poise lamps as futuristic spacecraft joysticks.

Jenna pilots the most advanced spacecraft in the galaxy
Works cited:
  • Einstein, A. (1917/2015). Relativity. The special and the general theory. (100th Anniversary ed.). Princeton: Princeton Univerity Press.
  • Treptow, R. S. (2005). E = mc2 for the Chemist: When Is Mass Conserved? Journal of Chemical Education, 82(11), 1636. doi:10.1021/ed082p1636
Notes:

1 To explain: For younger readers, television was first broadcast in monochrome (black and white – in effect shades of grey). My family first got a television after I started primary school – the justification for this luxury was that the teachers sometimes suggested programmes we might watch.

Colour television did not arrive in the UK till 1967, and initially it was only used for selected broadcasts. The first colour sets were too expensive for many families, so most people initially stayed with monochrome. This led to the infamous 'helpful' statement offered by the commentator of the weekly half-hour snooker coverage: "And for those of you who are watching in black and white, the pink [ball] is next to the green". (While this is well known as a famous example of misspeaking, a commentator's blooper, those of a more suspicious mind might bear in mind the BBC chose snooker for broadcast in part because it might encourage more people to watch in colour.)

Snooker – not ideal viewing on 'black and white' television (Image by MasterTux from Pixabay )

My father had a part-time weekend job supervising washing machine rental collections (I kid you not, many people only rented such appliances in those days), to supplement income from his full time job, and this meant on Monday evenings after his day job he had to visit his part-time boss and report and they would go throughout the paperwork to ensure things tallied. I would go with him, and was allowed to watch television whilst they did this – it coincided with Star Trek, and the boss had a colour set!

2 Assuming water had to be heated from 20˚C to 100˚C, and the cup took 200 ml (200 cm3) of tea then the calculation is 4.2 x 80 x 200

4.2 J g-1K-1 is the approximate specific heat capacity of water.

Changing these parameters (perhaps you have a small tea cup and I use a mug?) will change the precise value.

3 That is the energy needed divided by the energy released by each neutron: 67200 J ÷ 1.25 x 10-13 J/neutron = 537 600 000 000 000 000 neutrons

4 E=mc2

so m = E/c2 = 67 200 ÷ (3.00 x 108)2 = 7.47 x 10-13

5 E=mc2 = 1.67 x 10-27 x (3.00 x 108)2 = 1.50 x 10-10

6 Well, we could imagine that somehow Hoffal had devised a process where the neutrons somehow redirect energy provided to initially generate the magnetic field, and perhaps the weapon was actually an enormous field generator producing a massive magnetic field that the funnel somehow converted into a beam (of tachyons?) that could pass across vast amounts of space without being absorbed by space dust, remaining highly collimated, and intense enough to destroy a world.

So, perhaps the neutrons are analogous to the core of a laser.

I somehow think it would still need more than 8 of them.

7 That is the energy needed divided by the energy released by each neutron: 67200 J ÷ 1.50 x 10-10 J/neutron = 4.48 x 1014 neutrons

8 Of course molecules are not actually malicious and neutrons cannot be naughty as they are inanimate entities. I am not anthropomorphising, just alliterating.

Is mass conserved when water gets soaked up?

Setting up a thought experiment on plant growth and mass

Keith S. Taber

Image by truthseeker08 from Pixabay 

Sophia was a participant in the Understanding Science Project.

I was aware that research has suggested that children often do not appreciate how carbon obtained from the carbon dioxide in the air is a key source of matter for plants to build up tissue, so learners may assume that the mass increase during growth of a plant will be balanced by a mass reduction in the soil it is growing in.

"The extra [mass of a growing tree] comes from the things it eats and drinks from the ground. It's just like us eating and getting larger."

Response of 15 year old student in the National science survey carried out the Assessment of Performance Unit of the Department of Education and Science, as reported in Bell and Brook, 1984: 12.

During an interview in her first year of secondary education (Y7), Sophia reported that she had been studying plants in science, and that generally a plant was "a living thing, that takes up things from soil, to help it grow" (although some grew in ponds). Sophia was therefore asked a hypothetical question about weighing a pot of soil in which a seed was planted, with the intention of seeing if she thought that the gain in mas of the seed as it grew into a mature plant would be balanced by a loss of mass from the soil.

Sophia was asked about a pot of soil (mass 400g) in which was planted a seed (1g), and which was then watered (adding 49g of water).

The scenario outlined to Sophia

There seemed two likely outcomes of this thought experiment:

  • A learner considers that the mass of pot, seed and water is collectively 450g, and assumes that as the mass of plant grows, the mass of soil decreases accordingly to conserve total mass at 450g.
  • A learner is aware that in photosynthesis carbon is 'captured' from carbon dioxide in the air, so the mass of the plant in the soil will exceed 450g once the plant grows.

Of course, a learner might also invoke other considerations – the evaporation of the water, or the acquisition of water due to condensation of water from cold air (e.g., dew); that soil is not inert, but contains micro-organisms that have their own metabolism, etc.

I first wanted to check that Sophia appreciated we had (400 + 1 + 49 =) 450g of material at the point the seed was first watered. That was indeed her initial thought, but she soon 'corrected' herself.

Any idea how much it would weigh now?

[Four] hundred and fifty, no, cause, no cause it will soak it up, wouldn't it, so just over four hundred (400).

So we had four hundred (400) grammes of soil plus pot, didn't we?

Uh hm.

…And we had one (1) gramme of erm, of plant seed. Just one little seed, one (1) gramme. And forty nine (49) grammes of water. But the water gets soaked up into the soil, does it? So when it's soaked up, you reckon it would be, what?

Erm, four hundred and twenty (420).

Sophia's best guess at the mass of the pot with soil (initially 400g) after planting a 1g seed and adding 49g of water was 420g, as the water gets soaked up.

So, Sophia suggests that although 49g of water has been added to a pot (with existing contents) of mass 401g , the new total mass will be less than 450g, as the water is soaking into the soil. Her logic seems to be that some of the water will have soaked into the soil, so it's mass is not registered by the balance.

If you poured the water in, quite quickly, not so quickly that it splashes everywhere, but quite quickly. Before it had a chance to soak up, if you could read what it said on the balance before it had a chance to soak up, do you think it would say four hundred and twenty (420) grammes straight away?

No, it would probably be just under, erm, four hundred and fifty (450).

And it would gradually drop down to about four twenty (420) say, would it?

Yeah.

Might be four hundred and fifteen? (415) Could be four hundred and twenty five (425)?

Yeah.

Not entirely sure,

No

but something like that?

Yeah.

It appears Sophia recognises that in principle there would be a potential mass of 450g when the water is added, but as it soaks up, less mass is registered.

Sophia recognises that mass is initially conserved, at least before the water soaks into the soil.

In other words Sophia in the context of water soaking into soil is not conserving mass.

This is a similar thought experiment to when students are asked about the mass registered during dissolving, where some learners suggest that as a solid dissolves the total mass of the beaker/flask plus its contents decreases, as if the mass of the dissolved material is not registered (Taber, 2002). In that case it has been mooted that ideas about buoyancy may be involved – at least when it is clear that the learners recognise the dissolved material is still present in the solution.

However, that would not explain why Sophia thinks the balance would not register the mass of water soaked into the soil in this case. Rather, it sees more a notion that 'out of sight' is out of mass. Sophia's understanding of what is happening to mass here would be considered an alternative conception or misconception, and is likely based on her intuition about the scenario (acting as a grounded learning impediment) rather than something she has been told.

Sources cited:

Current only slows down at the resistor

Current only slows down at the resistor – by analogy with water flow 

Keith S. Taber

Students commonly think that resistance in a circuit has local effects, and in part that is because forming a mental model of what is going on in circuits is very difficult. Often models and analogies can be useful. However when an analogy is used in teaching there is also the potential for it to mislead.

Amy was a participant in the Understanding Science Project. Amy (when in Y10) told me she had been taught to use a water flow analogy for electric current. However, because her visualisation of what happens in water circuits was incorrect, she used the analogy to inform an alternative conception about circuits:

Do you have any kind of imagined sort of idea, any little mental models, about what (the flow of electricity round the circuit) might look like? Do you have a way of imagining that?

Erm, yeah, we've been taught the water tank and pipe running round it. … just imagine the water like flowing through a pipe, and obviously like, if the pipe becomes smaller a one point, erm, the water flow has to slow down, and that's meant to represent the resistance of something.

So, so if I had my water, er, tank and I had a series of pipes, they'd be water flowing through the pipes, and if I had a narrower pipe at one point, what happens then?

The water would have to slow down.

So would it slow down just as it goes through the narrow pipe, or would it slow down all the way round?

Erm – just through that part.

(Amy does not appreciate the implications of conservation of mass {that is, the continuity principle} here – at steady state there cannot be a greater mass flow at different points in the circuit).

…

And so how do you imagine that's got to do with resistance, how does that help you understand resistance?

…well resistance, it slows the current down, but then erm, once it passes a resistor or something it, the current is free to flow through the wire again

Analogies can be very useful teaching tools, but when using them it is important to check that the students already understand the features of the analogue that are meant to be helpful. It is also important to ensure that they understand which features are meant to be mapped onto the target system they are learning about, and which are not relevant.

Analogies are only useful when the learner has a good understand of the analogue. In this case, as Amy did not appreciate that the water flow throughout the system would be limited by the constriction, she could not use that as a useful analogy for why a resistor influences current flow at all points in a series circuit. This is an example of where a teaching model meant to support learning, which actually misleads the learner. That is, for Amy, with her flawed understanding of fluid flow, the teaching model acted as a pedagogic learning impediment – a type of grounded learning impediment.