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This is only true in the region 0 - 4°C, so you will need a freezer to demonstrate it.Fill a cup close to the brim with water, put it in the freezer, and stir it with a thermometer. When it reaches about 5°C, top it up to the brim, then watch what happens as it cools further. It will spill over the edge just before it freezes because cold water takes up more space (i.e. is less dense) than warm. If you use a conical bottle, say a wine bottle, the effect will be more spectacular.

However mass is different. According to Einstein's theory of relativity E =mc^{2} which means that given a finite amount of mass, when you add energy to it then it's mass will increase. However this increase is very very small. Too small in fact to measure in the laboratory.

How can i prove to my boss

But, cold water is more compressible than warm water. That is, it is easier to deform a cold parcel than a warm parcel. Therefore cold water becomes denser than warm water when they are both submerged to the same pressure.

However mass is different. According to Einstein's theory of relativity E =mc^{2} which means that given a finite amount of mass, when you add energy to it then it's mass will increase.

So please let me have your calculation of the relativistic increase in molecular mass over the phase range of liquid water,

Ok,So for a little applied math.E=mc^{2}m = E/c^{2}Now, let's assume that due to global warming, the oceans on Earth have uniformly warmed by 1°C (1 calorie/cc, or 4.184 joules/cc)Ok, so there is about 1,350,000,000 km^{3} of water in the oceans.10dm/m, 1000m/km, 10,000 dm/kmAnd one gets about:1,350,000,000,000,000,000,000 dm^{3} of water, so about 1.35 x 10_{21} kg, or 1.35 x 10^{24} ccAnd, raising the temperature by 1°C (1 calorie/cc, or 4.184j/cc), one gets 5.65 x 10^{24} joules.Now, for the speed of light, 299,792,458, and c^{2} = 9 x 10^{16} m^{2}/s^{2}.So,m/c^{2} = 5.65 x 10^{24} joules / 9x10^{16}m^{2}/s^{2}.And, you get that the oceans have increased in mass by about 62,800,000 kgWhew, more than I expected.---------------------------Perhaps one should look at it in grams per cubic km of water.So, one cubic km of water is: 1,000,000,000,000,000 (1x10^{15} cc)And, raising 1 cubic km of water by 1°C is about 4.184x10^{15} joules.So,4.184x10^{15} joules / 9x10^{16}m^{2}/s^{2} = 0.0465 kg/cubic km,or about 46.5 grams/cubic km of water per degree Celsius.

Except that the water is hotter. This means that the molecules have more kinetic energy, not that they have gained in mass - you can't have your cake and eat it!

I was pointing out an error in Clifford's arithmetic. I would be grateful for your estimate of the mass gain from heating say a gram of water through 1 deg C.

The momentum argument doesn't hold for more than one molecule as they are moving randomly so the net momentum is always zero, regardless of temperature. Momentum is a vector.

E.g. it weighs more and has greater momentum which in relativity goes into an increase in inertial mass.

How can i prove to my boss that hot water is heavier than cold water ...

... in water below about 4 °C, warmer water sinks whereas when above about 4 °C, warmer water rises ..."

Hey! I thought I was the King Nitpicker round here! So just to reassert my positionQuoteHow can i prove to my bossYou can't get away with theoretical physics, Pete! This guy is asking for proof.

A theorem is a statement that can be demonstrated to be true by accepted mathematical operations and arguments. In general, a theorem is an embodiment of some general principle that makes it part of a larger theory. The process of showing a theorem to be correct is called a proof.

If a body gives off the energy L in the form of radiation, its mass diminishes by the amount L/c^{2}. The fact that the energy withdrawn from the body becomes energy of radiation evidently makes no difference, so that we are led to the more general conclusion that - the mass of a body is a measure of its energy content: if the energy changes by the amount L, the mass changes in the same sense by L/9x10^{20}, the energy measured in ergs, and the mass in grammes.

Sorry, you have repeated Clifford's mistake. You have calculated the mass that would have to be annihilated to produce 1 calorie of energy.

Point conceded, I think.

Now the question revolves around the definition of "heavier" and "proof".

Once again, I have to ask you to do the calculation, ...

michael clark asked the Naked Scientists: How can i prove to my boss that hot water is heavier than cold water in such away that he can understand? As simple as possible please. Thank youWhat do you think?

Similarly you might consider that your cup of coffee is more massive when it’s hot – and gets measurably less massive when it cools down. Matter, in terms of protons, neutrons, electrons …and coffee, is largely conserved throughout this process. But, for a while, the heat energy really does add to the mass of the system – although since it’s a mass of m=e/c^{2}, it is a very tiny amount of mass.

Generally hotter water is less dense than cooler water , but it's the other way round between 0°C to 4°C ...Quote from: lsbu.ac.uk

Point conceded, I think.Now the question revolves around the definition of "heavier" and "proof". And then we have to consider the range of validity of that proof. Whilst it is obviously true that a single molecule gains mass as it gains velocity (but relative to what, in the case of an isolated molecule in space?), "hotter" is not defined for a single molecule - temperature is an ensemble property, but we'll return to that in a moment. Consider two isolated molecules, with different velocity vectors. How do we measure their mass? In principle (I'll grant you this bit is well outside the realm of experimental physics) the only way is to measure the gravitational force exerted on a test mass that is stationary with respect to each molecule. And to our (theorietical) astonishment, they both have the same mass. Back to the ensemble. Even if the said boss is indeed conversant with relativistic physics, he will still need an exceptional grasp of the Gibbs energy of dynamic Voronoi polyhedra if he is going to accept a theoretical proof, and despite advances in chaos theory since I last worked on shortrange order in water, the numbers remain very vague and entirely experimental. So we have to subtract the calculated relativistic contribution to the ensemble mass from the currently-incalculable density variation. Once again, I have to ask you to do the calculation, on the assumption that the said boss will understand it. I'll be generous here and let you use RD's (experimental) graph, just this once, to check your answer. And perhaps you will let us know, as an aside, how many angels can sit on the head of a pin!Frankly, I think putting a wine bottle in a freezer is a lot easier. But then I'm an experimentalist.PS - belated good Thanksgiving wishes. Thanks to the Pilgrim Fathers I now have plenty of wine bottles and an empty freezer!

Quote from: alancalverdExcept that the water is hotter. This means that the molecules have more kinetic energy, not that they have gained in mass - you can't have your cake and eat it! In Newtonian mechanics this is true. In relativistic mechanics its not true. However in the later case the increase is very very small. Too small in fact to be measured in a typical lab. One needs a mass spectrometer to measure such a small increase.

I haven't understood what you say here. Are you talking of a mass increase of a particle with speed or what?

And what are you saying about a mass spectrometer?

That you could measure a particle's mass increase with speed using that device? If you are saying this, it's incorrect, and not because it would be too small, but because you can't.

If you are saying this, it's incorrect, and not because it would be too small, but because you can't.

Quote from: alancalverdConsider two isolated molecules, with different velocity vectors. How do we measure their mass? Their relativistic mass does not depend on its direction so we only need to measure its mass as a function of speed.

Consider two isolated molecules, with different velocity vectors. How do we measure their mass?

To do that one can measure the mass of a moving particle such as a molecule by ionizing it and using a cyclotron to measure its mass by determining it's deflection by a magnetic field and measuring its radius of orbit. You're an experimental physicist aren't you? If so they why didn't you know this?

That's why I asked you to consider two particles with different velocity vectors.

But their masses, as determined by the gravitational force on a test particle travelling with each (the only way you can determine the mass of a free asteroid), must be constant and identical because there is no universal frame of reference and neither rock is moving with respect to its test mass. Or variable with a. Or fixed but different according to b and c. Please choose, and show your reasoning.

He asked for a proof that would convince his boss. I gave him a simple experiment that showed water in the range below 4°C was lighter than water at 4°C and this was unique,

The following statement was originally drafted by the Panel on Public Affairs (POPA) of the American Physical Society, in an attempt to meet the perceived need for a very short statement that would differentiate science from pseudoscience. Am. J. Phys. 67 (,

I derived some of the quantities on paper, but that is far from a rigorous empirical proof, and it would be difficult to weigh a cubic km of water or ice to an accuracy of a few grams.

I'm not sure a mass-spec has high enough of accuracy.

It also requires ions which may be problematic for a liquid water sample, but should still be representative.

Is, say, altering the temperature of a set of ions representative of the change in mass, or are there other confounding variables?

This is all fascinating stuff, but from the point if view of psychology and philology, not physics!

But the real point of argument is the meaning of "heavier".

Given that the original questioner didn't know how to demonstrate the effect, I think his enquiry was based on the common knowledge and observation of the anomalous expansion of water ...

..rather than the subtle timing corrections of a cyclotron driver - especially as you can't accelerate a water molecule in a cyclotron.

Therefore he was talking about bulk density, not molecular mass.

Quote from: lightarrowI haven't understood what you say here. Are you talking of a mass increase of a particle with speed or what?Yes. I'll restate it to make it clearer. In what follows what I mean by the use of the term mass I mean inertial mass in the Newtonian case and relativistic mass in the relativistic case.In Newtonian mechanics mass does not increase with speed.In relativistic mechanics mass does increase with speed.Simple, yes? Quote from: lightarrowAnd what are you saying about a mass spectrometer?That's one way to measure the relativistic mass of a charged particle.Quote from: lightarrowThat you could measure a particle's mass increase with speed using that device? If you are saying this, it's incorrect, and not because it would be too small, but because you can't.That's incorrect. In fact Wikipedia states that in 1901 Walter Kaufmann used a mass spectrometer to measure the relativistic mass increase of electrons. See ...sorry, you cannot view external links. To see them, please REGISTER or LOGIN).But if you intended that the mass spectrometer allows you to find the momentum |p| and so the total energy E and you identify the last with relativistic mass (multiplied c^{2}), then we again come back to what I have said before several times in various threads, even answering to you, that is that relativistic mass is nothing else than another name for total energy E.Edit: coloured in blue eq. (6) - 01/12/2014--lightarrow

Quote from: PmbPhy on 28/11/2014 21:16:39Quote from: lightarrowI haven't understood what you say here. Are you talking of a mass increase of a particle with speed or what?Yes. I'll restate it to make it clearer. In what follows what I mean by the use of the term mass I mean inertial mass in the Newtonian case and relativistic mass in the relativistic case.In Newtonian mechanics mass does not increase with speed.In relativistic mechanics mass does increase with speed.Simple, yes? Quote from: lightarrowAnd what are you saying about a mass spectrometer?That's one way to measure the relativistic mass of a charged particle.Quote from: lightarrowThat you could measure a particle's mass increase with speed using that device? If you are saying this, it's incorrect, and not because it would be too small, but because you can't.That's incorrect. In fact Wikipedia states that in 1901 Walter Kaufmann used a mass spectrometer to measure the relativistic mass increase of electrons. See ...sorry, you cannot view external links. To see them, please REGISTER or LOGIN).But if you intended that the mass spectrometer allows you to find the momentum |p| and so the total energy E and you identify the last with relativistic mass (multiplied c^{2}), then we again come back to what I have said before several times in various threads, even answering to you, that is that relativistic mass is nothing else than another name for total energy E.--lightarrowI am so glad that you posted this. I don't think many will appreciate the subtleties.

Quote from: lightarrowI haven't understood what you say here. Are you talking of a mass increase of a particle with speed or what?Yes. I'll restate it to make it clearer. In what follows what I mean by the use of the term mass I mean inertial mass in the Newtonian case and relativistic mass in the relativistic case.In Newtonian mechanics mass does not increase with speed.In relativistic mechanics mass does increase with speed.Simple, yes? Quote from: lightarrowAnd what are you saying about a mass spectrometer?That's one way to measure the relativistic mass of a charged particle.Quote from: lightarrowThat you could measure a particle's mass increase with speed using that device? If you are saying this, it's incorrect, and not because it would be too small, but because you can't.That's incorrect. In fact Wikipedia states that in 1901 Walter Kaufmann used a mass spectrometer to measure the relativistic mass increase of electrons. See ...sorry, you cannot view external links. To see them, please REGISTER or LOGIN).But if you intended that the mass spectrometer allows you to find the momentum |p| and so the total energy E and you identify the last with relativistic mass (multiplied c^{2}), then we again come back to what I have said before several times in various threads, even answering to you, that is that relativistic mass is nothing else than another name for total energy E.--lightarrow

Yup. It happens every time I make ice cubes. I'm very surprised that one got by an experimental physicist like Alan.

And what's a "cyclotron driver" anyway and how is it related to this topic?

There is absolutely no reason why a molecule of water, i.e. an H2O ion couldn't be accelerated in a cyclotron.

Quote from: PmbPhy on 29/11/2014 05:17:46Yup. It happens every time I make ice cubes. I'm very surprised that one got by an experimental physicist like Alan. The OP concerned water, not ice. Expansion on solidification is not unusual - printers' "hot type" and several other materials do it - but the expansion of liquid water over a significant range of cooling is remarkable and very important. Quote And what's a "cyclotron driver" anyway and how is it related to this topic? It's the electronic gubbins that turns the cyclotron from a drawing into a working machine. But don't worry your pretty head about that nasty engineering stuff - you'll get you hands dirty! QuoteThere is absolutely no reason why a molecule of water, i.e. an H2O ion couldn't be accelerated in a cyclotron. A molecule is not an ion. Which seems a good reason to me.

The OP concerned water, not ice.

But don't worry your pretty head ...

A molecule is not an ion. Which seems a good reason to me.

: an atom or group of atoms that has a positive or negative electric charge from losing or gaining one or more electrons

No need of relativistic mass.

A molecule yes but look at the hydrogen bonding.

A molecule is by definition electrically neutral and thus cannot be accelerated by a cyclotron.

While H_{2}O is not an ion, ...

A molecule yes...

To do that one can measure the mass of a moving particle such as a molecule by ionizing it and using a cyclotron to measure its mass by determining it's deflection by a magnetic field and measuring its radius of orbit.

Quote from: lightarrow on 30/11/2014 15:44:29...|p| = q*|B|*r... I am so glad that you posted this. I don't think many will appreciate the subtleties.

...|p| = q*|B|*r...

A benchtop mass spectrometer running at a few keV probably won't show the relativistic change of mass to a detectable level ...but a hospital cyclotron certainly will,

Quote from: lightarrowNo need of relativistic mass.So what? Do you realized just how many concepts there are in physics that aren't "needed"? In fact Hertz wrote an entire mechanics text without using the concept of force. I could write an entire textbook on relativity without using the concept of energy either. Let me show you how easy it'd be to derive the cyclone formula by using relativistic mass. Let B = Bk where B is a constant. Let v start out in the xy-plane. ThenF = qvxBmeans that the force is perpendicular to the velocity which means the charge moves in a circle and the force is perpendicular to the velocity. As such F = ma. If you're not aware that this is true then see...sorry, you cannot view external links. To see them, please REGISTER or LOGINF = ma = mv^{2}/r = qvBor mv = qrB = pwhich is the cyclotron relation, i.e.mv = qrBWhich requires a great deal less work that what you did.

lightarrow - I don't know why you're trying so hard to make this a debate about relativistic mass but I'm really not interested and I won't discuss it.

Why didn't you write the equation:F = ma = mv^{2}/r = qvBwith F, a, v, B in bold carachter?

... inertia and energy refer to conceptually different features of matter. On the one hand, inertia per se relates to the resistance to the change of state of motion of a body, as would be caused, in 'particle physics,' by an external force acting on the body (...). On the other hand, energy per se is defined in terms of the solutions of the conservation laws..... Thus we see that energy and mass are conceptually different (though complimentary) features of matter, according to relativity theory.