[thedoc (OP)]

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 you
What do you think?

[alancalverd]

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.   

[PmbPhy]

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.   

Depending on the exact meaning of what the OP's question really is, i.e. what he's seeking to learn, that's wrong. You appear to be addressing the question about the density of water and not the mass of the same amount of water. Water becomes denser when the temperature is between 0 and 4C as you said. That's a physical fact and its why ice doesn't form on the bottom on lakes but on the top.

However mass is different. According to Einstein's theory of relativity E =mc² 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.

[evan_au]

You could try it with a clear container and a jug of hot and cold water, each containing a different food dye.

Pour water from one jug into the clear container, and wait for circulation to almost stop. Then very carefully add water from the second container. There will be some mixing at the interface, but overall the less dense fluid will remain at the top.
Placing the more dense fluid on the top will cause more mixing.

But most temperatures that you try for this experiment (especially around "room temperature") will show that the cooler water is more dense.

[alancalverd]

However mass is different. According to Einstein's theory of relativity E =mc² 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.

Hey! I thought I was the King Nitpicker round here! So just to reassert my position

How can i prove to my boss

You can't get away with theoretical physics, Pete! This guy is asking for proof. So please let me have your calculation of the relativistic increase in molecular mass over the phase range of liquid water, and an indication of how he can measure it against the background of the variable density of the material. You have until nightfall (London time) to comply, otherwise .... off to the Tower with you, and may the executioner's blade be sharp!

[Roadrunner]

Will he believe that all scuba divers know that cold water sits on the bottom of a body of water, and warm water sits on top of the cold water?

Even on cloudy days - the New England Atlantic Ocean is one example of this, though this principle holds true in fresh water too. The water on top of a body of water can be 4 to 11 C, and the water at the bottom (all others variables being equal) will measure colder.

Tell him to scuba dive to find his "proof,"  or just pull out any oceanography book. It is a very basic fact that cold water is heavier than warm water.

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.

newbielink:https://www.rsmas.miami.edu/personal/lbeal/MPO%20503/Lectures%203%264.html [nonactive]

That's about as simple as one can have it.

[lightarrow]

However mass is different. According to Einstein's theory of relativity E =mc² which means that given a finite amount of mass, when you add energy to it then it's mass will increase.

...if that amount of mass is still, otherwise it's not true ( I know that you knows and I wouldn't give the idea that I'm correcting you because I believe you weren't aware of it 😊).

--
lightarrow

[lightarrow]

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

Sorry, what do you mean with "relativistic increase in molecular mass over the phase range of liquid water"?

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lightarrow

[CliffordK]

Ok,
So for a little applied math.

E=mc²
m = E/c²

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 km³ of water in the oceans.
10dm/m, 1000m/km, 10,000 dm/km
And one gets about:
1,350,000,000,000,000,000,000 dm³ of water, so about 1.35 x 10²¹ kg, or 1.35 x 10²⁴ cc
And, raising the temperature by 1°C (1 calorie/cc, or 4.184j/cc), one gets
5.65 x 10²⁴ joules.

Now, for the speed of light, 299,792,458, and c² = 9 x 10¹⁶ m²/s².

So,
m/c² = 5.65 x 10²⁴ joules / 9x10¹⁶m²/s².

And, you get that the oceans have increased in mass by about
62,800,000 kg

Whew, 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 (1x10¹⁵ cc)
And, raising 1 cubic km of water by 1°C is about 4.184x10¹⁵ joules.

So,
4.184x10¹⁵ joules /  9x10¹⁶m²/s² = 0.0465 kg/cubic km,

or about 46.5 grams/cubic km of water per degree Celsius.

[jeffreyH]

Ok,
So for a little applied math.

E=mc²
m = E/c²

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 km³ of water in the oceans.
10dm/m, 1000m/km, 10,000 dm/km
And one gets about:
1,350,000,000,000,000,000,000 dm³ of water, so about 1.35 x 10₂₁ kg, or 1.35 x 10²⁴ cc
And, raising the temperature by 1°C (1 calorie/cc, or 4.184j/cc), one gets
5.65 x 10²⁴ joules.

Now, for the speed of light, 299,792,458, and c² = 9 x 10¹⁶ m²/s².

So,
m/c² = 5.65 x 10²⁴ joules / 9x10¹⁶m²/s².

And, you get that the oceans have increased in mass by about
62,800,000 kg

Whew, 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 (1x10¹⁵ cc)
And, raising 1 cubic km of water by 1°C is about 4.184x10¹⁵ joules.

So,
4.184x10¹⁵ joules /  9x10¹⁶m²/s² = 0.0465 kg/cubic km,

or about 46.5 grams/cubic km of water per degree Celsius.

Use what he said that should impress the boss and scare the hell out of him thinking he is about to drown.

[alancalverd]

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! What you have proved is that the energy required to raise the temperature of the oceans by 1 degree could be generated by the annihilation of 62.8 kilotons of matter. 

[PmbPhy]

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!

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.

When the kinetic energy of a particle increases then so does it's mass since the particle is moving faster and mass depends on speed. Or to look at it another way (for those who abhor velocity dependent mass) The rest mass of liquid is a function of its energy content. When the liquid absorbs energy the kinetic energy of its atoms/molecules increases making them move faster. When they move faster the system of all the molecules has a greater inertia than when it was cooler. E.g. it weighs more and has greater momentum which in relativity goes into an increase in inertial mass.

I'm surprised you didn't know this alancalverd!? Or do you but neglected to mention it for some reason?

[alancalverd]

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.   

[PmbPhy]

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.

That's trivial since that's defined to be an amount of energy equal to one calorie which equals 4.148 J. Since E = mc² we have m = E/c² = (4.184 J)/(3x10⁸)² = 4.649x10^-17 kg

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.

That is incorrect. I'm sorry but you misunderstood what I said. When I wrote

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

it meant that if it was being weighed then it would weigh more after it was heated than it was before and when it was moving then its momentum would be greater after it was heated than before it was heated.

Also, I obviously know that momentum is a vector. In relativity the relevant momentum is the 4-momentum. Don't forget that I'm a physicist too.

[alancalverd]

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

[RD]

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

Generally hotter water is less dense than cooler water , but it's the other way round between 0^oC to 4^oC  ...

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

http://www1.lsbu.ac.uk/water/density_anomalies.html#density

[PmbPhy]

Hey! I thought I was the King Nitpicker round here! So just to reassert my position

How can i prove to my boss

You can't get away with theoretical physics, Pete! This guy is asking for proof.

I just saw this. I missed it when it was first posted. When someone asks for proof it by no means experimental proof. One would have to specify "experimental proof" for that to be a fact.

The reason I can "get away with theoretical physics" is that one can provide poof theoretically in cases such as this.

Recall that we use math heavily in physics to derive results. Recall what a theorem is

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.

All the results that we have in physics that aren't laws are theorems of this nature and have been "proved" in this way.

To do so means to apply logic. Start with one or more propositions. These propositions serve as the premises of the argument. One then applies the appropriate reasoning which leads to the final proposition that one was seeking to arrive at known as the conclusion. This is how theorems are said to be proved. Such conclusions are quite general. Otherwise for an experimental "proof", i.e. if one demonstrates something in the lab, then that's more of a confirmation of a proposition rather than a proof of it.

Here I'm assuming that you knew that theorems were used in physics as well as in math too, correct? This is obvious because the language of physics is math. They appear when one uses math to prove a physical result. For example; Newton's Shell Theorem and Poynting's theorem theorem are two such example

Here is a longer list of examples: http://en.wikipedia.org/wiki/Category:Physics_theorems

In the case E = mc² is a physical theorem that was proved by Einstein in 1905. He assumed that in the inertial rest  frame S of an object an amount of radiation was emitted in opposite directions in equal amounts. The total amount of radiation emitted was L. Then the system was analyzed by a frame S' which was moving at a constant velocity relative to S where the speed of that frame moved very slowly with respect to the speed of light.

Einstein then showed that the kinetic energy of the body as measured in S' was reduced by the amount (1/2)(L/c²)v². From this Einstein concluded that

If a body gives off the energy L in the form of radiation, its mass diminishes by the amount L/c². 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/9x10²⁰, the energy measured in ergs, and the mass in grammes.

For different but  similar detailed derivation of E = mc² please see my website at
athttp://home.comcast.net/~peter.m.brown/sr/mass_energy_equiv.htm

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

Clifford made no mistake, you did. Nothing personal Alan. However I'm quite surprised by this given your statement that you're an experimental physicist.

[alancalverd]

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!

[PmbPhy]

Point conceded, I think.

I'm delighted to hear you say that. It demonstrates just how well you understand the mass-energy relation and the scientific method.  My compliments!

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

Those terms are well known and as such there shouldn't be any confusion about it. However I'll take these as challenges and go on. First off the term weight is well defined as the force on the object due to gravity. To be precise, consider a scale which is at rest in a gravitational field (e.g. sitting on a bench top in a lab). We then place a container of water of it. Then the force exerted on the scale to the force of gravity pressing down on the scale. That force is defined as the "weight" of the object. Please see my derivation on this in my website at
http://home.comcast.net/~peter.m.brown/gr/weight_moving_body.htm

Regarding "proof" I already elaborated on that in my last post. I think it only applies to theorems but I'm still studying the philosophy of science. In the meantime please see
http://www.psychologytoday.com/blog/the-scientific-fundamentalist/200811/common-misconceptions-about-science-i-scientific-proof

Misconceptions about the nature and practice of science abound, and are sometimes even held by otherwise respectable practicing scientists themselves.  I have dispelled some of them (misconceptions, not scientists) in earlier posts (for example, that beauty is in the eye of the beholder, beauty is only skin-deep, and you can’t judge a book by its cover).  Unfortunately, there are many other misconceptions about science.  One of the most common misconceptions concerns the so-called “scientific proofs.”  Contrary to popular belief, there is no such thing as a scientific proof.

http://en.wikipedia.org/wiki/Scientific_evidence#Concept_of_.22scientific_proof.22

While the phrase "scientific proof" is often used in the popular media,[13] many scientists have argued that there is really no such thing. For example, Karl Popper once wrote that "In the empirical sciences, which alone can furnish us with information about the world we live in, proofs do not occur, if we mean by 'proof' an argument which establishes once and for ever the truth of a theory,"[14] and Satoshi Kanazawa has argued that "Proofs exist only in mathematics and logic, not in science."[15]

This is why I only spoke of proof in the mathematical sense in my last post.



See also

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,...

When an author speaks of the velocity or speed of an object in the context of relativity they're referring to the speed relative to a particular frame of reference. For example; if I said instead "the speed of the particle with respect to frame S" then that's very specific. However it has the same content as the other way since in this way I didn't say with respect to what S is defined relative to either. So I leave it out and assume that the reader knows what it means. Especially ones who appear educated in relativity as you appear to be.

... "hotter" is not defined for a single molecule - temperature is an ensemble property, but we'll return to that in a moment.

See again http://home.comcast.net/~peter.m.brown/gr/weight_moving_body.htm

Notice that in that page I refer to a container containing a large number of particles moving randomly. For the weight of the box to increase with an increase in energy then the weight of each and every particle must increase. Since the box contains a large number of particles moving randomly its safe to speak of temperature and an input of heat.

Consider 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. 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. See
http://home.comcast.net/~peter.m.brown/sr/cyclotron.htm

You're an experimental physicist aren't you? If so they why didn't you know this?

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

I don't see the point. I'm going to assume that "michael clark" is satisfied.

[PmbPhy]

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 you
What do you think?

See also Astronomy Without A Telescope – Mass Is Energy
http://www.universetoday.com/91132/astronomy-without-a-telescope-mass-is-energy/

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/c², it is a very tiny amount of mass.

I'd like to note that the assertion in the title, Mass Is Energy, is wrong. Mass is most certainly not energy!! To explain why takes an entire thread/physics journal article. I can upload one to my website and post the link to it here if anybody seriously wants to read it? Mendel Sachs wrote a paper on this subject.