# Is the molecule of hot water heavier than that of cold water?

We know that the molecule of hot water($H_2O$) has more energy than that of cold water (temperature = energy) and according to Einstein relation $E=mc^2$ ,this extra energy of the hot molecule has a mass. Does that make the hot molecule heavier?

$E^2 = m^2c^4 + p^2c^2$, where $m$ is rest mass and $p$ is momentum. If a molecule is moving faster it would have more momentum and more energy, but the same rest mass.

Some have defined "relativistic mass" as opposed to "rest mass" as $E=m_rc^2$, so yes the faster moving molecule would have a greater so-called relativistic mass.

• How evil not to provide a ROM value for the change in mass :-) – Carl Witthoft Apr 21 '14 at 14:44

According to http://www.verticallearning.org/curriculum/science/gr7/student/unit01/page05.html , the average velocity of a water molecule at 20C is approximately 590 m/s:

If you now calculate the difference in mass between water molecules "at rest" and moving at 590 m/s, this ratio is given by

$$\gamma=\sqrt{\left(1-\frac{v^2}{c^2}\right)}$$

Expanding this for the case where $v<<c$, we get

$$\frac{\delta m}{m} = \frac12\left(\frac{v}{c}\right)^2 \sim 2\cdot10^{-12}$$

The difference is there; it might be measurable; but it is very, very tiny. Any condensation on your colder sample will be many times heavier than the difference in mass.

Note that the above calculation is for "cool water" vs "water at absolute zero". The difference between "hot" and "cold" water will be even smaller. Also note that to do this calculation properly, you really need to evaluate the expression for $\gamma$ at every velocity, and integrate. This doesn't change the basic answer: "yes it changes; but it is a very very small change".

I am going to take a different approach from DavePhD and Floris.

"Hot"ness or temperature more generally is a thermodynamic idea, and can't really be applied to an individual molecule. Dave and Floris have avoided the issue by simply comparing and energetic molecule with a less energetic one, and that is reasonable, but it makes their answers frame-of-reference dependent. Presumably they are working in the rest frame of the macroscopic samples from which they drew their test particles. All very reasonable.

I'm going to make my usual argument about scale of inspection.

A mole of hot water is more massive than a mole of cold water because when examined at the human scale you can't differentiate the kinetic energy of the molecules from any other form of internal energy (and energy is mass). The scale of this change is that figured by Floris.

Examined at the level of a single molecule than each particle has the same mass--properly defined as the Lorentz scalar formed by contracting it's four-momentum with itself: $m^2 = \bar{p}\cdot\bar{p}$--and the fast molecule has more kinetic energy--$T = E^2 - m^2 = (\gamma - 1)m$--than the slow one. This approach follows the conventions of particle physics where we don't use the term "relativistic mass".

The upshot is a "Yes and no" answer, or perhaps a "You're not quite asking the right question".

• I respectfully disagree: temperature is a measure of the logarithm of the number of available energy states, so even a single molecule can have a temperature. I'm of the position that your linked Q&A does not support your position. – Carl Witthoft Apr 21 '14 at 17:08

Hot water is lighter. Hot water molecules take up more space.

• This is cute, if false. Is a pound of bricks heavier than a pound of feathers? – user121330 May 17 '17 at 15:03