Paradox of $E=mc^2$ in different states of matter We agree that energy is higher in liquids than solids and is higher in gases than liquids. How can we fit this into $E=mc^2$. As an example; 1 kg of ice has less energy than 1 L of water, right? But masses are the same specially in the case of water with density of 1,000 kg/m³. Could you please help me out with this paradox?!
All answers are appreciated.
 A: When one gram of ice becomes one gram of water, it gains 80 cal of energy, sometimes called latent heat.  80 cal is 335 J.  Einstein tells us that $$m=\frac{E}{c^2}$$ so the increase in the mass of the sample is $3.7 \times 10^{-12}$ grams.
Hardly measurable.
A: $E=mc^2$ is probably the most known and misunderstood equations in physics by the general population.  Lets look where it comes from.  The relativistic energy equation is $E=\gamma mc^2$ where $\gamma$ is the Lorentz factor $\gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$.  We set $\beta = \frac{v}{c}$ giving $\gamma = \frac{1}{\sqrt{1-\beta^2}}$.  Doing a Maclaurin expansion of the Lorentz factor around $\beta$ gives you get $\gamma\approx 1+\frac{1}{2}\beta^2+\frac{3}{8}\beta^4+...$
When we combine the expansion back into the energy equation we get:
$E\approx mc^2+\frac{1}{2}mc^2\frac{v^2}{c^2}+\frac{3}{8}mc^2\frac{v^4}{c^4}+...$
So velocity is $0$ this becomes the equation $E=mc^2$.  This is the mass energy conversion equation that everyone knows and loves.  It indicates how much energy you have if you convert mass into energy.  As velocity increases the other terms are important.  The first term simplified to $\frac{1}{2}mv^2$ is the kinetic energy term giving the connection to Newtonian physics.  This is also the term for things like temperature as temperature is the average random molecular kinetic energy in a system.  The third term $\frac{3}{8}m\frac{v^4}{c^2}$ is the first relativistic correction in special relativity.  Note that velocity must be very high ($\approx 0.75c$) before this term comes into noticeable effect.
A: To compare apples with apples, let's consider a particular substabnce that can exist in the three well-known phases: solid, liquid and gas. The thermal energy per unit volume (thermal energy density) in the gas phase would in general be larger than that of the liquid phase, which in turn would have a larger thermal energy density than the solid. However, the thermal energy is not the only form of energy that would be present in a particular volume of a articular substance. There can also be other forms of energy such as chemical energy. 
If we would consider the total mass of a particular volume of some substance in terms of energy through Einstein's relation $E=m c^2$ then we end up with an extremely large amount of energy that exceeds to amount of thermal energy by so many orders of magnitude that the contribution of thermal energy to the total amount of energy is completely insignificant. Therefore, within the accuracy, intrinsic in the statement that water has a density of 1000 kg/m$^3$, it would not be possible to notice the contribution of thermal energy.
A: In short: this is not the correct equation to describe the physical scenarios you have in mind.
$E=mc^2$ is probably the most misunderstood and misapplied equation in all of physics.  This equation gives some conversion factor ($c^2$) if some mass is converted to energy - think atomic bombs, and is not useful to describe the scenarios you're considering.
A more complete version would contain the kinetic energy term:
$E^2 = (mc^2)^2 + (pc)^2$.  
From this, you could calculate the average kinetic energy of your solid/liquid/gas, and would indeed find the total energy higher.  Alas, this is still incomplete, because it doesn't account for the potential energy of the system, which is important in this case.
