Mass-Energy Equivalence and First Law of Thermodynamics Einstein showed mass can be converted into energy and vice versa.
$E=mc^2$
However, in school we are taught that according to the First Law of Thermodynamics, energy can neither be created nor destroyed.
Are they not contradicting each other? I already tried finding it on other sites but was surprised that there was little information regarding this.
 A: 
Are they not contradicting each other?

Yes, there is a contradiction, but not between $E=mc^2$ and thermodynamics. The contradiction is between the actual meaning of $E=mc^2$ and its usual pop-science description. Unfortunately, although $E=mc^2$ is very famous, it is also very misunderstood. The usual English description of that equation is, as you stated, "mass can be converted into energy". However, that English description is obviously incorrect due to the conservation of energy, as you stated.
If scientists had wanted to write an equation that did state "mass can be converted into energy" then the equation would be $\Delta E = -c^2\Delta m$. This equation says that a negative change in mass gives a positive change in energy. That is what it would mean to convert between the two. So not only does $E=mc^2$ not describe a conversion, it is not intended to do so.
$E=mc^2$ is a statement at any given time. At that time a mass with no momentum has energy. There is no sense of conversion between the two. The mass and the energy are both present at the same time.
Note that I specifically said "a mass with no momentum". $E=mc^2$ only applies when $m$ has no momentum ($p=0$). In any scenario where $m$ has momentum then the more general formula is $m^2 c^2 = E^2/c^2 - p^2$.
A: I think the best way to resolve this question is to forget about more colloquial descriptions of $E=mc^2$ and just take a look at special relativity. SR imposes that the famous mass-energy relation (in natural units): $$ p_{\mu} p^{\mu} = E^2 - \vec{p}^2 = m^2$$ holds for all particles in all reference frames since $p_{\mu} p^{\mu}$ is a Lorentz invariant quantity. Furthermore, in SR we also have a conservation of total 4-momentum: $$p_{\text{initial}}^{\mu} = p_{\text{final}}^{\mu}$$ for any closed system.
In the nonrelativistic limit, the mass-energy relation reduces from $E^2 = \vec{p}^2 +m^2$ to $E = mc^2$ since the 3-momentum is small relative to the $mc^2$ term. However, the fundamental invariant quantity in SR is $p_{\mu} p^{\mu}$.
An example to think about mass and energy is found in nuclear physics. The proton and neutron can form a bound state called the deuteron with a mass $m_d = m_p + m_n - 2.2 MeV$. Thus, to break apart the deuteron, you need to put in at least 2.2 MeV of energy and you'll get out two particles with "greater mass than you started with" (seemingly $m_d \rightarrow m_p + m_n $, which apparently violates mass conservation). But to correctly understand what happened you must take both the energy and mass into account and you'll see conservation: $m_d + 2.2 MeV \rightarrow m_p + m_n$.
A: Mass is not converted to energy.  Mass is energy.  Sometimes mass energy can be converted into heat energy or light energy, or vise versa (as in chemical or nuclear reactions).  Thermodynamics always applies regardless to these transfers or conversion of energy.  In most thermodynamics and chemistry contexts, the amount of heat or mechanical (pressure) or chemical energy released or absorbed is incredibly small compared to the mass energy of the reactants and products - so when making computations, it is typically assumed that mass is constant, and only the other forms of energy are accounted for.
For example, in a reaction of 1 kg of hydrogen and oxygen gas:
$$ H_2+\frac 1 2 O_2 \rightarrow H_2O$$
the resulting water vapor product weighs about 0.1 micro grams ($1\times 10^{-10}$ th of a kg) less than the original mixture.  This minute difference accounts for the mass energy released as heat and light energy.  But in the context of chemistry, this is usually accounted for by saying the mass is constant (which is nearly true, and certainly to the limits of our measurements), while the heat and light energy came from the change in "chemical potential energy" associated with the reaction.
Bottom line is: $E=mc^2$ and thermodynamics are both always true, but the former is typically not relevant in chemistry or most everyday thermodynamic problems.
