Internal energy of a body I have a really fundamental question regarding the internal heat energy of a body. From textbooks, we know:
$$\Delta Q = m \cdot c \cdot \Delta T$$
where $m$ is the mass of the body,
$c$ its specific heat and
$\Delta T$ the temperature difference.
The last part of this equation implies that their need to be a difference in temperature to get a difference of the heat. Is it possible to say that the "absolute" amount of internal heat is defined as:
$$Q = m \cdot c \cdot T,$$
where $T$ is the absolute temperature?
Think about $1\,\mathrm{kg}$ of water with a specific heat of $4.19\,\mathrm{\frac{kJ}{kgK}}$ at a temperature of e.g. $400\,\mathrm{K}$. Is it correct to state the internal heat (or internal heat energy) then equals to
$$Q = 1\,\mathrm{kg} \cdot 4.19 \mathrm{\frac{kJ}{kgK}} \cdot 400K = 1676\,\mathrm{kJ} \quad ?$$
 A: There are a few problems with your proposal:


*

*The specific heat capacity is temperature dependent (and approaches zero as the absolute temperature approaches zero). So at least the equation must be written
$$ Q = m \int_0^T dT\, c(T). $$
The $c$ usually given is the $c$ at a specific temperature (and it is assumed to change slowly around this (e.g. room) temperature). But this assumption does not always hold: For example at phase transitions (e.g. when water freezes) the heat capacity may even jump.

*There is another effect that may occur at phase transitions – latent heat. When water freezes (or condensates) it will release heat without a temperature change. This latent heat has to be included in the equation above (it can be modelled crudely by a $\delta$ spike in the heat capacity).

*You have to specify what happens with the other variables (is pressure or volume constant), therefore the correct quantity is not the "total internal heat $Q$" but the so called internal energy $U$. This is the quantity occurring in the first law of thermodynamics (also known as energy conservation):
$$ dU = dQ + dW. $$
Only $U$ is a quantity that is independent of the process (a "state function").
The heat $Q$ and the work $W$ are process dependent ("process functions"), while $U$ is a function of the state given by entropy $S$, the volume $V$ and the particle number $N$.
So in summary: There is such a quantity that is kind of a "total internal heat" called the "internal energy" $U$. It is constructed in a way that encodes energy conservation in the occurring processes and it is independent of the process that leads to a given state. But one cannot simply calculate it as $U = m c T$, because of some subtleties.
Note, that the change in internal energy is given by
$$\Delta U = c_V m \Delta T $$
if $c_V$ is constant in this temperature range and is the heat capacity at constant volume (this formula is of course limited if there are other thermodynamic variables, such as magnetizations).
