# 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

$$Q = 1\,\mathrm{kg} \cdot 4.19 \mathrm{\frac{kJ}{kgK}} \cdot 400K = 1676\,\mathrm{kJ} \quad ?$$

• Not in classical thermodynamics. $Q$ is defined as a class of equivalence of functions with the same $\mathrm{d}Q$. The interesting part is its increment, or differential, but $Q$ and $Q + k$, with $k \in \mathrm{R}$ contain the same information. Dec 3 '15 at 22:13

There are a few problems with your proposal:

1. 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.

2. 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).

3. 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).

• I might add that the symbol $\Delta Q$ in this context is meaningless, since $dQ$ is not an exact differential, or, more physically, because $Q$ is a process-variable/path-function, not a state-variable/state-function. Dec 4 '15 at 1:21