From my past studies I have had the presumption that for an ideal gas the equation:
$du=c_v dT$
( where $du$ is the density of internal energy, $c_v$ is the specific heat capacity at constant volume and $T$ is temperature)
is only valid when the process is isentropic. However searching the internet I'm doubting if this is really true. For example the wikipedia entry for the subject and this MIT blogpost, mention the equation without any assumptions such as being reversible, adiabatic or isentropic. So my question is if the equation above is valid all the time or just unders certain assumptions? if later then what are those assumptions and what is the general form? and if former is the case how we can prove that?
Best Answer
I assume by the "density of internal energy" $u$ you mean
$$u = \frac{E_{therm}}{n},$$
where $E_{therm}$ is the thermal energy of the gas, and $n$ is the number of moles.
In this case your equation is true for any ideal gas process. Here is how we can show that this is true.
We begin by considering an isochoric process (V=const). By definition, the amount of heat $Q$ needed to increase the temperature of gas by $\Delta T$ is
$$Q = n C_v \Delta T\tag{1},$$
where $C_v$ is the molar specific heat at constant volume. Since there is no change in volume, the work done is zero. Therefore, from the 1st law of thermodynamics:
$$\Delta E_{therm} = Q + 0 = Q.$$
Combining with Equation (1) we get:
$$\Delta E_{therm} = n C_v \Delta T. \tag{2}$$
Next we apply a key idea from thermodynamics: the change in internal energy $E_{therm}$ of gas is the same for any two processes that results in the same change in temperature $\Delta T$. Therefore, Equation (2) is true for any ideal gas process, and not just the isochoric process.
If we divide both sides by $n$, we get
$$\Delta u = C_v \Delta T,$$
which is what we needed to show.