Why is there a spike in the heat capacity of a diatomic gas, at around the rotational temperature of the molecule? While studying for my Statistical thermodynamics test, I encountered this graph 

Source: https://www.physicsforums.com/threads/variation-of-specific-heat-with-temperature.399514/
I know this isn't the best graph you'll ever see, but the "bump" was present on several other graphs as well.
So just to clarify my question once more; I'm looking for a physically intuitive explanation of the "bump", which occurs at temperatures, where rotational degrees of freedom become relevant. 
Thank you!
 A: The bump can be observed through an explicit calculation.
If $\it l$ is the angular momentum quantum number of a molecule, then the rotational energy levels are
$$ \varepsilon_\text{rot} = \frac{\hbar^2}{2I}\it l(\it l + 1) = \frac{k\theta_\text{r}}{2}  \it l(\it l + 1) \,, \quad \textrm{ where }\quad \theta_\text{r} \equiv \frac{\hbar^2}{Ik} \,,$$
and $I$ is the moment of inertia of the molecule.
Since each $\it l$ is $(2\it l + 1)$-fold degenerate, the partition function over each rotational mode reads
$$ 
Z_\text{rot} = \sum_{l=0}^\infty (2\it l+1)\exp\left( - \frac{\theta_\text{r}}{2T}  \it l(\it l + 1) \right) = \begin{cases}
1 + 3 e^{-\theta_r/T} + 5e^{-3\theta_r/T} + \mathcal O\left( e^{-6\theta_r/T} \right) &\text{for } T \ll \theta_r \,, \\
2\frac{T}{\theta_r} + \frac13 + \frac1{30}\frac{\theta_r}{T} + \mathcal O\left( \left( \frac{\theta_r}{T} \right)^2 \right) &\text{for } T \gg \theta_r \,.
\end{cases}
$$
Using this, we can calculate the contribution to the internal energy per rotational degree of freedom.
$$
E_\text{rot} = NkT^2\frac{\partial}{\partial T}\ Z_\text{rot} = \begin{cases}
3Nk\ \theta_r\left( e^{-\theta_r/T} - 3e^{-2\theta_r/T} + \dots \right) &\text{for } T \ll \theta_r \,, \\
NkT\left(1 - \frac{\theta_r}{6T} - \frac{1}{180}\left(\frac{\theta_r}{T}\right)^2 + \dots \right) &\text{for } T \gg \theta_r \,.
\end{cases}
$$
Therefore, the contribution to heat capacity at constant volume by each rotational mode is
$$
C_V^\text{rot} = \frac{\partial }{\partial T}E_\text{rot} = Nk \begin{cases}
3 \left(\frac{\theta_r}T\right)^2 e^{-\theta_r/T} \left( 1 - 6 e^{-\theta_r/T} + \dots \right) &\text{for } T \ll \theta_r \,, \\
1 + \frac{1}{180}\left(\frac{\theta_r}{T}\right)^2 + \dots  &\text{for } T \gg \theta_r \,.
\end{cases}
$$
The above function has a maximum of $1.1\ Nk$ at about the temperature $0.81\ \theta_r/2$. As the temperature is increased way above $\theta_r/2$, it settles down to $1\ Nk$ and we recover the classical flat result.
A: You asked for a physically intuitive explanation, so here we go. For the full derivation, see Nanashi No Gombe's answer.
I believe this "bump" phenomenon is related to the Skottky anomaly, which is explained nicely for a two state system in this wikipedia article: https://en.wikipedia.org/wiki/Schottky_anomaly. I will begin by explaining the "bump" phenomenon for a two-state system and then generalise to a rotational system.
Here is the heat capacity for a two state system:

Here are the energy levels for a two state system:

For $k_BT << \Delta$, only the ground state is occupied, and increasing $T$ slightly isn't going to change this, hence $C \rightarrow 0$
For $k_BT >> \Delta$, both states are equally occupied, and increasing $T$ slightly won't make much difference to the average energy, hence $C \rightarrow 0$
In between these two extremes, increasing T will have a dramatic effect on the average energy, since it is now possible to excite transitions from the lower energy state. This is the cause of the large bump in the heat capacity.
Now, let's consider a rotational system. Here are the energy levels of a rotational system:

Since $E_2 - E_1 > E_1 - E_0$, the system looks somewhat like a two state system at low temperature. Therefore, you also get this "bump" behaviour in the heat capacity at low temperature.
Images are from wikipedia and "Concepts in thermal physics" by Blundell.
A: Graph is wrong.  Translation has three degrees of freedom, specific heat 3/2 R.
Rotation (excited at low temperature) adds two degrees of freedom, so specific heat becomes 5/2 R.  Vibration (excited only at temperatures of thousands of degrees) adds another two degrees of freedom (one kinetic, from relative motion, and one potential, from the interatomic potential) for a total of 7/2 R.
