If $Q = mC_p\Delta T$, shouldn't $C_p$ be updated every increment? Considering that the specific heat capacity of a material is a function of temperature, is it wrong to assume that $C_p$ is constant for notable differences in temperature ($>50\mathrm{K}$)?
It is something that I just noticed but can not find a conclusive answer to and it bothers me that I just do not know this already!
 A: It depends on what you mean by "wrong" and what range of temperatures you are considering. When you treat the gas as an ideal gas and the heat capacity as constant, this is commonly called the calorically perfect gas approximation. When the heat capacities vary with temperature but the gas is still treated as an ideal gas, you get what is called a thermally perfect gas approximation.
Both can be valid depending on the regimes and gases in question. In essence, what the heat capacity represents is the transfer of energy between molecules of the gas during collisions. There are different modes that energy can be stored and transferred -- a molecule has translational modes (moving in straight lines in space), a molecule might have rotational modes (moment of inertia about varies axes), a molecule might have vibrational modes (changes in the length of the bonds between atoms in the molecule), and a molecule might have electronic modes. A collision can transfer energy between modes and between molecules, all with different efficiencies, and this is what the heat capacity represents.
When an energy mode has absorbed all of the energy it can, then its contribution to the heat capacity stops changing with temperature. It effectively becomes constant. So, if you have a very simple molecule, say atomic helium, that will have translational modes but negligible rotation (the moments of inertia about the axes is negligibly small), no vibration (there's no bonds to vibrate), and no electronic modes. So once you get above a temperature where translational modes are fully saturated, which is on the order of $T > \approx 100 \text{K}$, then the heat capacity is constant. So modeling helium as a calorically perfect gas at room temperature is perfectly fine.
On the other hand, if you take something like oxygen and nitrogen in air, those both have translational modes, rotational modes, vibrational modes, and electronic modes. These modes all excite at different temperature ranges, from on the order of 20-50 K up to 2000-3000 K, and so as different modes activate, the heat capacities will change. This means it is not suitable to model is as a calorically perfect gas over a large range of temperatures. However, eventually at high enough temperatures, all the modes do saturate and then treating it as calorically perfect is okay again -- this happens in hypersonic flow regimes, for example.
Depending on your gas, and your temperature ranges, you will have to evaluate whether the heat capacity can be held constant or not. If you have gas molecules with multiple atoms in each, and the change in temperatures is large, then variations in heat capacity will matter for sure. Assuming the heat capacity is constant (and frozen at its room temperature values) will result in an over-prediction of temperature when the heat capacity is frozen.

If you're talking about other materials besides gases, the same thing holds but the energy transfer modes can be more complicated. In general, if the heat capacity has significant changes over the range of temperatures you are considering, then holding it constant over the temperature change will introduce errors. Ultimately, your equation is an approximation of:
$$ Q = \int_{T_0}^{T_1} m c_p dT $$
which can be approximated a number of different ways. The heat capacity can be held constant throughout the integration, or constant over small intervals with the integral treated as a Riemann sum, or other variations depending on the numerical quadrature rule used.
A: To reflect the dependence of $c_p$ on temperature, the equation should be $Q = m\int_{T_0}^{T_f}c_p(T)\enspace dT$.  You perform the integration to obtain Q.  Yes, $c_p(T)$ is a function of temperature, in general, but for solids over a small temperature range it is relatively constant.
