To what precision is the heat capacity an extensive quantity We know that heat capacity is an extensive quantity, basically meaning for double the amount of substance you need double the energy to increase temperature. To what extend is this actually true, like:

*

*Are there e.g. (measurable) surface effects?

*Have (precision) experiments been done on this?

 A: Heat capacity $C$ is related to the specific heat $c$(heat per unit mass per  degree) by
$$C=mc$$
The specific heat $c$ is a material property. The specific heats at constant pressure, $c_p$ and constant volume $c_v$ are defined by
$$c_{v}=\biggl (\frac{\delta u}{\delta T}\biggr)_{V}$$
$$c_{p}=\biggl(\frac{\delta h}{\delta T}\biggr)_{P}$$
Where $u$ and $h$ are the specific internal energy and enthalpy, respectively. Internal energy and enthalpy are both extensive thermodynamic properties.
Since the specific heats are defined entirely by extensive thermodynamic properties, heat capacities must also be extensive properties.

Thanks. Great perspective. Unfortunately it does not really help that
much, since you could rephrase then my question, in C=mc to what
precision is c independent of m. Let me give you an analogy where it
is maybe clearer (because it doesnt hold). I could ask how well do
resistors follow Ohms law. Then you could have said,well R:=U/I so
U=R*I, which is true, but R is not independent of I. (generally
resistors follow ohms law only approximatively)

That's a different question than is heat capacity truly an extensive property. Extensive property means it depends on the mass. The specific heat does not depend on mass as can be seen by the definitions. But it does depend on temperature. It can only be considered constant over a limited temperature range.
In terms of your Ohm's law analogy, $R$ is dependent on $I$ to the extent that $R$ is dependent on temperature and temperature varies with $I$ due to resistance heating $I^2R$. But generally $R$ does not depend on voltage $U$. An example of an exception is the solid state device called a varistor, or voltage dependent resistor, such as a metal oxide varistor (MOV). Similarly, specific heat depends on temperature but does not depend on mass. As far as I know, increasing or decreasing the mass of an object does not alter the specific heat. Nor does the amount of mass alter the specific internal energy or specific enthalpy. But of you can provide an example where it does, please do so.
UPDATE:
I would add, based on my discussions with @Hyportnex, that if you mean by "precision" you mean at the microscopic (e.g., molecular) level, then let me say I am approaching this from the macroscopic thermodynamics point of view, i.e., that is from the perspective of observable mass. I would acknowledge, however, that when you get to the molecular level, there can be variations in specific properties such as specific heat, specific internal energy, specific enthalpy, etc.
Take, for example, specific internal energy. It consists of sum or total amount of the microscopic kinetic and potential energies. At the molecular level, however, the kinetic and potential energies may vary from the average since speeds and separations of the molecules vary and are distributed about the average value.
For example, individual atoms and molecules, or small quantities of atoms and molecules, can have kinetic energies that are distributed about the average. But it is the average that determines the kinetic energy component of specific internal energy.  Since the average translational kinetic energy of the molecules of an object determines the overall temperature of the object, that, in turn, implies specific heat can vary with mass at the molecular level. However, as a practical matter, in macro thermodynamics specific heat is independent of the amount of mass.

I am not looking at a molecular level, but maybe in the range of
mg-mug-ng whatever data is available, basically to understand when
does it break down (with current technology)

So I guess what you are really asking is to what degree of granularity is specific heat independent of the amount of mass.
It seems to me that would depend on to what degree is the material ishomogenous, that is, to what degree does the material or system have the same properties at every point. Is it is uniform without irregularities? Or to put it another way, is it sufficiently homogenous that it cannot be mechanically separated into different materials?
Another possible consideration is to what degree is the material isotropic? That is, to what degree are the physical properties independent of the orientation of the orientation of the materiel or system. That could have an impact on the test conditions used to determine specific heats.
If in fact you have had a material that was both homogenous and isotropic, then it would seem that the specific heat (and other specific thermodynamic properties) would be independent of the amount of mass or its orientation, at least at the level of ponderable masses.
A Wikipedia article on variations in specific heat states that "specific heat can be defined and measured for gases, liquids, and solids of fairly general composition. These include gas mixtures, solutions and alloys, or heterogenous materials such as milk, sand, granite, and concrete, if considered at a sufficiently large scale" (bold italics mine)
I believe if you want to know what a "sufficiently large scale" is, you would need to do research for the specific material(s) to find out.
Hope this helps.
A: Experimentally, I think it is easier to do the opposite: cooling bits of heated material in water inside a insulated container and measure the temperature change for example.
When trying to increase the mass of material heated in a furnace to the same temperature, and measure the associated increment of energy (say gas or electric), the losses through the furnace walls may change, complicating the computation.
Surface effects act on the kinetics of heating / cooling, not in the temperature change, and plays no role in the heat capacity, as I understand.
A: Classically, the heat capacity of all substances would follow the law of Dupont and Petit with the same value per atom, so then there are no surface effects.
In the quantum regime, at low temperatures, one would expect a surface effect because the vibrational modes are different at surfaces. For measurements one would need a substance with a large surface area. But then a signal might be due to ordering or desorption of adsorbed gasses.
That is why one studied $c_v$ of things like very fluffy graphite: the 2-dimensional phase transitions. The difference between the surface and the bulk of graphite is not large and I would not expect that a difference in $c_v$ between graphite and fluffy graphite would have been studied. There are other things in such systems that are much more interesting than a small difference in the Debye temperature.
