It will predict how other certain experiments with the combined gas will behave. That's what people mean when they say that the entropy is not an "objective" property of a physical system: that it depends on how we choose to describe that physical system - and what experiments we can perform acting on that description.
By my understanding, even if we have no idea what gas we have, if we put it into a calorimeter and measure the amount of heat we need to transfer to it to change its temperature to some value, we will get a value that will be different for a gas made up of only argon versus one that contains both neon and argon. Doesn't this show that there is some objective definition of the entropy of the gas that doesn't care about an observer's knowledge of it?
Actually the molar heat capacity for neon, or argon, or a mixture thereof, is the same. These are monotomic ideal gases as far as your calorimeter measurements can see.
If the number of particles is the same you’ll need the same heat to increase the temperature by some amount and the entropy increase will be the same. Of course you could do other things to find out what it is, like weighing the container or reading the label.
No, they are not. The entropy of an ideal monatomic gas depends on the mass of its atoms (see the Sackur–Tetrode equation). And a gas mix is not an ideal monatomic gas; its entropy increases at the same temperature and volume compared to an equal volume divided between the two gases.
Also, entropy is not the same thing as heat capacity. It's true that I didn't describe the entropy measurement process very well, so I may have been ambiguous, but they are not the same quantity.
I'll leave the discussion here but let me remind you that you talked (indirectly) about changes in entropy and not about absolute entropies: "if we put it into a calorimeter and measure the amount of heat we need to transfer to it to change its temperature to some value".
Note as well that the mass dependence in that equation for the entropy is just an additive term. The absolute value of the entropy may be different but the change in entropy is the same when you heat a 1l container of helium or neon or a mixture of them from 300K to 301K. That's 0.0406 moles of gas. The heat flow is 0.506 joules. The change in entropy is approximately 0.0017 J/K.
> And a gas mix is not an ideal monatomic gas; its entropy increases at the same temperature and volume compared to an equal volume divided between the two gases.
A mix of ideal gases is an ideal gas and its heat capacity is the weighted average of the heat capacities (trivially equal to the heat capacity of the components when it's the same). The change of entropy when you heat one, or the other, or the mix, will be the same (because you're calculating exactly the same integral of the same heat flow).
The difference in absolute value is irrelevant when we are discussing changes in entropy and measurements of the amount of heat needed to increase the temperature and whether you "will get a value that will be different for a gas made up of only argon versus one that contains both neon and argon".
