Entropy is quite often introduced with the help of a deck of cards. Because there are so many cards, and thus an enormous number of microstates (orderings of the cards), if we assume equipartition of "energy" (equal distribution of the cards among the available spots in the ordering), then the number of "ordered macrostates" (e.g. microstates in which cards are perfectly ordered within their respective suits) is incredibly small compared to the total number (52!).
This got me thinking though: is it possible to derive a sort of "temperature" for a deck of cards?
Let's say that the number of cards functions as a sort of "energy" (number of cards is conserved, after all). If we have two different decks
- with 2 cards
- with 9 cards
then it is possible to establish a "temperature":
If we change the energy by one card in each deck, then the entropy change is:
- $\log(3!) - \log(2!) = \log(3!/2!) = \log(3)$
- $\log(10!) - \log(9!) = \log(10!/9!) = \log(10)$
Temperature is defined as the change in energy over the change in entropy -- so whatever unit we choose for the energy the "temperatures" will be proportional to:
And thus the larger deck is at a "lower temperature" than the smaller one.
However, this is sort of at odds with my understanding of the connection between energy, entropy, and temperature.
For instance, let's consider two boxes of gas instead.
If we put the same amount heat energy in to the two different boxes of gas, the "lower temperature" box will be the one for which that heat energy increases the entropy more. I.e. the increase in the number of microstates made available to the gas by the introduction of the heat energy will be larger for the cooler box of gas than it will be for the hotter box of gas, given the same heat -- if there are already a large number of microstates available (the gas is "hot"), then the additional energy will not change in the number of microstates as much as if there were a smaller number of microstates available to begin with (the gas was "cold").
The deck of cards seems to behave in an opposite way: the introduction of the same amount of "energy" to each deck increases the entropy more for the larger deck than the smaller one. The larger deck has more microstates available to begin with and its entropy was increased more by the introduction of the same amount of energy.
Where is the flaw in this analogy?
Is the "energy" not properly defined?
Is the temperature improperly defined?
Am I wrong in my way thinking about how to interpret the temperature of a box of gas?
Where does the analogy break down?