Clarification on Entropy Entropy is loosely defined as order becoming disorder and I just was wondering how this concept translates to physics. Is disorder characterized by the atomic stability or energy somehow? Is there any method for calculating entropy? Furthermore doesn't the idea of entropy presented in the second law of thermodynamics contradict with the idea of electrons being drawn to orbitals with lower energy to maximize 'order' of the atom?
 A: Mathematically, entropy (in the sense of Gibbs and Shannon) measures the spread of a distribution. I tend to think of statistical entropy as microscopic indeterminacy (instead of, say, 'missing information') and thermodynamic entropy as microscopic freedom (instead of, say, 'disorder'). In case of reasonably uniform systems, the latter is basically the volume of phase space available for system evolution.
To derive the thermodynamic entropy of an isolated classical system from first principles, it is my understanding that you'd have to do something like this: Take the microscopic description, wait for Poincaré recurrence, divide phase space into cells, weight them by relative time spent within, let cell size go to zero and compute the mathematical entropy of the resulting distribution.1
Now, as to your question, lowering the electron's energy and the  accompanying release of energy is still a process that makes the system access a new region of phase space and thus compatible with the entropy concept.

1 I've never seen it spelled out quite this way in the literature, so feel free to correct me if there's anything wrong with this picture.
A: As far as i remember reading, solitary atoms do not qualify as a thermodynamic systems. A thermodynamic system is a macroscopic volume in space that can be adequately charecterised by thermodynamic state variable like pressure,temperature,entropy ,volume. An atom can't be described using these. So the concept of entropy which applies to a thermodynamic system does not apply to a solitary atom. 
To give you a semi-mathematical explanation of entropy, consider a box with a partition in the middle and some gas in only one half of this box. Microstates of a system is the various microscopic configurations of that system. That is, if you click a picture of the particle's location and momentum at some random time, it resembles a possible microstate of the system. Let $\Omega _1$ denote the number of such microstates in this half box example. Then the entropy is given by $S = k_b .log \Omega _1$ where $k_b$ is the boltzman constant. Now if you remove the partition, the gas spreads out. Since now it can occupy positions in the other half as well, the number of possible microstates increases to $\Omega _2 (\gt \Omega _1)$. So the entropy also increases as shown above in the formula. This is thus a measure of disorder of the system. By removing the partition, we removed the constraints(order), makig available other equally possible configurations(microstates) of the system.
