Entropy and the Second Law Lecture 2
Getting to know Entropy • Imagine a box containing two different gases (He and Ne) on either side of a removable partition. • What happens when you remove the partition? • Did the energy state of the system change? • Is the process reversible? (All the kings horses and all the kings men…) • What changed?
The Second Law • It is impossible to construct a machine that is able to convey heat by a cyclical process from one reservoir at a lower temperature to another at a higher temperature unless work is done by some outside agency (i.e., air conditioning is never free). • Heat cannot be entirely extracted from a body and turned into work (thus car engines always have cooling systems). • Every system left to itself will, on the average, change toward a condition of maximum probability. • The entropy of the universe always increases.
My Favorite • You can’t shovel manure into the rear end of a horse and expect to get hay out of its mouth. The Second Law states that there is a natural direction in which reactions will tend to proceed.
Statistical Mechanics and Entropy The Microscopic Viewpoint
Back to our Box Imagine there were just two atoms of in each side of our box. Only one possible arrangement If we remove the partition, there are 24 = 16 possible arrangements. Basic Postulate of Statistical Mechanics: a system is equally likely to be found in any of the states accessible to it. Odds of atoms being arranged the same way: 1 in 16. When we removed the partition, we simply increased the number of possible arrangements. Suppose we had a mole of gas. What are the odds of atoms being arranged the same way?
Another Example Imagine a two copper blocks at different T separated by an insulator. What happens if we remove the insulator? Suppose we initially had 1 unit (quanta) of energy in the left block and 5 in the right. How will they be distributed after we remove the insulator?
How many ways can we arrange the energy? • Each of these arrangements is equally possible. • Since we can’t tell the individual quanta apart, some of these arrangements – combinations -are identical. • How many combinations are there that correspond to the first block having 1 quantum? 26 = 64
How many when the left block has 2? Is there a rule we can use to figure out questions like this? • Where E is the total number of energy units and e is the number the left block has.
Calculating Probabilities • Suppose there are 20 quanta of energy to distribute. Too many to count individual combinations! We need an equation. • The equation will simply be the number of combinations corresponding to a given state of the system times the probability of that combination occurring. P(f) = Ω(f) × C, where C is a constant giving the probability of the combination occurring (0.520) in this case.
Entropy and Thermodynamics • Unlike a simple physical system (a ball rolling down a hill), in thermodynamics whether or not the system is at equilibrium depends not on its total energy, but on how that energy is internally distributed. • The function that predicts how energy (or atoms) will be distributed at equilibrium is entropy. • Our experience is that for blocks of equal size and composition at equilibrium, energy will be distributed equally between them. • How do we develop this mathematically?
Entropy • Consider again the energy distribution. • What is maximized when the system is at equilibrium? • What is a mathematical characteristic of maxima?
Defining Entropy • For our blocks, equilibrium occurs where • We define entropy, S, as S = k lnΩ • Entropy is a measure of the randomness of a system. • An increase in entropy of a system corresponds to a decrease in knowledge of it. • We can decrease the entropy of a “system”, but only by increasing the entropy of its surroundings.
The Second Law • Entropy also has the interesting property that in any spontaneous reaction, the total entropy of the system plus its surroundings must never decrease. • In our example, this is a simple consequence of the observation that the final probability, P(E), and therefore also Ω, will be maximum and hence never be less than the original one. • Any decrease in entropy of one of the blocks must be at least compensated for by an increase in entropy of the other block. • Formally, the Second Law is written as: • In the case of a (fictive) reversible process, this becomes an equality.
Integrating factors and exact differentials • Any inexact differential that is a function of only two variables can be converted to an exact differential. • dWis an inexact differential, and dV is an exact differential. Since dWrev = -PdV, dWrevcan be converted to a state function by dividing by Psince dV =-Wrev/P • Similarly for heat • dQrev/T = dS • Here we convert heat to the state function entropy by dividing by T.