Download Presentation
## Entropy and the Second Law

- - - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - - -

**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.