Entropy

[9.1] The Direction and Irreversibly of Thermodynamics Processes

The second law of thermodynamics deals with the direction taken by spontaneous processes. Many processes occur spontaneously in one direction only—that is, they are irreversible, under a given set of conditions. Although irreversibility is seen in day-to-day life—a broken glass does not resume its original state, for instance—complete irreversibility is a statistical statement that cannot be seen during the lifetime of the universe. More precisely, an irreversible process is one that depends on path. If the process can go in only one direction, then the reverse path differs fundamentally and the process cannot be reversible.

For example, heat involves the transfer of energy from higher to lower temperature. A cold object in contact with a hot one never gets colder, transferring heat to the hot object and making it hotter. Furthermore, mechanical energy, such as kinetic energy, can be completely converted to thermal energy by friction, but the reverse is impossible. A hot stationary object never spontaneously cools off and starts moving. Yet another example is the expansion of a puff of gas introduced into one corner of a vacuum chamber. The gas expands to fill the chamber, but it never regroups in the corner. The random motion of the gas molecules could take them all back to the corner, but this is never observed to happen.

[9.2] Entropy and the Second Law of Thermodynamics

The second law of Thermodynamics:-

The second law of thermodynamics is one of the most fundamental laws of nature, having profound implications. In essence, it says this:

The second law - The level of disorder in the universe is steadily increasing. Systems tend to move from ordered behavior to more random behavior.

One implication of the second law is that heat flows spontaneously from a hotter region to a cooler region, but will not flow spontaneously the other way. This applies to anything that flows: it will naturally flow downhill rather than uphill.

The second law also predicts the end of the universe: it implies that the universe will end in a "heat death" in which everything is at the same temperature. This is the ultimate level of disorder; if everything is at the same temperature, no work can be done, and all the energy will end up as the random motion of atoms and molecule.

The second law of thermodynamics states that the total entropy of an isolated system can never decrease over time, such that, dS ≥ 0. A spontaneous (irreversible) process takes the system towards a state of higher entropy (dS > 0). If the process is reversible (at all), then the entropy will remain constant (dS = 0).

[9.2.1] Entropy and Interacting Systems

If a certain system interacts with another system or with the entire universe, the original system may not be isolated. However the combined systems, or the system and the universe as a whole, would be isolated and the second law will apply to the total system. Under these circumstances, even if the original system may indicate a decrease in the entropy, the entropy of the combined systems, or the system and the universe, will be increased.

[9.2.2] Entropy and the Disorderliness

Entropy:-

A measure of the level of disorder of a system is entropy, represented by S. Although it's difficult to measure the total entropy of a system, it's generally fairly easy to measure changes in entropy. For a thermodynamic system involved in a heat transfer of size Q at a temperature T , a change in entropy can be measured by: ΔS= Q/T

The second law of thermodynamics can be stated in terms of entropy. If a reversible process occurs, there is no net change in entropy. In an irreversible process, entropy always increases, so the change in entropy is positive. The total entropy of the universe is continually increasing.

There is a strong connection between probability and entropy. This applies to thermodynamic systems like a gas in a box as well as to tossing coins. If you have four pennies, for example, the likelihood that all four will land heads up is relatively small. It is six times more likely that you'll get two heads and two tails. The two heads - two tails state is the most likely, shows the most disorder, and has the highest entropy. Four heads is less likely, has the most order, and the lowest entropy. If you tossed more coins, it would be even less likely that they'd all land heads up, and even more likely that you'd end up with close to the same number of heads as tails.

[9.2.3] Microstates and the Disorderliness of a System

A simple system of two distinguishable gas molecules can use to illustrate the disorderliness associate with the number of available microstates. The two gas molecules contained in a twin ask vessel that has a tube connecting the two asks. At rest the connecting tube was closed and both molecules were conned to the right-hand ask as shown in the left hand side (a) of the image. In this state, the probability of finding both molecules in the left ask is 1. Once the connecting tube opened the two molecules travel randomly throughout the entire space. The configuration space allows four possible arrangements for the two molecules as shown in the right hand side (b) of the image. Each of these configurations is a microstate available to the system. The probability of finding both molecules in the left ask has now decreased to 1/4, while the probability that the two molecules scatted across the two asks is 1/2. If there were three molecules, there will be 9 microstates to the system and the probability that all three molecules will be in the left ask will become 1/9. In this situation, the probability that the three molecules assuming a microstate where they are distributed in different ways across the two asks becomes 7/9 (7/9>1/2). This indicates that more microstate becomes available to the system more likely the system to remain disordered.

[9.4] Incorporating Entropy into the First Law of Thermodynamics

The first Law of thermodynamics states,

∆U = Q − W,

where ∆U is the change in the internal energy of the system, Q is the heat added to the system, and W is the work done by the system. Then a small change in the internal energy dU can represent in terms of the associated amounts of heat, and the work such that,

dU = dQ − dW.

The pressure volume work integral implies that, dW = p dV. Also the definition of the change in entropy indicates that,

dS = dQ T ⇒ dQ = T dS The first law of thermodynamics can be rewritten using these results, such that,

dU = T dS − p dV.

**Note that this statement assumes no change in macroscopic energy, and no work done by external forces acting the macroscopic energy of the system. This statement for the rst law appears in terms of state variables only (p, V, T,S), therefore the relationship is applicable to both reversible quasi-static processes as well as to irreversible processes. **

[9.4.1] Revised First Law of Thermodynamics for an Idea Gas System

The heat capacity at constant volume is given by,

Cv = ∆U ∆T and therefore, dU = Cv dT.

Then the rst law takes following form for an ideal gas system,

Cv dT = T dS − p dV

[9.5] T-S Diagrams

A temperature–entropy diagram, or T–S diagram, is a thermodynamic diagram used in thermodynamics to visualize changes to temperature and specific entropy during a thermodynamic process or cycle as the graph of a curve. It is a useful and common tool, particularly because it helps to visualize the heat transfer during a process.