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Waves Atomic Theory Heat and Temperature The Ideal Gas and Kinetic Theory Chemical Thermodynamics and the Equilibrium State | Let us work on our understanding of entropy, because not only the disposition of energy, but also the disposition of order is crucial to understanding the behavior of a chemical system. This discussion will be a bit advanced, to make you think. Don't worry too much. Treat this discussion as a bit of reading comprehension.
First define the system in terms of its macroscopic properties (pressure, volume, temperature), then in terms of the states of the particles of which it is composed. We want to develop a conceptual sense of the basis of entropy in statistical mechanics.
What do we mean by 'states of the particles'? The number of states for particles to occupy is the system's multiplicity. Like quantized electron orbitals, possible states of molecular motion exist as a series of waveforms which in one dimension would be described like a standing wave on a fixed string. The entire series represents the probability distribution for the particles in a system.
Okay. Don't panic. What we are trying to get at is the sense that each particle has a number of definite ways to be. The number of quantum states of a particle in an energy range for a gas depends on the space available and the temperature. So at a higher temperature, the gas particles have more states available, while a low temperature, less. For example, the reduced heat capacity of crystals at low temperature is evidence that quantum theory describes molecular motion (in this case vibrational) as a series of energy levels. At low temperatures, the amounts of energy transferred through molecular collisions are less than sufficient to raise atoms between vibrational states, so the heat capacity is close to zero.
The equation giving the fraction of gas molecules with velocities in a certain range is the Maxwell-Boltzmann distribution law. It gives the most probable way of distributing the total energy of the system among the molecules, the distribution with the maximum number of molecular states. Now here is the key concept. The greater the multiplicity of the system, the greater the entropy. A system with a great deal of multiplicity, has many ways to achieve the same macrostate, making that macrostate very likely (!!!)
If a restraint prevents a system from achieving the state with the greatest multiplicity and this restraint is removed, the system will move spontaneously toward maximum multiplicity through simple probability. Increase in entropy occurs in expansion or mixing because of the increase in the multiplicity of quantum states. At absolute zero, only one quantum state would represent the system of an ideal gas, so the entropy is zero.
That's tough stuff. If it is a bit hard to chew, read it again slowly, and your mind will digest it while you are sleeping the next few weeks. What is really important for this MCAT course at this stage is that your intuitive feel be gathering steam. These concepts will help you understand a great deal about chemistry, although they are one step beyond premedical level.
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Heat and Temperature The States of Matter Chemical Thermodynamics and the Equilibrium State | At absolute zero, only one quantum state would represent the system, so the entropy is zero. As heat flows in, the multiplicity increases. Because entropy increases with the logarithm of the temperature, the rate of change (dS/dT) is proportional to 1/T. The molar entropy of water at zero Celsius equals the molar entropy of zero Celsius ice plus the heat of fusion divided by 273 Kelvin. That's what dS/dT tells us. The entropy of a gas can be calculated theoretically as a state function, depending on Temperature, Volume, and MW (or T, P and MW) or it can be determined experimentally by the addition of heat from temperatures close to 0 K through melting, vaporization to the final temperature.
The residual entropy of nitrous oxide at 0K is an interesting problem. Because the molecules in the crystal can be oriented in one of two ways, there is still entropy at absolute zero because of the random alignment of nitrous oxide molecules in the perfectly motionless lattice.
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