31. What is entropy?

            Entropy is a measure of the amount of disorganization of a (large) group of particles, the greater the organization the smaller the entropy.

            In a kinetic particle universe any value can be taken for the entropy for an assembly of kinetic particles. The differential change of entropy “ ” for a thermodynamic process is given by

where is the differential amount of heat energy (per unit mass of the thermodynamic system) added reversibly and T is the gas temperature at the time the heat is added. The temperature in the kinetic particle theory is  since all the particles have the same mass “m”. The thermal velocity is the rms velocity “ ”. A simple set of units can be to use q in joules per kilogram, T in joules, and  in meters per second. Then “s” then will be per kilogram. The unit of this temperature scale is very small, since “m” is so small, but it is useful here to use this scale to elucidate the meaning of the entropy “s”.

            The change of entropy from one state to another is independent of the path taken, since entropy is a state function. This fact provides a procedure for determining entropy change for reversible as well as irreversible processes. We illustrate the computation of entropy change for several cases.

            We consider the expansion from one value of volume to double the volume of a piston/cylinder for three cases. We use one kilogram of gas in all cases.

Case 1. Isothermal Reversible Process

 the internal energy doesn’t change

Case 2. Adiabatic Reversible Process

 The internal energy decrease is the same as the work done

 (since adiabatic)

 (since adiabatic and reversible)

Case 3. Adiabatic and Irreversible Process

            Let the piston expand to  faster than the particles move – which is the maximum irreversibility. In this case the temperature (and internal energy) remain the same. Since this process is irreversible to compute the entropy change we will use a reversible isothermal expansion (which facilitates computation of ) followed by an adiabatic expansion (where ). Let state 1 be the starting state, state 3 be the intermediate state, and state 2 be the final state.

 since

Thus

            In these cases we note that the entropy increased or remained constant. In the isothermal reversible process the gas absorbed energy from the environment, did work with the energy, and used part of the available cylinder volume so that less cylinder volume remains available for subsequent work. (Note that  becomes infinite as the expansion grows without bound.) In the adiabatic reversible case (useful) work is done at the expense of internal energy decrease and the entropy remains constant. The adiabatic irreversible case did no work, transferred no energy, and kept the internal energy constant. The volume expanded so that there was less organization in the cylinder so the entropy increased. A significant thing about this process is that it is irreversible and the analysis illustrates the computation of entropy where, for the analysis the heat is added reversibly (while no heat is actually added).

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