Thermodynamic Entropy

Introduction

Physics properties are defined by equations. Clausius correctly defined thermodynamic entropy by an equation that is simple. Yet, it does not have an explanation for what it is physically. Entropy, temperature, Boltzmann's constant, and Planck's constant will each have physical explanations.

Thermodynamic properties are properties such as pressure, temperature, and volume for which we make macroscopic measurements that pertain to measuring internal energy. The measurement of these properties must be done on a medium that is in equilibrium. Temperature is commonly explained as a property that demonstrates when two systems are in thermal equilibrium. When the systems are touching with no barrier, no measurable exchanges of heat occur between the systems.

When external forces act on a system, or when the system exerts a force that acts on its surroundings, then the forces must act quasi-statically. This means forces must vary so slowly that any thermodynamic imbalance is infinitesimally small. In other words, the system is always infinitesimally near a state of true equilibrium. If a property such as temperature changes, it must occur so slowly that there is no more than an infinitesimal temperature variation between any two points within the system.

In the work that follows, all parts of a system are in states of equilibrium with one another. All changes that occur between systems or parts of systems occur sufficiently slow that each part of all systems remain infinitesimally close to equilibrium.

Definition of Thermodynamic Entropy

Thermodynamic Entropy is defined by an equation that establishes a direct relationship between the heat entering the engine at a constant temperature and an increase in entropy. The equation is:

ΔS = ΔQT

Where ΔS is a change in entropy, and ΔQ is the corresponding change in heat — energy in transit into or out of a system — and T is the temperature of the system in degrees Kelvin. By convention heat entering the gas is positive. Heat leaving the gas is negative. The equation that defines thermodynamic entropy is based upon a Carnot engine operating in a four-part cycle.

There is a steady source of heat and a steady heat sink. The heat source is at temperature T_high, the heat sink is at temperature T_low. The Carnot engine operates cyclically between these two temperatures, absorbing heat from T_high and rejecting heat to T_low. For this example, the working substance is an ideal gas.

The cycle begins with the engine in unrestricted contact with the heat source T_high. The gas volume expands pushing a piston against an outside resistance quasi-statically, so slowly that the temperature of the ideal gas remains constant at T_high. The engine is then removed from contact with T_high. The heated gas continues to expand adiabatically until its temperature falls to T_low. The engine is put in contact with T_low and the gas is compressed while remaining at temperature T_low. Finally, the engine is separated from the heat sink while the returning piston adiabatically compresses the ideal gas, causing its temperature to rise back to T_high. This ends the four-part cycle.

It is known for the reversible Carnot engine that:

ΔQ_hT_h = ΔQ_lT_l

The temperatures T_h and T_l may vary, but the equality of this relationship remains true. This relationship is the basis of the definition of thermodynamic entropy. The major difference between Clausius's entropy and the later invented entropy is that Clausius's entropy is independent of temperature while the invented entropy is dependent upon temperature.

For Clausius's entropy of a reversible Carnot engine, entropy S increases at a constant rate. Heat Q increases at a constant rate. This condition is due to the temperature T remaining constant. For a reversible Carnot engine, the increase in entropy exactly equals the decrease in entropy, so their sum is zero — the cycle is brought back to its initial condition with no net change in entropy.

Clausius's definition of entropy shows how thermodynamic entropy is calculated but does not make clear what entropy is. It is temperature that masks the identity of entropy. Temperature is an indefinable property in theoretical physics. If the physical action that is temperature was identified, then entropy would be explainable.

The answer can be found in the operation of the Carnot engine. The Carnot engine is the most efficient engine, theoretically speaking. Its efficiency is independent of the nature of the working medium. Efficiency depends only upon the values of the high and low temperatures in degrees Kelvin. Something very important happens during the derivation of the engine's equation of efficiency that establishes a definite rate of operation of the Carnot cycle. The engine is defined as operating quasi-statically. There is one specific rate, above which, the equilibrium will be lost. This is the rate that becomes fixed into the derivation of the Carnot engine.

In this new theory, temperature is identified and defined as the rate of exchange of energy between molecules. Temperature is not quantitatively the same as that rate because temperature is assigned the units of degrees and its scale is arbitrarily fitted to the freezing and boiling points of water. This discrepancy can be moderated with the introduction of a constant of proportionality:

dQdt = k_T · T

Multiplying by dt and recognizing that the differential of entropy is dS = k_T · dt. Both dS and dt are variables. It is necessary to determine the value for the constant k_T. This value may be contained in the ideal gas law. For a single molecule, the kinetic energy is:

ΔE = (3/2) · k · T

Where k is Boltzmann's constant. This suggests that for an ideal gas molecule:

ΔS = (3/2) · k

In other words, the entropy of a single ideal gas molecule is constant. The condition under which this is true is when the gas molecules act like billiard balls and their pressure is very close to zero. Near zero pressure for any practical temperature requires that the gas molecules be low in number and widely dispersed.

I interpret this to mean, under these conditions, that the thermodynamic measurement of temperature and kinetic energy approach single molecule status. The ideal gas law written for a single gas molecule gives reason to consider that for a single molecule ΔS = (3/2)k. Substituting for Boltzmann's constant:

ΔS = (3/2)(1.38×10^-23 j/K) = 2.07×10^-23 j/K

I have defined entropy as ΔS = k_T · Δt. Therefore:

k_T · Δt = 2.07×10^-23 j/K

If I could establish a value for Δt, then I could calculate k_T. Since this calculation is assumed to apply to a single gas molecule and is a constant value, I assume that in this special case, t is a fundamental increment of time. In this theory, there is one fundamental increment of time:

Δt_c = 1.602×10^-19 s

Substituting this value and solving for k_T:

k_T = 2.07×10^-23 j/K1.602×10^-19 s = 1.292×10^-4 j/(s·K)

Substituting the units for each quantity as determined by this new theory, k_T = 1.292×10⁻⁴ is a unit-free constant of proportionality. It also follows that Boltzmann's constant is defined as:

k = (2/3) · k_T · Δt_c

For the ideal gas equation, the entropy of each molecule is constant: ΔS = k_T · Δt_c. However, thermodynamic entropy is defined as an aggregate macroscopic function. There are a great number of molecules involved, and their interactions overlap and add together. Expanding the meaning of entropy into its more general form and substituting k_T into the general thermodynamic definition of entropy:

ΔS = k_T · Δt

The Δt in this equation is not the same as the Δt_c in the equation for a single molecule. In the macroscopic version, it is the time required for a quantity of energy, in the form of heat, to be transferred at the rate represented by the temperature in degrees Kelvin. Substituting this equation for entropy into the general energy equation and solving:

ΔS = ΔE/T = ΔQ/T = k_T · Δt

This function of Δt is what would have become defined as the function of entropy if temperature had been defined directly as the rate of transfer of energy between molecules. The arbitrary definition of temperature made it necessary for the definition of entropy to include the proportionality constant k_T.

Now, I consider an engine that operates infinitesimally close to equilibrium conditions but has heat loss Q_z that does not result in work. The entropy of the engine is not changed by this loss of heat. The entropies that are affected are those of the high heat source and the low heat source. Entropies are measures of time required for the lost heat to be released by the high heat source and later absorbed by the low heat source. The net change in entropy is:

ΔS = Q_z/T_low − Q_z/T_high

The quantity of heat transferred is the same in both cases. The rates at which that heat will be transferred are different. The low temperature represents a slower rate of exchange of heat than for the high temperature. This means it takes longer for the low temperature source to absorb the quantity of lost heat than it does for the high temperature source to emit the heat. This time difference is the change that occurs, and it is what is represented by the measure of change of entropy.

Interpreting Planck's Constant

In this theory, Boltzmann's constant has acquired the definition of k = (2/3) · k_T · Δt_c. There is a relationship between Planck's constant h and Boltzmann's constant k:

h = k · Δx_c

Substituting for k and rearranging terms:

h = k_T · (2/3 · Δx_c) · Δt_c

The fraction is moved into the parenthesis with photon length because, as has been shown earlier in the theory, this term demonstrates the definition includes a remote measurement. In other words, we determine the value of Planck's constant by making remote macroscopic measurements of the energy of photons.

This interpretation of Planck's constant allows for a modification to the definition of entropy. Using ΔE = ΔS · T and substituting T = (2/3) · Δx_c · ω, and ΔS = k_T · Δt_c, through substitution and rearrangement:

ΔE = k_T · (2/3) · Δx_c · Δt_c · ω = h · ω

Planck's constant is a part of the above equation so long as it applies to an ideal gas. For the entropy definition Δt_c is replaced with the variable Δt in order that the equation may apply to more general cases. Defining an analogy to entropy for frequency:

ΔS_p = k_T · (2/3) · Δx_c · Δt

So, Planck's constant is the constant ΔS_p for an ideal gas, while the form above is the variable form for general cases.

Analyzing Planck's Constant

The potential energy of the hydrogen electron in its first energy level is ΔE_eH1 = h · ω_eH1. The angular frequency of the electron is:

ω_eH1 = v_c · α2π · Δx_c = α2π · Δt_c = 12π · α^-1 · Δt_c

The denominator on the right side is the period of the frequency. Therefore:

ΔE_eH1 = h2π · α^-1 · Δt_c

Also, the potential energy for a circular orbit can be expressed as ΔE_eH1 = f_eH1 · Δx_c. Setting these equal and solving for Planck's constant:

h = f_eH1 · Δx_c · Δt_c · 2π · α^-1

This result defines Planck's constant in terms of properties of the hydrogen atom.

Defining Temperature

I have defined temperature as T = k_T · ΔE / Δt_c. I have also derived h = k_T · (2/3 · Δx_c) · Δt_c. Solving for k_T and substituting the expression for Planck's constant:

k_T = f_eH1 · Δx_c · Δt_c · 2π · α^-1(2/3 · Δx_c) · Δt_c = 3 · f_eH1 · π · α^-1

Defining Boltzmann's Constant

I have established a relationship between Planck's constant and Boltzmann's constant in the form of k_B = h / Δx_c. Substituting for Planck's constant:

k_B = k_T · (2/3) · Δt_c = f_eH1 · 2π · α^-1 · Δt_c

Or, in terms of momentum:

k_B = ΔP_eH1 · 2π where ΔP_eH1 = f_eH1 · α^-1 · Δt_c = f_eH1 · Δt_eH1

Defining Frequency

I have defined h = k_B · Δx_c. Substituting for k_B:

h = ΔP_eH1 · 2π · Δx_c = ΔP_eH1 · λ_eH1

Also:

ΔE_eH1 = h · ω = ΔP_eH1 · λ_eH1 · ω_eH1 = ΔP_eH1 · v_eH1

And, from an earlier result, substituting h = f_eH1 · Δx_c · Δt_c · 2π · α⁻¹:

ΔE = h · ω = (f_eH1 · Δx_c)(α^-1 · Δt_c)(2π · ω)

The first set of parentheses contains the potential energy of the hydrogen electron in its first energy level. The second set is the period of time required for the electron to complete one radian. The third set is the angular velocity of the electron in units of radians per second. This theory's definition of Planck's constant first changes frequency into radians per second, then converts radians per second into a measure of the number of radians traveled during the period of time required for the hydrogen electron to travel one radian, and finally multiplies by the potential energy of the hydrogen electron in its first energy level. In other words, Planck's constant uses fundamental properties of the hydrogen atom as the standard by which to convert frequencies into quantities of energy.

Boltzmann's Entropy

This theory introduces the idea that a consequence of defining thermodynamic entropy using an ideal gas is that, as the pressure approaches zero, the exchanges of energy between molecules theoretically reduce to single exchanges. A point is reached where exchanges occur at a rate that can be modeled as one at a time without delay between them. That is an equilibrium point where the temperature is close to a constant value. Clausius's thermodynamic entropy applies to that low pressure where the exchanges that occur can be ideally represented as each molecule taking its turn, without delay, to pass on average molecular kinetic energy.

This process can be modeled by considering all the gas molecules lined up in a single file and the average molecular kinetic energy of one of them is transferred down the line from molecule to molecule until the energy has been transferred to the last molecule. The time required to complete this process is 'internal' thermodynamic entropy.

Temperature is proportional to the rate of transfer of average molecular kinetic energy between molecules. The modified temperature is the rate at which energy is transferred between molecules. It was shown that the average kinetic energy divided by modified temperature equals Δt_c. In the equation below, Boltzmann's constant is defined as the first equal term and by thermodynamics as the second equal term:

k = (2/3) · k_T · Δt_c = RN

N is Avogadro's number, the number of molecules in a mole of gas. R is the universal gas constant. Solving for R and substituting the appropriate values:

R = (2/3)(6.02×10²³ mol^-1)(1.292×10^-4)(1.602×10^-19 s) = 8.31 s·mol^-1

The universal gas constant R is directly proportional to the total time required for a mole of ideal gas to transfer average molecular kinetic energy from molecule to molecule without delay between exchanges until the number of molecules in a mole of gas is reached. R must be divided by k_T so that it becomes defined using modified temperature, and multiplied by 3/2 to remove the 2/3 that resulted from the kinetic theory of an ideal gas:

(3/2)(R / k_T) = N · Δt_c = (6.02×10²³ mol^-1)(1.602×10^-19 s) = 96,440 s·mol^-1 = 26.8 hrs·mol^-1

Boltzmann's constant is the time represented by the universal gas constant R reduced to single molecule status:

k = RN = 8.3126.02×10²³ (s/mol)/mol = 1.38×10^-23 s

Boltzmann's constant is directly proportional to the time necessary for a single exchange of average kinetic energy to occur between two molecules of an ideal gas, independent of temperature. The actual time, devoid of the molecule indicator, is given by:

Δt_c = (3/2)(k / k_T) = 1.602×10^-19 s

The reason for eliminating the kinetic theory of an ideal gas fraction of 2/3 is that it pertains to macroscopic properties while the time of exchange of kinetic energy between individual molecules is a microscopic property.

The number of possible arrangements for a mole of ideal gas is infinite. Boltzmann's entropy requires there to be a limited number of possible arrangements. In quantum theory, there are a naturally limited number of available arrangements — microstates which particles might occupy. If the concept of microstates is idealized so that all microstates are equally likely to be occupied, then the inclusion of Boltzmann's constant causes the calculation to be analogous to that of thermodynamic entropy. The number of microstates simulates ideal gas molecules. The entropy calculation simulates the calculation of internal entropy of an ideal gas.

The calculation of the entropy for any number of microstates will yield a solution identical to an analogous calculation for an equal number of ideal gas molecules. However, Boltzmann's entropy is defined as S = k log Ω. Therefore, Boltzmann's entropy is proportional to the time of a single transfer of ideal gas molecule energy times the logarithm of the number of microstates. Boltzmann's entropy is not an expression of simulated internal thermodynamic entropy. Boltzmann's entropy is no longer a direct measure of time. The units of seconds carried along by Boltzmann's constant have become irrelevant. Boltzmann's constant can be set to unity without units. Its connection to thermodynamic entropy is already lost.

Conclusion

Clausius's thermodynamic entropy is the time it takes for heat Q₁ to be absorbed into an ideal gas at the rate of temperature T_high, or for Q₂ to be released out of an ideal gas at the rate of temperature T_low. The time it takes for a single ideal gas molecule to pass its kinetic energy off to another ideal gas molecule at a distance equal to the radius of the hydrogen atom is the unit of absolute time derived in the article A Unit of Absolute Universal Time.