The First Law of Thermodynamics

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    Because force displays two fundamentally different aspects, so too does energy display two fundamentally different aspects. Consider the fact that Newton’s second law of motion tells us that when a body A exerts a force on a body B that force stands equal to the rate at which the body B’s linear momentum changes; in other words, applied force equals inertial reaction, which equality we express algebraically as

(Eq’n 1)

Thus we have the two aspects of force, applied force on the left side of the equality sign and inertial reaction on the right.

    To obtain an appropriate description of energy subtract the applied force from both sides of Equation 1, multiply the result, in the manner of a vector dot product, by the minuscule element of distance over which body B moves, and then integrate with respect to distance. We get

(Eq’n 2)

In that equation EB, which we call the total energy of body B, represents a constant of integration, so we know that it cannot go to infinity (either positive or negative) because we require the constant to have a definite value and infinity does not represent a definite value. On the left side of that equation TB represents body B’s kinetic energy, the energy manifested in body B’s motion and based on its inertia. And UAB represents body B’s potential energy vis-a-vis body A, energy that we associate not so much with body B directly but with body B’s relationship with the field of force surrounding body A.

    The same analysis applies to body A vis-a-vis body B. In this case Equation 2 takes the form

(Eq’n 3)

As we can also do for body B, we can derive an explicit formula for calculating the kinetic energy carried by body A. The body ponders a mass mA, so we describe its linear momentum as pA=mAvA and calculate the kinetic energy as

(Eq’n 4)

    Considering bodies A and B as a single system raises a question about potential energy. We would like to think that the total energy of the system equals the sum of the energies that we calculate for the bodies separately, EAB=EA+EB. In order to justify that thought we need to rederive the energy formula from scratch. We have the sum of Equations 2 and 3 in the form

(Eq’n 5)

in which I have made the substitution, by way of Newton’s third law of motion, FBA=-FAB. We get a description of the potential energy as

(Eq’n 6)

In that equation drAB represents the differential change in the vector distance between the bodies, so the equation represents the fact that, although the overall potential energy in the system equals the sum of the individual potential energies calculated separately for the two bodies, that overall energy reflects work done by or against a change in the distance between the bodies.

    If we add more bodies to the system, bringing them in as if from an effectively infinite distance, each new body brings more energy into the system. It brings its own kinetic energy in accordance with Equation 4. And it gains or loses potential energy vis-a-vis each of the other bodies separately in accordance with the work done on or by it as it moves toward that body.

    Note here the linearity in the algebraic description of the energy. That description consists of a sum of terms that all stand independent of each other. The kinetic energy contained in the N-body system depends on N terms, each of which depends only on the properties of a single body. The potential energy contained in the system depends on N! terms, each of which depends on the distance between two of the bodies and remains unaffected by the presence of the other bodies. We can thus calculate the total energy in the system by generalizing Equation 5, vis-a-vis Equation 6, as

(Eq’n 7)

Thus we can calculate the total energy contained in any N-body system, whether it consists of N atoms or molecules in a gas or N stars in a globular cluster, and know that the energy remains constant so long as the system remains isolated from other dynamic systems. In that statement we have encoded the first law of thermodynamics, also known as the law of conservation of energy.

    That law necessitates the existence of a state of absolute zero of energy. Because energy obeys a conservation law, it must have a definite value at all times; thus, energy can never exist in an infinite quantity, which has an indefinite value. Kinetic energy always takes a positive value, so it exists in a finite quantity because the Universe could only have come into existence with a finite amount of kinetic energy. But potential energy can take negative values, which means that a body could acquire infinite kinetic energy if its potential energy were to go to negative infinity. Because that cannot happen, according to the law of conservation of energy, the laws of physics must have such a form that no potential energy can go to negative infinity. Thus, we infer that the Universe has a state of absolute zero energy, from which all energies, kinetic and potential, differ by a finite amount.

    In thermodynamics we do not concern ourselves with small numbers of particles, as we do in our basic mechanics. Instead, we want the physics of an uncountably large number of particles, as we might have in a large volume of gas. Here we encounter the physicists’ version of the soros paradox: at what number of particles does mechanics become thermodynamics? The answer to that question depends upon the mathematical methods available to us. But we can feign not having the techniques to describe systems with more than a certain number of particles mechanically. Then we can choose any large number arbitrarily as the number where thermodynamics begins. For convenience we might say that the number N of particles is so large that its logarithm, to base ten, has more than one digit, which means that thermodynamics properly applies to systems containing at least ten billion particles.

    In mechanics we use Newtonian equations of motion to work out a description of the trajectory followed by each and every particle in the system. In thermodynamics, on the other hand, we use statistical methods to work out a description of the system in terms of collective properties. If we use that latter approach, though, we must rely heavily on the first law, on the fact that the energy in an isolated system remains constant.

    Even when a system reaches equilibrium and appears not to change, changes occur within it at the level of individual particles. The kinetic energy of a particle changes as it collides with other particles and/or moves through the forcefields emanating from them, which makes the particle’s potential energy also change. Instead of trying to track those changes, we calculate a kind of average value for the particle’s properties.

    If the center of mass of a system does not move, then the average velocity of the particles zeroes out; <v>=0. Thus, average particle velocity does not give us a useful number for describing the dynamics of the system. The next kind of average, the variance of the velocity, the second moment of the velocity about the average value, gives us a more useful number. Mathematically we have

(Eq’n 8)

In the first line of that equation vi represents the i-th possibility in a finite set of discrete possible velocities at which a particle can move and p(vi) represents the probability of a particle actually moving at that velocity. We sum over all M of the velocity states available to the system. In the second line of the equation the discrete set of possible velocities has become a continuum and the probability of a particle moving at a given velocity has become the product of a probability density with respect to velocity, f(v), and a differential element of velocity. To a good approximation, the latter expression provides an accurate accounting of the variance of the velocities in a real thermodynamic system.

    If N identical particles constitute a system whose center of mass lies at rest and if we multiply the variance of the velocities of those particles by the mass of a single particle, the result equals twice the average kinetic energy of the particles,

(Eq’n 9)

in which the average kinetic energy gets its definition through T=N<τ>. We thus have a means of calculating the kinetic energy term in Equation 7.

    From here the study of thermodynamics consists, to a large extent, of working out proper descriptions of the various kinds of potential energy manifest in the system. We can describe the potential energy represented by pressure (in a gas), chemical potentials, thermoelectric potentials, and so on. Adding those descriptions to the randomized kinetic energy called heat gives us a number that must remain constant in an isolated system, regardless of how the system shifts its manifestation of those energies. And that leads us to the more conventional statement of the first law of thermodynamics: no real process can create or destroy energy, but can only transform it.

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