Energy and mass are the most basic concepts of physics. As new discoveries are made about energy transformations, the results are put in the form of equations. Physics without math is like marriage without sex, or breakfast without coffee. The main reason that the newer physics of relativity is not well understood is that many people do not know the old physics. Einstein did not discard classical physics; he modified it to describe events inside of atoms and in outer space. For events between these extremes, Archimedes and Newton do very well, thank you.

Both energy and mass are conserved; within a closed system, there is no net gain or loss. When a home run is hit, there is a loss of the mass of one baseball, because it has left the system. When a hot dog is eaten, its physical state has been altered, but it is still within the system. The work and heat produced by the base runner are not created by him; they are derived from the chemical bond energy of the food he has eaten.

The basic equation for energy relations is: ΔG = ΔH - TΔS.  Δ is the standard symbol for an increment or amount of change. ΔG stands for change in free energy, which means energy available to do work, not a free lunch. The concept was developed by Willard Gibbs, a professor of physical chemistry at Yale, and probably the best scientist ever born in America. H is heat; T is temperature on the absolute or Kelvin scale, and S is entropy, an indicator of disorder in a system. There is no device for measuring S; it is always calculated.  ΔG can be obtained from redox potentials and from equilibrium concentrations of chemicals in a reaction mixture. ΔH is found from temperature measurements. And I am not making this up. It is best not to use the word entropy unless you are actually working on problems in thermodynamics.

Energy budgets are expressed in joules (jewels, not jowls), the SI units comparable to calories. 1 joule is the amount of work done when 1 kg is lifted 1 meter, in other words, the energy expended to move this weight in opposition to the force of gravity. It is such a small amount that the answers to energy problems are usually in kilojoules (kJ).

Glucose burned in a calorimeter yields the same products, CO₂ and H₂O, and the same amount of energy, 17 kJ per gram, as glucose metabolized by living cells. Mass and energy are conserved in both processes.

For those accustomed to working with calories, the conversion factor is:

1 kcal = 4.186 kJ.

The Calories listed on cereal boxes are really kcal. The label states that 30 g toasted oats contains 110 Cal. To compare with the content of glucose above:

110 kcal / 30 g = 3.667 kcal / g;  3.667 X 4.186 kJ / kcal = 15.35 kJ / g.

Since cereal is mostly carbohydrate, it is not surprizing to find a value equal to 90 % of the glucose value.

Of course, the supply of energy is not the whole story of human nutrition.  Adequate water and salts, essential amino acids, vitamins, all are important. Also desirable are a good cook and a pleasant dinner companion.

Power is the rate of doing work, expressed as 1 J / s = 1 Watt. Much of the early work on thermodynamics emerged from the work of Watt on steam engines.  Joule was recognized for his work on the mechanical equivalent of heat. Cecie Starr wrote that heat production by humans is similar to the heat output of a 100-Watt light bulb. Since the number of seconds in a day is: 60 X 60 X 24 = 86 400 s / day, the BMR above becomes  7 410 kJ / 86 400 s = 0.08576 kJ / s = 85.8 W.

In spite of the equal net effects of combustion and metabolism, the chemical reactions are totally different. There is no such thing as a “fat-burning” fitness scheme.  The living machine makes ATP (adenosine triphosphate) at ordinary body temperature, a molecule that powers muscular contraction, breaks down in the process, and is then resynthesized. The ΔG of ATP hydrolysis is -30.56 kJ/ mol (-7.3 kcal).  By convention, ΔG is negative when energy is available to do work, and positive when energy must be put into the system to produce a reaction. As Mathews wrote, “evolution has created an array of enzymes that preferentially bind ATP, and use its free energy of hydrolysis to drive endergonic reactions.”

The empirical formula of ATP is C₁₀H₁₆O₁₃N₅P₃. Adding up all the atomic weights gives a molecular weight of 507.488; thus 1 mole weighs 506.488 g. Compared to glucose, the energy of ATP is very small:

1 g / 506.488 g/mol X 30.56 kJ/mol = 0.0602 kJ / g.

This emphasizes the point that ATP is not a fuel; it is part of the cellular machinery, a carrier of chemical bond energy to the sites where energy is needed.

Large amounts of kinetic energy are released in explosions. 1 ton of TNT has been reported to release 5 X 10⁶ kJ. A ton of 2 000 lbs. is the same as 907.19 kg. Therefore:

E = 5.51 kJ / g,

about 1/3 as much energy as glucose, but much greater power, because it is released suddenly.

Fig. 6-1

photo credit: Wikipedia

Until 1900, mass and energy were entirely separate concepts, and the motion of electrons around the nucleus of an atom resembled a miniature solar system. First the quantum mechanics of Max Planck, and then the relativity of Albert Einstein, changed all that. The general public left quantum theory to the physicists, but they loved the idea of relativity. Einstein did not like the word; he was interested in absolutes. He liked women, and disliked haircuts (Fig. 13-1), which made caricatures almost too easy.

This book is about useful calculations, not theoretical physics.   Interconversions of mass and energy are not conspicuous features of everyday life, but they do occur. The production of C-11 in a cyclotron (Chap.10) is an example of conversion of energy to mass. Conversion of mass to energy occurs in the radioactive decay of uranium to lead; one intermediate step is the radium-to-radon transition (Chap. 10). The conversion was first demonstrated conclusively in a nuclear reactor, then shown in a spectacular way in atomic bombs, and now occurs regularly in nuclear power plants.

In the equation E = mc², E is energy (kJ) and m is mass (kg) and c is a constant equal to the speed of light, 3 X 10⁸ m/s. Ever since it became clear that the speed of light can be measured, some physicists have been interested in obtaining more and more precise values of c, a recent one: c = 2.99792458 X 10⁸ m/s. This is only slightly more practical than calculating π to one thousand places.

The fission of U²³⁸ has been reported to liberate 200 MeV (million electron volts) per atom. After several conversion factors are applied, this works out  to 8.097 X 10⁷ kJ/g.

According to the basic equation:

E = 0.001 kg (3.0 X 10⁸ m/s)² = 8.988 X 10¹⁰ kJ/g.

From this, it is apparent that only a small fraction of the mass of uranium is converted to energy, but that fractional amount, compared to other energy sources, is enormous.

There is some conversion of mass to energy outside of nuclear power plants. If a 10-Watt flashlight is left “on” for a long time, there is a small loss of mass:

10 W = 10 J/s

m = E/c² = 10 / (3 X 10⁸ m/s)² = 1.1126497 X 10^-16 kg  per second.  

Seconds in one year = 60 s/min X 60 min/hr X 24 hr/d X 365.2 d/y =

3.1553 X 10⁷ s/y.

m/y = 3.1553 X 10⁷ X 1.113 X 10^-16 = 3.5107 X 10^-9 kg = 3.5107 µg/y.

An analytical balance can measure a change of 0.1 mg or 100 micrograms. The time required to make the loss of mass from the flashlight perceptible:

t = 100µg / 3.5107 µg/y = 28.48 years.

This is really just playing with numbers, a game that theoretical physicists love to play.