# Energy Units

© Fernando Caracena, 2016

The concept of energy was among those concepts developed early in the history of physics. At first, energy was encountered in studies of mechanical motion, and was not known to be operating in other areas of physics such as in thermodynamics. But today, we know that energy is present throughout the universe in many forms. Meanwhile, we have a plethora of units used to measure energy in a variety of specialized area. Here, we discuss some of these various units, and relate them to the SI unit of energy known as the Joule.

## Energy and Information

Everything is made of energy, which takes on various shapes. Energy is the heart beat of physical reality. The combination of energy and information as substance and form, respectively, enter into the existence of all material bodies. Information specifies the state of a physical system, and that state manifests as physical form that exists in a framework of time and space. Everything is made up of states of energy that composite into shapes in space. And the objects thus created form the fabric of space, their motion the appearance of time.

On the microscopic level, all the parts of physical reality are in constant motion; and this motion happens on several levels. The gross motions of the atoms and constituent molecules of a substance are what we sense as heat. The motion of the outer components of atoms, the electrons, make up electric currents. The internal motions of elementary particles, such as spin, give these objects their basic properties. We have learned that a set of various properties of motion that we measure in our environment are really all the same thing—energy. As a result there are many units of energy, which we keep segregated in different fields: Joules (J), ergs, watt-hours(Wh), calories(cal), British Thermal Units (BTU)s, barrels of oil (BOE), tons of dynamite(t), electron volts(eV), etc.

## The SI Unit of Energy

This unit of energy was defined from mechanics in terms of work. Work, which is the sum of the product of the component of force acting parallel to any displacement it causes times the magnitude of that displacement, is just the transfer of energy.

The following excerpt is from the post, The Concepts of Energy and Momentum Conservation:

To describe the useful application of a force, physicists defined a quantity called work as the product of a component of force (F) acting in the direction of a displacement (dr) that it produced in an object:

dW = F•d.                                (1)

Note that this expression represents an infinitesimal amount of work, the work done by a small part of the application of a variable force, which therefore, lends itself to the methods of calculus (see below). Calculus simplifies the mathematics in favor of clearly presenting the physics.

The amount of work (or energy transferred by a force f acting in the x-direction and displacing an object a distance d12(= x2 - x1) is therefore given by the integral,

Work= 12 f(x) dx  .             (2)

It is clear then that in SI units, the unit of energy is the Newton-meter (N m), which is called the Joule(J). In words, the Joule is defined as the amount of energy transferred by a force of one Newton exerted parallel to a displacement of one meter.

## Heat Units

Mankind has known how to make fire by rubbing a piece of wood (say a dowel) on another (a grooved block of wood). In Europe in the late 1700s, Benjamin Thompson noted that a lot of heat was generated in the process of boring a cannon.  Based on experiments conducted on these effects, in1798 Thompson published "An Experimental Enquiry Concerning the Source of the Heat which is Excited by Friction", Philosophical Transaction of the Royal Society, p.102. James Prescott Joule (among others) conducted a number of experiments, in which a measured amount of mechanical energy that disappeared was always accompanied by the production of an equivalent amount of heat. He published his results in a paper entitled, "The Mechanical Equivalent of Heat". In time, physicists realized that heat and mechanical energy are not only equivalent, but that they are the same thing. energy.

The unit of heat developed by people working in thermodynamics , the calorie, was defined as the amount of heat required to increase the temperature of 1 gram of water 1 °C. Later experimenters discovered that this definition was insufficient, because this amount varied with varying ambient conditions of pressure and temperature, being very close to a value of about 4.2 J per calorie. The current standard is chosen as the following equivalence:

1 Joule (J) = 4.184 calories (cal),

or

1 cal= 0.2390 J.

A similar unit of heat used by the English is called the British Thermal Unit (BTU), which was originally defined as the amount of heat required to raise the temperature of 1 lb of water 1 °F. An American pint of pure water is just about a pound (16 ounces of mass). The BTU is as variable as the calorie, so it has to be standardized as a unit of energy in Joules by the same standard as for the calorie(above), which yields

1 BTU = 1054.35026444 J .

This is an approximate unit anyway, and could have been defined a rounded number, such as 1054 J. However, the calorie and BTU are not independent of each other, so that only one of them can be chosen as a conveniently rounded off value.

Incidentally, it so happens to be that one ordinary wooden match, when burned completely, generates approximately         1 BTU of heat.

## Quads

In dealing with very large energy usage, such as the annual US energy consumption, the ordinary BTU is too small a unit, and for the public, the numbers have to be expressed using a lot of zeroes. To make it more convenient to express these numbers, text-oriented, left-brained people use a unit called quadrillion BTUs, or quads, where

1 quad = 1015 BTUs.

or

1 quad = 1015  *1054.35026444 J .

1 quad = 1.054X1018  J .

That is a lot of matches burning!

## Power and Energy

Power is defined as the time rate of energy transfer, the unit of which as the Watt (W), where∫

1 W = 1 J/ s.

People normally associate the Watt with electric power consumption, in a unit of energy called the kilowatt-hour (KW h). Since 1 h=60. (min)*60. (sec./min.) or 3600. sec.

1 KWh =1000. W* 3600. sec. =3.6 x 106 J.

By dividing one quad by a KWh, one obtains the number of KWhs in a quad  as

1 quad=1.054X1018  J/3.6 x 106 J/KWh

1 quad=2.928 x 1011 KWh

## Barrel of Oil Equivalent

Petroleum companies measure energy delivered in terms of a Barrel of Oil Equivalent Equivalent (BOE), which is the approximate amount of energy by burning a barrel of oil, equivalent tometric prefixes

1 BOE = 6.1178632 × 109 J

as defined by the Internal Revenue Service of all things.

## Kilotons of TNT

When enormous amounts of energy are involved, such in the detonation of nuclear devices, energy involved is normally given in kilotons of Trinitrotoluene (TNT). By convention the output of one ton of TNT (t) is defined as

1 T = 4.184 gigajoules or 4.184 x 109 J.

A kiloton (KT) is therefore,

1 kt= 4.184 terajoules or 4.184 x 1012 J .

(To see definitions of numerical prefixes click here.)

1 mt= 4.184 x 1015 J.

The atomic bomb dropped on the Japanese city of Nagasaki during WWII had a blast yield of

21 KT=8.786 x 1013 J,

which is equivalent to annihilating a total of mass consisting of of 0.9762 grams of equal parts of matter and antimatter (see gram).

## Mass Equivalent of Energy

One way to cut the size of numbers of energy units is to express energy in terms of the equivalent mass using Einstein' s equivalence formula for mass and energy

E = m c2

or

m=E  c-2 .

As currently measured, the speed of light in the vacuum is

c=2.99792458 x 108 m s-1 .

It is easy to cut and paste these numbers from one Web reference into a python command line with a slight modification of the power of 10 (c=2.99792458e8 m s-1). However, it is easier to remember the approximate value of

c ≈ 3.00 108 m s-1 .

In any case, this implies that a kilogram mass-equivalent of energy is a huge number,

1 kg * C2 =  8.98755179 x1016  J,

or

1 kg * C2 ≈ 9e16J

1 kg *C2 =8.987551787368176e+16 J.

1 kg * C2 ≈ 21481 KT =21.481 Megatons of TNT.

## Solar Radiative Energy Striking the Earth

Life on Earth depends on solar radiation, and even most of the energy that we use originated in the solar flux of energy striking the earth. It is natural to ask about how much solar energy falls on the earth each day, or each year. The number must be enormous, right?

The amount of solar energy incident on top of the earth's atmosphere is called the solar constant,  as historically so named, but is not really a constant,

Sflux ≈ 1.361 kilowatts per square meter (kW/m²).

Consider that the sun is always shining upon the earth. Sun light strikes a cross sectional area of the earth equal to that of a circle having the same approximate radius as the spherical earth, which is

re ≈ 6371 km

or

re ≈ 6.371 x106 m.

This cross sectional area is

Aerth ≈ π re 2 ,

which is easily computed using python. The result is

Aerth ≈ 1.275 x1014 m2

The total solar power falling on the earth is therefore8.98

SFT=Aerth * Sflux

SFT ≈ 1.736  x1017 Watts,

which is equivalent to a stream of matter annihilating into radiation of about 1.931 kg per second.

The annual input of solar energy into the earth is enormous. Consider how many seconds there are in a year,

1yr=3000. sec/hr *24 hr/day *365.25 day

1yr=3.156 x 10sec.

The amount of annual solar energy received by the earth is

Eats=6.094 x 10kg.

The energy consumption in the United States in 20014 was 27,050 TWh or 9.738 x 1019 J. Converted into mass equivalent, this amount of energy would be equivalent to 1083 kg, which is equal to about 0.0018 % of Eats. The International Energy Agency estimates that, in 2013, total world energy consumption was 5.67 × 1020 J  or about 6310 kg mass equivalent. This is just about 0.010 % the total solar energy input for a year.

## The Electron Volt

The electron volt (eV) is an important unit of energy in physics used in particle physics. This is the amount of energy acquired by a particle having a unit of elementary charge (e) when accelerating through a potential difference of one volt (V). The electronic unit of charge in Coulombs (C) is

e =1.602176621x10-19 C.

The electron volt is therefore

1 eV=1.602176621x10-19 J,

Which is used in multiple using the metric prefixes: mega(106 ), giga(109 ), tera(1012), etc. As a example,

1 Mev=106 eV =1.602176621x10-13 J.

1 GeV=109 eV =1.602176621x10-10 J.

1 TeV=1012 eV =1.602176621x10-7 J.

In the LHC protons circulate in tight, needle-like clusters, each proton having a kinetic energy of  6.5 TeV, which in terms of Joules is 1.0414 x10-6 J. Now how does that energy compare with something in the range of human experience?

Consider a mosquito in flight, how muck kinetic energy does it develop?

The mass of a mosquito (from a Wikipedia article )is about

mo =2.5 x10-6 kg

; and (from another, “Speed of Animals”) it flies at about

vo = 0.73 m/s .

The kinetic energy of a flying mosquito is then about

½ mo v2 = 6.7 10-7 J , or 0.67 x10-6 J.

Comparing this number with the average energy of a proton circulating in the LHC, you can see that the kinetic energy of a proton in a LHC beam is about 1.56 times that of a mosquito in flight! That is a huge amount of energy packed in a particle only 0.8418±0.0007 fm ( 10-15m) across (source)!

## Nutritional Energy Requirements of Humans

Nutritionists measure food energy in kilocalories:

1 kcal = 239.0 J.

The daily energy requirements of adults (HERA) run around 2000 Kcal:

HERA=2000 kcal = 4.780 x 105 J.

The power requirement for an adult human is therefore

1 HERA /day = 4.780 x 105 J/ 86400 s

PHERA=1 HERA /day = 5.53 Watts.

For fun, translate the HERA into kilowatt hours (kWh)s. Consider that

1 kWh = 3.6 x 106 J.

HERA=4.780 x 105J/ 3.6 x106J  kWh.

HERA=0.133  kWh. or 133 Wh.

Note that this same number is obtained by multiplying PHERA by 24 h. The human body is indeed a low wattage machine! According to a Wikipedia article,

"Most of the world's population live in areas with insolation levels of 150-300 watts/m², or 3.5-7.0 kWh/m² per day."

Obviously, there is a lot of room to meet the nutritional requirements of humanity, despite the pessimistic views of the club of Rome.

## The Sun is Losing Mass!

The amount of energy reaching the top of the Earth's atmosphere, called "the solar costant”, which is about 1.361 kW/m², is not really a constant, but has slight variations. The planet is bathed with an enormous amount of power, as is stated in a Wikipedia article:

"for the whole Earth (which has a cross section of 127,400,000 km²), the power is 1.730×1017 W (or 173,000 terawatts),[9] plus or minus 2 W/m2."

Further,

"The angular diameter of the Earth as seen from the Sun is approximately 1/11,700 radians (about 18 arc-seconds), meaning the solid angle of the Earth as seen from the Sun is approximately 1/175,000,000 of a steradian. Thus the Sun emits about 2.2 billion times the amount of radiation that is caught by Earth, in other words about 3.86×1026 watts."

Total solar energy radiation of 3.86×1026 W in mass energy units is 4.29 x 109 kg/s. In one year (1yr=3.156 x 107 sec.) this flux adds up to 1.354X1017 kg of solar mass converted into radiant energy. Compare that with the mass of the earth, 5.9736 x 1024 kg, which allows one to compute the total solar energy radiation in Earth masses per year, 2.27 x 10-08. At this rate, the sun will radiate away 1 earth mass in about 50 million years. “The Sun has 333,000 times more mass than the Earth.”—source. So you see, we are not in danger of losing the sun in the near future.

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