Popular Definition of Energy

When most people refer to energy, they mean the amount of energy in a substance such as coal or petroleum that can be transformed into useful work. For example, a liter of gasoline can be combusted to power an automobile to drive uphill for several kilometers. Or, they are referring to electric energy, which can likewise be used to perform work.

Physics Definition of Energy

The physics definition of energy includes the popular definition, but it is broader. In physics, energy can also refer to energy that cannot be used but is nevertheless present. For example, ambient heat contains energy. Likewise, there is tremendous nuclear energy contained in the atoms comprising ordinary household objects such as forks and spoons but which cannot be accessed for useful purposes.

Energy is often defined as the ability to do work. However, to be useful, we need to be more specific. Ultimately, in physics, the  term energy \(E\)ultimately refers to kinetic energy involving  the motion in a system, where \(m\) is the mass of an object and \(v\) is its velocity:

\(E = \frac{1}{2}mv^2\).

An important point is that motion can be on a visible, macroscopic level, such as that of a rushing train, or on a microscopic level, such as the motion of molecules in the atmosphere. Randomly-moving molecules are represent thermal energy or heat.

Energy can also refer to stored energy that can be transformed into motion. Energy occurs in several forms:

• Batteries, springs and raised objects store potential energy. For example, the energy stored in a spring, where \(x\) is the distance a spring is compressed or stretched, and \(k\) is a constant of proportionality, is:

\(E =kx^2\)

• Molecular bonds can store chemical energy
• Atomic nuclei can store nuclear energy

Units of Energy

There are several units in which energy can be expressed.

• The standard scientific unit is the Joule, or \(J\).
• A less standard science unit is the calorie, or \(cal\). One calorie is the amount of energy required to raise one gram of water by one degree Celsius at standard atmospheric pressure.
• In the USA, the unit for energy in food is also called the Calorie (with a capital C), but the food Calorie \(Cal\) is equal to 1000 science calories.
• The energy involved in heating systems is often expressed as in terms of the British Thermal Unit, or \(Btu\). One Btu comprises about 1055 Joules.
• Electric energy, such as that delivered to your home, is often expressed in terms of the kilowatt-hour \(kWh\). One Watt, \(W\), is equal to one Joule per second. The Watt is a unit of power. Multiplying power by time results in an expression of energy.

Photovoltaic panels (source: U.S. Dept. of Energy)

Power

As mentioned, when most people refer to energy, they mean useful energy, for example a quantity of coal or petroleum. However, when people want to use energy, they nearly always mean power, which is how much work (essentially useful energy) can be delivered per unit of time. Watts, kilowatts, and Horsepower are units of work.

Production of Power

Power can be produced by many means. Animals such as horses convert foodstuffs such as hay into muscular force that can be used to pull carts and plows. Wood or coal can be turned to create a thermal difference vis-a-vis the atmosphere that can be used to drive steam engines. Petroleum can drive combustion engines. Photovoltaic cells can convert high energy solar photons into electrical power. Wind mills convert the pressure differentials manifested by  wind into electrical motion.

Introduction

Thermodynamics is a branch of physics that concerns the flow of heat energy and the ability to convert energy into work . (Work is also expressed as force x distance.). Thermodynamics is extensively used in chemistry, atmospheric science, geology and engineering.

Thermodynamics is part of a yet larger branch of physics called statistical mechanics, which bridges thermodynamics with modern physics. In fact, the discovery of quantum mechanics was an outcrop of thermodynamics. Josiah Williard Gibbs first utilized a quantum approach to express chemical reactions. Max Planck then utilized a quantum approach to express energy levels in photons being emitted from a hot object.

Motivation and Applications

An important motivation for development of thermodynamics as a discipline was to understand how to make engines more efficient, and what the maximum efficiency for engines could be. Another application was to express the energy involved in chemical reactions. Today, thermodynamics is useful for a wide range of applications from energy efficiency to economics.

The Law

The First Law of Thermodynamics states that the total energy of an isolated system shall not change. In other words, the First Law of Thermodynamics requires that energy can neither be created nor destroyed. In other words, energy is conserved. This simply means that if heat flows from one object to another, the quantity of heat leaving the first object must equal the quantity of heat entering the second object.[1]

Discussion

The total energy shall neither increase more decrease. In physics, we say that the energy of an isolated system is conserved. (In physics, the term conserved has nothing to do with the environmental term of conservation).

What is an isolated system? Not surprisingly, a characteristic of isolated systems is that energy can neither enter or leave them. For example, a perfectly well-insulated container of hot water would be an isolated system. Although that particular hot water system is impossible in real life, it can be fairly well approximated using a reflective vacuum chamber, such as in an old-style coffee thermos. The entire Universe is considered to be an isolated system.

Mixing example

A simple example that demonstrates the First Law is to mix a quantity of cool water with an equal quantity of hot water. If the water is kept in insulated containers before and after the mixing, then the temperature of the final mixture will be the mean of the temperatures of the original constituents (there may be a slight variation due to evaporation or escaped heat). In other words, the total amount of heat energy remained the same despite the mixing and temperature changes.

Conduction example

Another simple example concerns a thermal conductor through which thermal energy flows from a warmer body of to a cooler body. The rate of thermal energy transfer Q is known as Newton’s Law of Cooling. We call the bodies reservoirs. The “hot” reservoir has a temperature of T_H, and the “cold” reservoir has a temperature of TC. The conductor is of area (cross section) A, and of length L.

Conductor bridging a thermal difference

The flow of thermal energy is:

\(\frac{dQ}{dt}=kA\frac{T_H – T_C}{L}\).

The quantity of heat energy lost by the warmer body is identical to the quantity of heat energy gained by the cooler body. This example can be easily replicated by using a U-shaped aluminum conductor to bridge two well-insulated cups of water of different temperatures.[2] (The conductor should be appropriately insulated as well for best results).

Conversion of Form of Energy

The First Law allows energy to be transformed from one form into another, such as from potential to kinetic energy. Yet the total amount of energy must remain the same.

Consider the case of a swinging pendulum (in a vacuum). As the pendulum bob falls, it will speed up, and its kinetic energy shall consequently increase, while its potential energy (due to gravity) shall decrease. When the bob rises, its potential energy increases at the cost of its kinetic energy. In all cases, though, the total energy remains the same:

\( Total~Energy = Kinetic~Energy+Potential~Energy+Thermal~Energy\).

Likewise, there is no change in the total energy of the system:

\(\Delta T + \Delta V + \Delta Q = 0\),

where

\(\Delta T\) is the change in kinetic energy, \(\Delta V\) the change in potential energy, and \(\Delta T\) the change in thermal energy.

What if we now have the pendulum operate in air, instead of a vacuum? The pendulum will gradually slow down due to air resistance. The total of potential + kinetic energy shall decrease! However, according to the First Law, the energy must go somewhere. It cannot merely disappear. As friction continues to operate and the pendulum continues to hit air molecules, the average (mean) velocity of the individual air molecules increases. So the air heats up a bit, and the pendulum’s energy is gradually transferred into heat energy. Eventually, the motion of the pendulum will stop, but energy in the air will have increased, as exhibited by an increased air temperature.

Modern Physics Modification

Who has not heard of Albert Einstein’s famous equation \(E = mc^2\)? Energy can be changed into mass and vice-versus. So the First Law must be modified to take into account this mass-energy equivalence. This is the realm of nuclear reactors and nuclear bombs, and its effect is insignificant on most systems we encounter in our lives.

Resources

• Paper concerning Fourier’s Law of Thermal Conduction
• Paper concerning Newton’s Law of Cooling and a simulation.
• Newton’s Law of Cooling simulation.
• Newton’s Law of Cooling program code.

Notes & References

[1] The phrase “conservation of energy” has a much different meaning than the common phrase “conserving energy”. The latter refers to consuming less of useful forms of energy such as coal or petroleum.

[2] Such demonstration kits are commonly sold by science education equipment firms. If ice water is used, then energy due to the phase change of melting ice must also be accounted for.

Introduction

Recall that one form of energy is thermal energy, which comprises the random motion of individual molecules that are part of a larger system. For example, water molecules in a tea kettle are moving back and forth in all directions and at many different speeds. Some water molecules might be moving slowly, and they would have a lower energy than those that are moving quickly.

Temperature refers to the intensity of thermal energy. Regardless of the energy of individual molecules, if their average (mean) velocity is high, then the temperature of the collection of those molecules is high. The the average is low, then the temperature is low.

Measuring Temperature

Analog thermometers (source: U.S. govt.)

One cannot easily measure the individual velocities of molecules within a large collection of such. Fortunately, there exists easier ways to measure temperature. The traditional device for measuring temperature is the thermometer. Thermometers can operate by measuring the expansion of a fluid such as mercury or alcohol. Other thermometers operate by comparing the expansion on one metal to another.

More modern devices can measure temperature by detecting infrared radiation emitted from an object. Such devices only measure surface temperature, but adjustments can be made to infer the internal temperature of an object such as a human.

Units of Temperature

Temperature measuring devices (source: U.S. government)

• In the USA, the degree Fahrenheit, F, is used to express temperature.
• In the metric system, the degree Celsius (or Centigrade)  expresses temperature.
• The preferred unit by physicists is the Kelvin, K. One unit of Kelvin is equal to one degree Celsius, except that the Kelvin system starts at absolute zero temperature. 0 degrees Celsius is equal to 273.15 Kelvin.

Distinction Between Thermal Energy and Temperature

Thermal energy is the kinetic energy contained in the random movement of molecules. Temperature is the measure of the strength or intensity such thermal energy. Temperature does not concern the amount of thermal energy. For example, a cup of water may be quite hot, yet contain much less thermal energy than the near-freezing Arctic Ocean. The relationship between temperature T and thermal energy Q is as follows:

\(T = \frac{Q}{MC}\),

where M is the mass under consideration and C is the heat capacity of the substance containing the thermal energy.

Resource

U.S. National Institute of Standards and Technology (NIST), SI Units: Temperature.