# Heat and Energy

### Written by tutor Anthony G.

The world is composed of many particle systems interacting in space and time via four forces – gravity, electromagnetism, the strong force, and the weak force. In modern physics, the mechanism by which these forces function is described mathematically as a “field.” The field theory of gravity is General Relativity, in which a mass distribution determines the contours of space and time, and space and time determine how the mass distribution moves. The quantum field theories of electromagnetism, the strong force, and the weak force are known, respectively, as quantum electrodynamics, quantum chromodynamics, and flavor dynamics, and these last three theories, along with the zoology of particles composing our universe, are grouped together as the standard model of particle physics. While both the standard model, and general relativity, have met with extraordinary experimental success, they are believed to be incompatible. In this situation, at a very basic level, the questions “what is matter” and “what is energy” are not yet answerable.

In everyday terms, energy is the capacity to do work. Energy is said to be conserved. This means that:

- The total energy of the universe is a constant
- The total energy of some subsystem is equal to that system’s energy at an earlier time, plus the energy that entered the system, minus the energy that left the system

So, for instance, if a car crashes into a wall, the energy of the car before the collision is approximately equal to the energy needed to crush the car, damage the wall, heat up the road, generate a noise during the crash, and shake up the contents of the vehicle.

In classical mechanics work is expressed as:

where P is a path from one point in space to another point in space. Without calculus, the amount of work done by a constant force in one dimension is

Work=Force*displacement in the direction of the force

=Force in the direction of the displacement*displacement

A force that acts solely parallel to the motion does not do any work.

A simple illustration of conservation of energy follows. A body is raised a distance H above the ground, acquiring a potential energy of MgH. If that body is then allowed to fall, its final kinetic energy, ignoring losses to friction, is going to equal its initial potential energy. We can thus compute its final speed:

The work we did lifting the body up was stored as energy in the gravitational field. When we released the body, gravity did work on it, accelerating it and causing it to acquire kinetic energy equal to the work we originally did lifting the body against gravity.

The unit of energy, and of work, is the unit of force, the Newton, multiplied by the unit of distance, the meter. A Newton-meter is also known as a Joule, in honor of physicist James Prescott Joule. Joule demonstrated that work done by gravity or by an electric coil could be transformed into an equivalent amount of heat.

In Special Relativity, mass, even mass that is not moving, is equivalent to an amount of energy, called its rest energy, given by the Einstein formula:

E = m_{o}c^{2}, where

m_{o} is the rest mass of the object, and c is the speed of light, 3 x 10^{8} m/s

The energies released in fission and fusion reactions come from the transformation of mass into energy. Mass is approximately conserved in chemical reactions because the energies involved in chemical reactions, when divided by c^{2}, yield small numbers.

In electromagnetic theory, energy is stored in electric and magnetic fields, and is transported from one place to another via electromagnetic radiation, or light. In classical electromagnetism, the energy in an electromagnetic wave is smeared over the entire wave, and even a mathematically small portion of the wave contains some energy. In a semi-classical picture, the energy arrives in chunks, called photons, and the energy of a photon is given by the Einstein-Plank formula:

E = hf

High frequency radiation, such as ultraviolet light, x-rays, gamma rays, and cosmic rays, is referred to as ionizing radiation because a single photon of such light can ionize an atom. The semi-classical picture of light-matter interactions is further refined in Feynman, Schwinger, and Tomonaga’s quantum theory of light, quantum electrodynamics.

In thermodynamics, heat is the energy associated with the apparently random motion of particles. Heat can be transferred by conduction, convection, or radiation. Conduction involves bringing two bodies into physical contact. Heat flows from the higher temperature body to the lower temperature body until both bodies have reached the same temperature, at which point the bodies are said to be in thermal equilibrium. Convection involves the transfer of heat from one object to another via an intermediary substance, such as an air current. Radiation involves the transfer of energy via electromagnetic radiation.

Work can be converted into heat with 100% efficiency. Heat is not so easily converted to work. The Second Law of Thermodynamics, as stated by Claussius, holds that:

“Heat can never pass from a colder to a warmer body without some other change, connected therewith, occurring at the same time”

The Kelvin-Plank statement of the 2nd Law is:

“It is impossible to devise a cyclically operating device, the sole effect of which is to absorb energy in the form of heat from a single thermal reservoir and to deliver an equivalent amount of work.”

The first law of thermodynamics is simply a statement of conservation of energy. The first law would not be violated were we to heat a room by taking a cube of ice, extracting heat from it until it was at a temperature near 0 Kelvin (absolute zero), and then discarding it. However, the 2nd law states that such an process is impossible.

The thermodynamic entropy is

with dQ an infinitesimal of heat transfered during a reversible process at temperature T. In a process involving heat transfer, it can be shown that the total thermodynamic entropy remains the same, or increases. This condition then prevents the spontaneous transfer of heat from an object to a warmer object. In statistical mechanics, the entropy can be formulated as the number of states accessible by a system, or as the log of the number of states accessible by a system:

S = k_{B}ln*w*, where k_{B} = 1.3806488 x 10^{-23} ^{Joules}/_{degree Kelvin}, and w is the number of states.

The tendency of a gas to fill all of the volume available to it, or of heat to flow from a warm body to a cold body, can be thought of as a natural tendency to maximize the entropy. According to the Central Limit Theorem, we would not expect to flip a fair coin a trillion trillion times and get anything very different from exactly 50% heads. Similarly, we cannot expect heat to spontaneously flow from a cold object to a warm one. Collisions between particles tend to transfer energy from the higher energy particle to the lower energy particle, and such a flow of heat would require trillions of trillions of collisions that transfer energy from lower energy particle to higher energy particles. Even small deviations from the 2nd law have never been observed for macroscopic systems.