First things first – what is energy?

Ha! Great question.

Energy is one of the most powerful, unifying, and—yes—abstract concepts ever developed in science. At its core, energy is a conserved property of matter, historically measured by the work it could perform. It’s a bit more nuanced than this, as you’ll soon see, so let’s first start with the easier concept of work.

Work – perhaps an easier starting point

Work itself is not abstract. It is physical and immediate. When you lift a weight, you do work, you consume energy. You feel it in your muscles. You are burning fuel–consuming energy–in your body to raise that weight against gravity.

We first learned how to quantify work long before the arrival of modern physics, through the analysis of simple machines—levers, pulleys, and other mechanical devices developed thousands of years ago.

In its simplest form, work was defined as weight multiplied by vertical distance. In modern notation, this becomes:

W = mgΔh

where W = work, m is mass, g is the gravitational acceleration, mg is weight, and Δh is the change in height.

More generally, work is defined through Newtonian mechanics as the integral of force through distance. That is the formal definition—but the physical idea remains the same: work results by pushing or lifting something through space against resistance, e.g., gravity.

Some history

Historically, work became the reference point for energy. The units of work—such as foot-pounds—became the standard units of energy. One foot-pound is the energy required to lift one pound of weight one foot high.

This origin is not arbitrary. The concept of work emerged from very practical problems—most notably in mining. When deep mine shafts filled with water, that water had to be removed. Teams of horses were used to haul buckets upward. Later, steam engines replaced the horses. The effort required to lift those buckets—weight through vertical distance—was called “work.” Steam engines proved to be more cost effective than horses for producing a given amount of work.

Work played a critical role in Sadi Carnot’s 1824 theoretical analysis of the steam engine and became one of the key forms of energy, alongside heat, in Rudolf Clausius’s ground-breaking 1850 publication that replaced the incorrect caloric paradigm with an entirely new paradigm called energy.

From there, the idea expanded. Different physical processes designed to produce work—mechanical, thermal, electrical—could all be compared by asking a simple question: what is required for the process to produce a given amount of work? Such questions gave rise to the concept of energy as a universal currency. And the answers to such questions provided the basis for economic cost comparisons.

Many ways to do work

Consider that there are many ways to lift a weight.

• A horse or a person can do it using muscle.
• A high-pressure gas inside a piston can push the weight upward.
• A high-temperature gas can heat that same system, raising pressure and enabling work—this is where Carnot’s and Clausius’s analysis and, more generally, thermodynamics itself entered the picture (see here). (Note: the original name thermodynamics was meant to capture the ability of heat (thermo) to do work (dynamics).)
• A battery can power an electric motor in which moving electrons generate magnetic forces that turn a shaft and lift the weight.

These are very different physical processes. They involve chemistry, heat, electricity, and mechanics. And yet, in every case, the same thing is happening:

• Work is being done, and energy is being consumed.

But what does “energy is being consumed” mean?

In the world of energy, there must be a cause behind the effect. Buckets of water don’t magically come out of the mine shafts on their own. So what is it about energy that prevents this magic from happening? Answer: energy is a conserved quantity. This is what makes energy such a powerful concept.

Energy — a conserved quantity

Energy is a property of matter, defined as U for a closed system. In practice, only the change in energy matters. In fact, thermodynamics is built on change—which is why it makes extensive use of calculus and differential relationships.

The concepts of energy, work, and heat are all tied to change.

Energy is conserved, meaning that the sum of all energy changes must equal zero. Not approximately zero—exactly zero.

Work provides a clear starting point.

When a force acts through a distance, work is done. When you lift a weight, you change its position in a gravitational field. That change has an associated energy increase. But that increase does not come from nowhere—the source of the motion must lose energy by exactly the same amount.

When you compress a gas by pushing a piston into a cylinder, you again do work. The energy of the gas increases, often observed as a rise in temperature. And again, the source of that work must lose an equal amount of energy.

This is what conservation of energy means.

Consider an ideal case.

Consider a gas confined by a movable, weighted piston. Initially, the system is at equilibrium: the piston is stationary because the gas pressure balances the external pressure.

When the cylinder contacts a hot reservoir, heat flows into the cooler gas. The gas warms and expands at approximately constant pressure, raising the piston and weight. The reservoir’s energy loss equals the increase in the gas’s internal energy plus the gravitational energy gained by the piston and weight. Thus, total energy is conserved.

Notice how we describe this process by saying that heat flows from the hot reservoir into the cooler gas.

This language is convenient, but it can be misleading. Heat is not a substance, and nothing physically “flows” in the sense of a material transfer. Heat is simply a way of describing energy transfer driven by a temperature difference.

It is important to distinguish clearly:

• Energy is a property of a system
• Work and heat are two different ways that energy can change
• The thermodynamic symbols used for these terms are: U = energy, W = work, Q = heat

This bears repeating: the power of energy as a property is that it is conserved

In every example above, the increase in energy in one place is matched exactly by a decrease somewhere else—your body, a fuel source, a compressed gas, or a thermal reservoir.

This equality is what allows us to compare and convert different forms of energy using common units.

In real systems, additional effects—such as friction—must be considered. These do not violate conservation of energy. Instead, they redistribute energy into forms that are less useful for producing work, often appearing as a heat loss from the system, e.g., friction causing equipment to heat up, uninsulated pipes losing heat to the environment, etc.

As a result, while energy is always conserved, the amount of useful work obtained from a real-world, non-ideal process is often less than the total energy consumed in generating that work. If things were ideal, the amount of work obtained would equal the energy consumed. But, and this is critical, the amount of work obtained would NEVER be greater than the energy consumed. That is a hard limit.

This concept deserves further discussion.

Energy as a conserved quantity establishes limits for what is and is not possible

The fact that energy is conserved means that an increase in energy somewhere must be accompanied by a decrease in energy somewhere else. Energy change does not happen on its own. There must be a cause behind the effect.

This forces a strict accounting.

The minimum amount of energy required to lift a weight is equal to—never less than—the increase in energy of the lifted weight itself. In practice, the required energy is usually greater due to real-world effects such as friction and other inefficiencies.

Conversely, the maximum amount of work that an elevated weight can produce is equal to—never greater than—the decrease in its energy as it is lowered from a higher to a lower position.

These are not approximations. They are hard, exact limits.

Energy conservation tells us what is possible and what is impossible. It sets the boundaries within which all physical processes must operate.

As will be covered later, additional limits—more subtle, but just as powerful—emerge when entropy enters the picture.

Kinetic Energy — The Energy of Motion

In earlier chapters, we introduced energy as the sum of kinetic and potential components. While that idea is straightforward, the details require closer attention.

Kinetic energy is typically written as: 1/2 m v²

While this expression works well for a single atom moving through space, most real systems consist of molecules made of multiple atoms. In such systems, motion becomes more complex.

A molecule can:

• translate (move from one location to another)
• rotate
• vibrate internally
• move collectively, as in bulk flow

Each of these contributes to the total kinetic energy.

As a result, a multi-atom molecule carries more total kinetic energy than a single atom of the same mass moving at the same translational speed. That additional energy is real, and it matters.

Tracking all of these motions explicitly is possible in principle but impractical for real systems containing vast numbers of atoms. Classical thermodynamics takes a simpler approach. It introduces heat capacity (discussed in depth in Chapter 9).

Heat capacity quantifies how much energy a system absorbs for a given rise in temperature. It depends directly on how many modes of motion — translation, rotation, vibration — are available to the atoms and molecules involved. More modes means more ways to absorb energy, and thus a higher heat capacity. Temperature alone cannot capture this. Temperature is an intensive property that tells you the average translational kinetic energy. Heat capacity is an extensive property that scales with the amount of matter present; per mole or per unit mass, it becomes intensive. Together they tell you how much total energy changes for a given change in temperature — or how much energy is required to change temperature by a given amount — and thermodynamics, as this chapter repeatedly emphasizes, is fundamentally about change.

Heat capacity is defined as:

C = δQ / dT

where δQ is the infinitesimal thermal energy added to the system and dT is the resulting change in temperature. When thermal energy is added to a system, its internal energy changes accordingly:

dU = δQ = C_v dT

The physical meaning of heat capacity — why it takes the values it does, and why it differs between gases, liquids, and solids — will be developed fully in Chapter 9.

A brief aside: the term heat capacity is unfortunate. It originates from an outdated view in which heat was thought to be a substance called caloric. As we know now, heat is not a substance—it quantifies the change in energy of a system caused by thermal contact with another system (see here). But the name remains, and it continues to confuse. We will return to this later.

Potential energy

If you hold a basketball outside a window on the 10th floor of a building, the ball isn’t moving—its kinetic energy is zero. But you have no doubt that it has the potential to deliver a great deal of kinetic energy to the sidewalk below.

When you let go, gravity takes over. The ball accelerates toward the Earth. Potential energy is converted into kinetic energy, and the total energy remains constant.

We are very familiar with this kind of behavior. Gravity-driven events are part of everyday experience.

But gravity is only one example of potential energy. The other is electromagnetic forces, which play a much greater role in thermodynamics. Events involving these forces are just as common—more common, in fact—but far less visible. They are happening all around us, all the time, but they are hidden at the atomic scale.

Consider gasoline.

When fuel burns in an engine, the potential energy stored in reactants’ chemical bonds transforms into kinetic energy in the products’ motions—manifested as high-temperature, high-pressure gases that push pistons and move your car down the road.

This is no different, in principle, from the falling basketball. Potential energy transforms to kinetic energy. The only difference is the source of the potential energy: electromagnetic rather than gravitational.

As discussed in the previous chapter, the electromagnetic potential energy of matter arises from several sources:

• interactions between atoms and molecules
• vibrational motion within multi-atom molecules
• interactions between electrons and nuclei within atoms and across chemical bonds

This quickly becomes complicated.

How do we keep track of all of this? Two approaches are used:

• heat capacity accounts for the energy changes involved with molecular motion (first two bullets)
• calorimetry measures energy changes associated with the third bullet (discussed in Chapter 13)

Together, these allow us to quantify total energy changes of a system.

Internal energy (U) represents the total kinetic and potential energy of a system. It is a core property used in thermodynamics, largely because it is conserved. Tabulated values of U are always measured from some chosen reference state, so what matters in practice is the difference between two states..

Absolute or Relative?

Kinetic energy and potential energy both look like fixed numbers, sitting inside a molecule. Neither one is. Each is measured from a reference.

Kinetic energy, ½mv², depends on v, and v depends on the frame in which motion is measured. Watch a still body of air go by from a moving car, and the kinetic energy you calculate for it changes — without a single atom inside it moving any differently relative to its neighbors. Thermodynamics settles this by measuring motion relative to the system’s own center of mass, setting aside any bulk motion of the system as a whole. That’s why the air inside a moving train has the same temperature as the air on the platform. With that frame fixed, kinetic energy has a genuine zero: no motion relative to the center of mass.

Potential energy depends on where its zero is placed, and physics places it most naturally at infinite separation, where atoms no longer interact. That reference isn’t merely convenient; it can be approached. Expand any gas far enough and it becomes ideal: its atoms spend essentially no time near one another, and its energy becomes entirely kinetic. Compare a real gas, liquid, or solid with that ideal gas at the same temperature, and the difference is its potential energy — a quantity determined from measured pressure, volume, and temperature data.

Property tables, by contrast, place their zero wherever is convenient, and so the absolute numbers they report carry no physical meaning on their own. None of this makes potential energy any less real. Potential energy exists at absolute zero; it is what holds a crystal together once atomic motion has all but stopped.

In the end, the key thing to remember is that thermodynamics is built on change. It’s the change in energy — kinetic, potential, internal, or otherwise — that matters, not the number you started from.

The 1st Law of Thermodynamics – simply put

This then brings us to the first law of thermodynamics, which in its most general form states:

• there is a property of matter called energy that is conserved

A more specific form of the 1st Law was created by Rudolf Clausius when he considered energy changes of a system contained inside a piston-in-cylinder assembly:

ΔU = Q − W

This equation states that the change in internal energy of a system (ΔU) is equal to the thermal energy (heat) entering the system (Q) minus the work done by the system (W). If no heat and no work, no change in energy. For more on this, see here.

Note: In (Planck and Ogg, 1990) p. 57, Planck pointed out that heat (Q) quantifies the difference between two numbers and that such terms as “δQ” are inappropriate in this situation since the differential of a specific number is not the same thing as the very small difference between two specific numbers.  Q should stand on its own as a measured difference in internal energy caused by the exchange of thermal energy between two bodies.  Q is a difference, not a thing.  A body doesn’t have Q.  See also (Lewis and Randall, 1923) p. 54 for further discussion.  In this book the author chose to remain with δQ to denote a very small change in Q. 

Planck, Max, and Alexander Ogg. 1990. Treatise on Thermodynamics. 3. ed. Transl. from the 7. German ed. New York: Dover.

Lewis, Gilbert Newton, and Merle Randall. 1923. Thermodynamics and the Free Energy of Chemical Species. New York: McGraw-Hill Book Company, Inc.

Mass too is a conserved property of matter

Mass is also conserved.

As stated in Chapter 1, atoms persist. They do not appear or disappear under ordinary conditions. When applied to large collections of atoms, this leads to conservation of mass.

This principle was established experimentally by Antoine Lavoisier in the late 1700s, who showed that for chemical reactions, the total mass of reactants equals the total mass of products.

Matter is neither created nor destroyed.

This principle, combined with energy conservation, forms the foundation of the mass and energy balance, a central tool in engineering and science (see Chapter 4).

Summary

Energy is best understood through something physical, such as lifting a weight. Raise the weight and its energy increases; lower it and its energy decreases. What matters is not how much energy the weight has, but how much it changes — and that change is exactly the work done to lift it, or the work it can do as it falls.

Energy exists in kinetic and potential forms; one may change into the other, but in an isolated system the total does not.

Because energy is conserved, every change in energy must be accounted for — an increase in one place is matched exactly by a decrease somewhere else. This sets strict limits. The energy required to produce a change can never be less than the change itself, and the work extracted can never exceed it. Real processes fall short of these limits, but never beat them.

The First Law of Thermodynamics formalizes this accounting: ΔU = Q − W. The change in a system’s energy equals the energy exchanged as heat and work. Mass, too, is conserved, and together the two form the basis of the mass and energy balance, the subject of the next chapter.

END