#### xaktly | Thermodynamics

Conservation of energy

### The First Law of Thermodynamics

The first law of thermodynamics is a conservation law. Simply put, it says that

#### The energy of a closed sytem is conserved.

But what does that mean and why is it important? A closed system is something we define. It could be the whole universe, a room or a reaction mixture in a beaker. For energy to be conserved means that the total energy of our system at any time is the same.

The first law says that energy can neither be created from nothing nor destroyed (turned to nothing). The universe or any closed system we choose within it contains a fixed amount of energy which can exist in different forms (work, potential energy, kinetic energy, heat), but the total amount of energy present is always constant.

There are four other conservation laws. They are

• conservation of mass (or matter)
• conservation of charge
• conservation of linear momentum
• conservation of angular momentum

The usefulness of conservation laws for solving problems is that they allow us to make equations – with an equal sign.

We can, for example, equate an expression of the total energy of a system before some change with the same expression afterward, allowing us to solve for a missing variable – maybe something we'd really like to know. In thermodynamic systems, we often form equations that reflect that

#### The heat lost by one part of the system = heat gained by the other part

The first law can help us to rule out bad ideas, too. Many proposed ideas "violate" the first law, and just can't happen. It's good to know that and not have to waste time considering them.

#### One caveat: relativity

One exception worth considering is the notion that mass can, in fact, be converted directly into energy, which does get around the laws of conservation of mass and energy. Einstein showed the equivalence of mass and energy with his famous expression

$$E = m c^2,$$

where $c$ is the speed of light.

#### The first law of thermodynamics

The energy of a closed system is conserved.

#### A caveat

Most of the time (the vast majority), the law of conservation of energy serves us well. However, the concept was modified by Albert Einstein, who showed an equivalence between mass and energy. Einstein's well-known formula,

#### E = mc2

showed that not only can mass be converted directly into energy, but that the amount of energy is tremendous. E is energy, m is mass and c is the speed of light, c = 2.99792458 x 108 m/s (exactly). The energy released by destroying mass is tremendous because c2 is on the order of 1016. So under special conditions (nuclear fission, fusion and radioactive decay) mass can be destroyed and energy can be created.

As I write this, data on the existence, nature and behavior of a subatomic particle called the Higgs boson, and possibly others related to it, are being examined. This new research might give us some insight into what mass is, why and how it is related to gravity, and the relationship between mass and energy.

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### caveat (cah·vee·ot)

A caveat is a warning that some specific conditions might exist on a statement. An example would be "I'm a pretty thin person – caveat: I sometimes gain weight during the holidays."

### Glossary

Before we go on, we need to get some basic thermodynamic definitions down:

An adiabatic process occurs with no heat transfer to or from the surroundings. The opposite is a diabatic process.

#### Energy

Energy in the thermodynamic sense (for the purpose of this section) will be the internal kinetic energy of atoms and molecules, which is in the form of atomic & molecular translation, and molecular rotation & vibration. Later we will include work and potential energy, crucial to a deeper development of thermodynamics.

#### Endothermic

An endothermic process takes energy, in the form of heat, from the surroundings in order to proceed. If the heat is not available, the process cannot proceed.

#### Exothermic

An exothermic process releases stored energy, in the form of heat, to the surroundings.

#### Equilibrium

The state of a system in which no properties are changing unless acted upon by some outside force. When all of the ice in a glass melts and the water comes to room temperature, the contents have reached equilibrium.

#### Heat

Heat is the sum total of all kinetic energy of that arises from the motion of atoms and molecules, and atoms in molecules (e.g vibrations of bonds).

#### Isothermal

An isothermal process is a change in a system that occurs without any change in temperature, ΔT = 0.

#### System (closed, open)

A system is the part of the universe under study. It could be a beaker on a hotplate or the air in a room, but each part needs to be defined. An open system is one that is free to exchange energy or matter with the surroundings - things outside of our system definition. A closed system is one that cannot exchange anything with the surroundings. A closed system might be a covered beaker, or an open beaker where we consider the atmosphere to be part of our system.

### Conservation of energy in chemical reactions

The total amount of chemical energy stored in the reactants of a chemical reaction is almost never exactly equal to the total amount stored in the products. The law of conservation of energy says that any excess energy must go somewhere and that any needed energy must come from somewhere.

Take a look at the reaction coordinate below. We can sketch one for any reaction. The vertical scale is energy and the horizontal scale is called "reaction progress," or the "reaction coordinate." It's a vague term that we use to capture all of the details of what goes on in a reaction – things that could, in fact, be quite a bit more complicated.

In the reaction coordinate below, the reactant(s), on the left, have a certain energy, and the product(s) have a lower energy. That means that in order for energy to be conserved, some will have to be released from the reaction in the form of heat. Such a reaction is called an exothermic reaction. Notice that the reaction has to proceed through a state of even higher energy, and we label that hump in the curve as the "activation energy," Ea.

Nearly all reactions require some extra energy "bump" or "spark" to get them going. That's the activation energy. After that, they proceed on their own. Consider, for example, a mixture that contains twice as many moles of H2 as O2. Left to its own, that gas mixture would sit there forever, but if we introduce a spark, the reaction

#### $$2 \, H_2 + O_2 \, \longrightarrow \, 2 \, H_2O$$

will proceed to completion.

The reaction coordinate below shows an endothermic reaction, in which the energy of the product(s) is greater than the energy of the reactant(s). In such a reaction the reactants must absorb some energy from the surroundings in order to proceed. One example of an endothermic process is dissolving urea, CH4N2O, in water. When a pile of solid urea is added to water, the beaker will get cold enough that atmospheric water condenses on it. The beaker feels cold because the reaction has removed heat from the water and the beaker.

### Energy and chemical bonds

It's important to understand how energy is "stored" in chemical bonds, and that can be a little tricky, so let's look at it carefully.

We'll start by considering the formation of a bond (it doesn't matter what kind) between two identical atoms, the green balls in the figure below. We'll consider the potential energy between these particles. At long range (the right if the diagram) there is very little attractive or repulsive force between the atoms. At very close range (far left), the atoms are so close that their electron clouds begin to overlap and they repel. Like charges repel.

But somewhere in between, the repulsive and attractive forces balance, and a stable bond is formed. That place on the potential energy (PE) vs. distance curve is the lowest PE between the particles. It's worth spending some time with this curve; it's one of the most important diagrams in chemistry.

Force and potential energy are related by a derivative in calculus: Force is the first derivative of PE. You don't need calculus to understand the concept, though. The force is proportional to the steepness of the PE curve.

At short range (left), the curve slopes steeply downward from left to right (a negative slope), so the force is pushing atoms apart – it's repulsive. At long range, the slope rises slowly from left to right, so the force is attractive and weak. In the middle, the force is small, and any excursion of the particles from the minimum will result in a restoring force back toward the balance point. Recall that the force betwen charged particles is the electrostatic force, also called the Coulomb force because it is modeled by Coulomb's law:

$$F_c = \frac{k q_1 q_2}{r^2}$$

where k is the electrostatic constant (k = 8.98 × 10-7 N·m2·C-2).The electrostatic constant just gets the units right. q1 and q2 are the charges on our particles and r is the distance between them. When the signs of q1and q2 are different, the force is negative, or attractive. When they're the same, the force is positive, or repulsive.

All binding curves have this feature in common – that there is a minimum in the potential energy that represents a balance between repulsive and attractive forces. Because the PE is reduced, bond formation always releases energy in some form.

Bond formation is always exoergic, meaning that it always releases some energy to the surroundings.

The attractive force between atoms is there because the nucleus of one atom exerts an attractive force not only on its own electrons, but (to a lesser extent) on the electrons of other atoms.

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### exoergic

Exoergic means occurring with the release of energy. This word is used because it allows for the possibility that the release of energy isn't in the form of heat.

#### "Energy storage" – it's a relative thing

We've established that when bonds form, energy is released, and the principle of conservation of energy demands that the opposite is true: that breaking bonds requires energy. But this often conflicts with a common notion that energy is released when bonds are broken. That's certainly often true, on the surface. For example, when one of the phosphate bonds of ATP is broken to convert ATP to ADP inside of living cells, we say (and we are correct!) that energy is released.

What's really happening, is that one bond is broken in order to form one or more stronger bonds. For example, when ATP → ADP, energy is consumed, but ultimately one or more molecules of H2O and CO2, both of which have stronger bonds than the O–P bond of ATP, are formed. The energy released in forming those bonds is larger than the ATP → ADP conversion energy, thus a net amount of energy is released.

Here is a quote that might help you understand all of this. I'm borrowing here from Prof. George Miller of UC, Irvine because he put it so well:

You exist, and you're made of atoms that are held together by forces we call chemical bonds. You're not going to just fall apart, and in fact it will take energy to do that. That's what a burn or an injury is.

The force holding atoms together is the electrostatic attraction between positively-charged nuclei and negatively-charged electrons. Nuclei exert attractive forces not only on their own electrons, but on the electrons of other atoms, although to a lesser extent. Not all forces are attractive, however. When atoms are brought too close together, their electron clouds repel, so there's a sweet spot, a distance between atoms that we call the bond length.

Energy is a relative thing, and energy can be effectively released from a bond if that process leads to the formation of stronger bonds.  Dr. Cruzan's Math and Science Web by Dr. Jeff Cruzan is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. © 2012, 2022, Jeff Cruzan. All text and images on this website not specifically attributed to another source were created by me and I reserve all rights as to their use. Any opinions expressed on this website are entirely mine, and do not necessarily reflect the views of any of my employers. Please feel free to send any questions or comments to jeff.cruzan@verizon.net.