#### xaktly | Physics | Chemistry

Conservation laws

### The basic idea

In the sciences, for a thing to be "conserved," means that it never changes, no matter what processes it goes through. We sum up these conserved quantities with "conservation laws," which are often the only foothold we have to solve some complicated problems in chemistry and physics .

In this section we'll briefly go over some of the most important conservation laws, and see how they work and how we can use them to solve problems.

Each is also incorporated into other sections where it's most relevant, but it might be helpful to see them all presented here.

Conservation laws are broadly grouped into two main categories, conservation of mass (mass can neither be created from nothing nor destroyed*), or conservation of energy, into which we can group the momentum conservation laws.

The basic conservation laws are →

1. Conservation of mass – mass cannot be destroyed in any process, with one exception (see #6), and mass cannot be created from nothing.
2. Conservation of charge – In any physical process, like a chemical reaction, the number of positive and negative charges remains the same after the process is complete.
3. Conservation of energy – The total amount of energy of the universe, or any isolated system in it, is always the same.
4. Conservation of momentum – The total amount of momentum present in a system is always the same after some event, like a collision.
5. Conservation of angular momentum – In rotating systems, the total amount of rotational angular momentum is the same before and after any event.
6. Equivalence of mass and energy – In certain reactions (nuclear), mass can be converted directly to energy (E = mc2).

### 1. Conservation of mass

Any chemical reaction is an example of conservation of mass. In a reaction, a number of reactants goes through some process resulting in a rearrangement of atoms, but the same number of each kind of atom present before the reaction must be present afterward.

Consider, for example, the combustion of octane. Octane is one of the liquid components of gasoline. You've probably seen these markings on a gas pump. They give a measure (it's a complicated measurement based upon the combustion properties of certain gas mixtures in a test engine) of the amount of octane present in gas, which is a mixture of various hydrocarbons.

Octane is a chain of eight carbon atoms, with 18 hydrogen atoms bound to them wherever possible, CH3-(CH2)6-CH3, which is usually abbreviated C8H18. To burn it in an engine means to combine it with oxygen to produce carbon dioxide (CO2) and water (H2O) in a combustion reaction. Here is the basic reaction: Now there's a problem: The number of like atoms on each side of the → isn't the same. For example, there are 8 carbons on the left, but only one on the right. That's why chemical reactions always need to be balanced by writing coefficients (red) in front of the reactants and products (a coefficient might be 1, in which case we just wouldn't write it).

The balanced reaction for the combustion of octane would be: Now take a closer look at that reaction below. It satisfies the law of conservation of mass. The same number of C, O and H appear on both sides of the →. Here's another way to think about that reaction and conservation of mass. Say we burn our octane in a sealed flask that contains (because we balanced the reaction) the precise amount of oxygen needed to react with (combust) some amount of octane. If we set up the reaction as shown, so that we can ignite it electrically, it should go to completion.

The result would be no more octane left in the beaker, and the sealed flask would contain only CO2 and H2O in the ratio we expect from the balanced equation – experiment should match theory. Furthermore, the mass would not change at all. The same atoms would still be in the flask, they'd just be arranged differently, into different molecules.

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### Hydrocarbons

Hydrocarbons are molecules composed only of the elements carbon (C) and hydrogen (H). Some examples are

methane     CH4

benzene     C6H6

acetylene     C2H2

undecane     CH3-(CH2)9-CH3

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### Combustion

In a combustion reaction, one or more reactants are combined with oxygen (O2) to create carbon dioxide (CO2) and water (H2O).

If the reactant combusted contains elements other than carbon and hydrogen, other products, such as SOx and NOx (x = 1, 2, 3) can also form.

Burning of wood or paper in the presence of oxygen is an example of combustion.

#### Conservation of mass

The total amount of mass in an isolated system, in which we can account for all mass, is always conserved.

Mass can neither be created from nothing nor destroyed, except when converted directly to energy in very special circumstances.

### 2. Conservation of charge

There's an old chemistry joke that goes like this:

Two hydrogen atoms are walking along the street. One bumps into something and says, "hey, I think I just lost an electron." His friend says, "are you sure?" He replies, "yeah, ... I'm positive."

That's what happens when something acquires a positive charge. It must lose one or more negative charges [almost always in the form of electron(s)].

Matter in the universe can be positively-charged, negatively-charged or, if it contains an equal number of positive and negative charges, neutral.

The law of conservation of charge says that charges can be neither created nor destroyed, or, more precisely, that the sum of all charges in a closed system is constant. That is, if we add up all negative and positive charges in any isolated system (which includes the whole universe), that sum is constant.

As an example, let's take a look at the phenomenon of polarization. Think about a metal sphere: The charges on a neutral sphere are balanced – just as many negatives as positives – and evenly distributed, so that each positive is paired with a nearby negative.

If we place a polarized object – an object with an uneven distribution of charges – near our neutral sphere, the overall sphere remains neutral, but the charges are redistributed so that the sphere has a negative "side" and a positive "side." The positive end of the green rod attracts the negative charges of the sphere and repels the positive charges away from it.

You might ask how the green rod got polarized in the first place. It turns out that we can cause negative charges to jump from one object to another – like getting a shock from a carpet in the dry air of winter. The rubbing of your feet on the carpet causes electrons to come off of the molecules of the carpet fiber. But what's important to remember is that negative charge transferred to one object is now absent from the other, leaving behind a net positive charge.

Charged particles may only be moved around in any process. The sum of the values of all charged particles in any isolated system has to remain the same, no matter what happens to it.

#### Conservation of charge

If an isolated system undergoes any process, the sum of the values of all charges must remain constant.

For example, if the charge of an electron is -1 and the charge of a proton is +1, then the sum of all charges in a neutral system of objects is zero and will always be zero, unless charges are let in or out from outside the system.

### 3. Conservation of energy

The law of conservation of energy, also known as the first law of thermodynamics says that the total energy of an isolated system must remain constant. Energy can't be created from nothing and it can't just disappear (though it has an annoying way of spreading out called entropy). Energy can, however, be transformed from one type to another.

The law of conservation of energy is what allows us to solve many problems in dynamics, and it forms the basis for the whole field of thermodynamics.

A good mechanical example of conservation of energy is a roller coaster – see the diagram below.

It takes work to lift a roller coaster up to the top of the first (highest) hill, where it acquires all of the potential energy it will ever have.

On the way down, the coaster gains kinetic energy –the energy of motion – and loses potential energy – the energy of position. The maximum potential energy that the coaster gained was exactly equal to the work done in getting it to the top of the hill.

In such a system, work, potential energy and kinetic energy are all equivalent and conserved, and they have the same units. #### Conservation of energy

The energy of an isolated system remains constant, but the energy may be redistributed as work, potential energy or kinetic energy. The kinetic energy of atoms and molecules is source of heat.

### 4. Conservation of momentum

Momentum is mass multiplied by velocity:

p = mv

Momentum is always conserved. We usually focus on the momentum in collisions, and you can read more about that in the section about momentum.

The law of conservation of momentum says that the momentum of an isolated system is conserved.

That is, if there are no outside forces at work on the system, the total momentum of all particles or objects in it will always be the same. Remember that a change in the momentum (p = mv) of some object requires a change of velocity, that a change of velocity is an acceleration, and that acceleration requires a force (F = ma).

We often solve collision problems in physics by equating the before and after momentum:

Pbefore = Pafter

### 5. Conservation of angular momentum

Angular momentum is the analog of linear momentum in the rotational world. When things rotate, the world becomes a little more complicated, so we usually separate the rotational and linear worlds.

Angular momentum is also conserved. Perhaps one of the best-known examples is the spinning skater. Run the video below a few times and notice how, as the skater moves her arms close to her center of rotation, she spins faster.

Here's what's going on. The angular momentum, L, is

L = ½Iω2,

where I is the moment of inertia and ω is the angular velocity, which would have units like degrees/s or radians/s.

The moment of inertia is the key to this phenomenon. As the skater moves her arms and legs closer to her center of mass, through which passes her axis of rotation, her moment of inertia (the rotational equivalent of mass) decreases. Therefore, to keep the angular momentum, L, constant, her angular velocity must increase. And you can see that it does.

You can do this experiment yourself. Sit in a swivel chair or on a swivel stool. Hold some weights – maybe big cans of beans – at arms length to the side. Start spinning, lift your feet and move the cans toward your torso. You should speed up. Science!

Just like conservation of linear momentum, conservation of angular momentum is what makes solving many problems even possible.

### 6. Equivalence of mass and energy

We have to go back a bit and re-address the law of conservation of mass, because in certain extreme situations, it has to be modified.

Albert Einstein showed that in certain nuclear reactions, mass can be converted directly into energy. His famous formula

E = mc2

allows us to calculate that energy, and it's immense. Let's calculate the energy in 1 gram of mass (1 g = 0.001 Kg). The energy is

E = (0.001 Kg)(2.99 × 108 m/s)2

E = 9 × 1013 Joules

which is about 21,000 tons of TNT, a tremendous amount of energy.

This energy, which comes from the extremely-strong attractive forces that hold nuclei together, is very difficult to release, however, and we don't ever observe it in everyday physics. So don't worry, you won't just disappear.  xaktly.com by Dr. Jeff Cruzan is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. © 2016-2019, 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.