Overview: In this section we review the key concept of Conservation of Energy. The procedure for determining vector sums is also outlined.
Skills:
- Calculations of potential, kinetic and total energy
- Calculation of momentum
- Adding vectors
- Determining the force on charged particles in an electric field
New terms:
| Energy | Momentum |
| Kinetic Energy | Displacement |
| Velocity | Force |
| Potential Energy | Weight |
| Conservation of Energy | Weight |
| Acceleration | Field |
To understand chemistry, it is very important to understand the energy and forces behind the chemistry. In many reactions, energy is either given off or absorbed. This is the driving force of many reactions. But what is energy? All of us have a general sense of what energy is, but in a scientific sense, energy is defined as the measure of the capacity to do work. Mechanical work is done when a force acts through a distance. Energy is typically measured in Joules (J), and 1 J = 1 kg.m2/s2. When working with chemical reactions involving large amounts of energy, it is often more convenient to work with kilojoules (kJ). We will discuss two types of energy: kinetic and potential.
Kinetic Energy
Kinetic energy is the energy a body possesses due to its motion. Because it is energy, kinetic energy is expressed in units of Joules.
Velocity is very similar to speed. The difference is that velocity is the speed in a given direction. Imagine a car driving down a straight highway at 55 mph due north. The speed is 55 mph and the velocity is also 55 mph. Now the car makes a 90 degree turn and is traveling due west. The speed is unchanged at 55 mph. However, the velocity did change since the direction of travel has changed. The velocity is now 55 mph west.
Example 1.
What is the kinetic energy of a ball with a mass of 0.050 kg that is traveling at a velocity of 25 m/s?
Potential Energy
Potential energy is the energy an object possesses due to its position in a force field. Potential energy, like kinetic energy, is expressed in units of Joules. Potential energy is associated with forces of attraction and repulsion. For instance, a book on a table has potential energy due to its position because of the attractive force of gravity pulling it to the ground. Although a number of different forces can give rise to a potential energy, one of the most important forces is gravity.
The acceleration due to gravity is equal to 9.80665 m/s2 on earth. Remember that other environments, such as the surface of the moon or other planets, have a different acceleration due to gravity.
Example 2.
A person, with a mass of 65 kg, walks up a flight of stairs between two floors of a building separated by 3.0 m. How much more potential energy does this person have by walking up the flight of stairs? In other words, what is the potential energy of the person on the upper floor relative to that they had while on the lower floor?
Where does this person get the energy to climb the stairs? The person needed kinetic energy to move up against the force of gravity and this energy is provided by the biochemical processes involved in extracting the energy stored in chemical bonds, most likely by the metabolism of glucose (a common biological energy source).
Law of Conservation of Energy
The total energy of the universe is constant due to the Law of Conservation of Energy. This means that the energy of the universe remains the same before an action, during the action and after the action. In an equation:
Take, for example, a bouncing ball. If you have a ball that makes a perfectly elastic collision with the ground, what would happen when you drop the ball? First the ball is in your hand and at rest. It has zero kinetic energy and an initial non-zero potential energy. Now think about what happens when you release the ball. When the ball is halfway to the ground it is moving, which means that it has some kinetic energy. Gravity continues to act on the ball because it is still above the ground. However, the ball does not have all of the potential energy that it had when it was in your hand. Since the total energy of the ball and the other forms of energy must remain constant, some of the initial potential energy has been converted into kinetic energy. What about at the instant before the ball hits the ground? At that point it has no potential energy left. The ball is in motion so it has a non-zero kinetic energy. The initial non-zero potential energy has been converted into kinetic energy. After colliding with the ground, the ball begins to bounce back up and the kinetic energy is converted into potential energy. Since the total energy of the ball is conserved it will continue upward until it has reached its initial height (achieving the same potential energy that it had initially). This, according to what we said, should continue forever.
However, if you actually drop a real ball this is not what will happen. The ball never quite makes it back up to its original height and eventually the ball even stops bouncing. What happened to this stored energy. Is this a violation of the Law of Conservation of Energy? Answer
Example 3.
A ball with a mass of 0.95 kg is dropped from a height of 13.9 m.
What is the kinetic energy when it hits the ground?
The total energy must stay the same throughout the process. So P.E.initial + K.E.initial = K.E.final + P.E.final. At the beginning the ball has only potential energy and no kinetic energy. At the end, when it hits the ground, the ball has kinetic energy and no potential energy. So this means that P.E.initial = K.E.final. So we will calculate the potential energy before the ball begins to fall.
All of the potential energy is converted into kinetic energy when the ball hits the ground so the kinetic energy then is 1.3 x 102 J.
What is the kinetic energy of the ball when it has fallen halfway to the ground?
Again the total energy must remain constant. So P.E.initial + K.E.initial = K.E.halfway + P.E.halfway = K.E.final + P.E.final.
Acceleration
We have talked about the acceleration due to gravity, but have not said what acceleration is. Acceleration occurs when an object experiences a change in velocity. When a car changes from 55 mph to 65 mph, the car is obviously accelerating. A not so obvious cause of acceleration is when a car traveling 55 mph changes direction while maintaining the same speed. The car changes velocity by changing direction and therefore accelerates.
Momentum
Momentum (p) is also related to velocity. Usually we think about momentum when dealing with collisions. Momentum is always conserved just as energy is. Think about two billiard balls. You hit one ball into the other. The ball that was originally stationary now moves while the ball that was originally in motion either stops or slows down. Why? The momentum from the first ball was transferred into the second ball putting it into motion. However keep in mind that the total momentum of the two balls after the collision is the same as the momentum of the ball in motion before the collision. Momentum is calculated by multiplying the mass times the velocity.
Vectors
Velocity is a vector quantity as it indicates both speed as well as the direction of travel. On the other hand, speed is a scalar quantity because it only shows the magnitude, not the direction. A vector shows both magnitude and direction and is represented by an arrow pointed in the appropriate direction of that vector. The length is proportional to the magnitude.
We can add vector quantities either graphically or by components.
For the graphical method, you will take the first vector (A) and leave it in place. Then take the second vector (B) and place the tail of that vector at the tip of vector A. The sum of the vectors will go from the tail of vector A to the head of vector B. You can add as many vectors as you would like by this head-to-tail method.
For the components:
Example 4.
A car goes west at 10 m/s for 6 s, then it goes north at a rate of 10 m/s for 5 s, and then it goes west again at 4 m/s for 15s. What distance does the car travel? What is the displacement?
First let’s start by drawing a diagram so we can see what is happening.
We know that the car goes 60 m to the west to start since it travels 10 m/s to the west for 6 s (10 m/s x 6 x = 60 m), then north for 50 m and west once again for 60 m. So the distance that the car travels is just the sum of the magnitudes of vectors (distance is a scalar quantity). So that is 60 m + 60 m + 50 m = 170 m. The second question asks about displacement. What is displacement though? How is it different from distance? Displacement is the shortest straight line between the starting and ending points of the motion of the object regardless of the path taken. In the drawing above, the displacement is represented by the dashed line. How do we find this vector? We add the other vectors together. So the car traveled a total of 120 m to the west and 50 m north.
Now we can use the components idea to find both the magnitude and direction of the displacement.
So the displacement is 130 m at a 23o angle from the horizontal.
Forces
Basically a force is a push or a pull on an object. If we want to move something, we exert a force on it. When an object falls, the force of gravity is the cause. If we want to change the direction an object is moving, we need a force to affect that change in direction. So we can think of force as an action that is able to accelerate an object. Does force always move an object? Answer
Force is measured in Newtons (N) and 1 N = 1 kg.m/s2. Force is a vector quantity. You have probably heard for every action there is an equal but opposite reaction. This has to do with forces. When you push on a wall, you exert a force on it. However, while you are pushing, the wall exerts a force equal in magnitude, but in the exact opposite direction on you.
Weight
One important force to know is weight. Weight is the gravitational force acting upon a mass. Because weight is a force it is measured in Newtons.
Recall that mass is the quantity of matter an object contains. Weight and mass cannot be used interchangeably. They are very different. If the gravitational force changes, weight changes, but the mass remains the same. Let’s take an object with a mass of 10.0 kg. Its weight on earth is 98.1 N (wt = 10.0 kg x 9.81 m/s2). If this object was taken to the moon, its mass is still 10.0 kg, but its weight is about 16 N because the acceleration due to gravity on the moon is approximately 1/6 that of the earth. It is very important to understand the difference between weight and mass.
Okay, okay, I know this is a lot of physics and this is supposed to be chemistry. So what do velocity, acceleration, and force have to do with chemical bonding and chemical reactions? Well, actually it has quite a bit to do with chemistry. Energy drives all chemical reactions; you either release energy or absorb energy. Kinetic energy is involved in where electrons are in the atom. In fact the kinetic energy of particles allows us to produce x-rays. Potential energy allows chemical bonding to occur. Potential energy has to do with the distance (or in the case of gravity, height) and distance has a lot to do with chemical bonding. Velocity and acceleration relate to force. Force is very important in chemistry. It drive electrical fields and these fields play an important role in ionic bonding. Momentum has to do with collisions and collisions are responsible for chemical reactions. So even though there is a great amount of physics, it is still very important to know in order to understand chemistry. You will learn more about the importance of physics in chemistry in Chemistry 111 and 112.
Electric Fields
Think about what happens when two like charges are next to each other. You have probably heard that “opposites attract”. Well this is true. Opposites attract one another and charges of the same sign repel. So if two like charges are next to each other, they move apart. If two opposite charges are next to each other, they move towards one another. But why does this happen? These particles are not touching each other, so how do they move? Forces. Forces can be exerted through space, there does not need to be contact between the two objects. Charged particles create an electric field, E, around themselves. This electric field exerts forces on other charged particles that just enter the field, not only those that touch the charged particle. E is a vector measured in Newtons/Coulomb (N/C).
When an electric field is drawn, by convention, the vector arrows point towards the negative charge.
Coulomb’s Law
So each charged particle has its own field and each field exerts its own force on other charged particles. Coulomb described the relationship between the charges, distance, and force in Coulomb’s Law that says:
When the force is attractive the charges are opposite and therefore the force is negative. When the Coulombic force is positive, the charges are the same and repel each other.
So what does this mean physically? Let’s look at two particles with opposite charges. We will begin with them at a very large distance apart. What is the Coulombic force? Answer
Now the two charges move closer together until they are very close to each other. What happens to the force? Answer
Let’s say we replace one of the particles with an oppositely charged particle in order to make both particles have the same charge. What happens to the force? Answer
Many times chemists apply electric fields to separate charged particles. Chemists typically place two charged plates, one negative and one positive, across from one another to create an electric field. A charged particle travels between the two plates. Once the particle enters the electric field, a force is exerted on it. It has now shifted from its original path. Scientists are able to predict which way the deflection will be as well as how large it should be.
We have illustrate such an apparatus above with an electron shot between two charged plates. How will the electron’s trajectory be affected by the charged plates? In other words is it deflected and if so which direction will it be in? Answer
Summary
Now you should have a good understanding of what potential and kinetic energy are and how the two are related. Also you should know about velocity, acceleration, momentum and force. You should be able to add vectors and know what a vector quantity is. You should also appreciate what an electic field is and how a charged particle will react in that field. You will learn more about the behavior of particles in electric and magnetic fields in Chemistry 111.
Practice Problems
Deep Impact
Background: On July 3 at 10:52:24 p.m. (PDT), the impactor from NASA’s Deep Impact space probe was positioned in the path of the comet Tempel 1. The comet, which is essentially made of dirty ice was moving with a speed of 23,000 mph (36,000 km/hr). Astronomers estimate the comet’s approximate dimensions as 14.0 km x 4.0 km x 4.4 km. The space probe’s impactor consisted of a 373 kg mass of copper.
- 1. Assuming that the comet’s density is approximately equal to that of ice at temperatures approaching absolute zero (~ 0.94 g/cm3), and using the dimensions of the comet quoted above, use the formula for the volume of an ellipsoid [V = (p/6)a x b x c, where a,b and c are the ellipsoid dimensions] to obtain an estimate of the comet’s kinetic energy.
- 2. The Deep Impactor space probe traveled a distance of 431 x 106 km in approximately 173 days. Assuming these figures can be used to estimate the impactor’s velocity, what was the impactor’s momentum at the time of impact?
- 3. The gravitational acceleration constant at the surface of the Tempel 1 comet is estimated to be between 0.027 and 0.04 cm/s2. What was the change in the impactor’s gravitational potential energy in the last 5 m of its trajectory towards impact with the comet surface?