Law of Conservation of Energy
The law of conservation of energy is one of the basic laws of physics, along with the conservation of mass and the conservation of momentum. The law of conservation of energy states that energy can change from one form into another, but it cannot be created or destroyed. Or the general definition is:
The total energy of an isolated system remains constant over time.
Energy can be defined as the capacity for doing work. It may exist in various forms and may be transformed from one type of energy to another in hundreds of ways.
For example, burning gasoline to power cars is an energy conversion process we rely on. The chemical energy in gasoline is converted to thermal energy, then converted to mechanical energy that makes the car move. The mechanical energy has been converted to kinetic energy. When we use the brakes to stop a car, that kinetic energy is converted by friction back to heat or thermal energy.
A consequence of the law of conservation of energy is that a perpetual motion machine of the first kind, which produces work without the input of energy, cannot exist.
Example: Conservation of Mechanical Energy
Assume a pendulum (ball of mass m suspended on a string of length L that we have pulled up so that the ball is a height H < L above its lowest point on the arc of its stretched string motion. The pendulum is subjected to the conservative gravitational force where frictional forces like air drag and friction at the pivot are negligible.
We release it from rest. How fast is it going at the bottom?
The pendulum reaches the greatest kinetic energy and least potential energy when in the vertical position because it will have the greatest speed and be nearest the Earth at this point. On the other hand, it will have its least kinetic energy and greatest potential energy at the extreme positions of its swing because it has zero speed and is farthest from Earth at these points.
If the amplitude is limited to small swings, the period T of a simple pendulum, the time is taken for a complete cycle, is:
where L is the length of the pendulum and g is the local acceleration of gravity. For small swings the period of swing is approximately the same for different size swings. That is, the period is independent of amplitude.
Example: Elastic Collisions
What are the velocities of the neutron and carbon nucleus after the collision?
Solution:
This is an elastic head-on collision of two objects with unequal masses. We have to use the conservation laws of momentum and kinetic energy and apply them to our system of two particles.
We can solve this system of equations or use the equation derived in the previous section. This equation stated that the relative speed of the two objects after the collision has the same magnitude (but opposite direction) as before the collision, no matter what the masses are.
The minus sign for v’ tells us that the neutron scatters back of the carbon nucleus, because the carbon nucleus is significantly heavier. On the other hand its speed is less than its initial speed. This process is known as the neutron moderation and it significantly depends on the mass of moderator nuclei.
Example: Inelastic Collisions
A ballistic pendulum is a device for measuring the velocity of a projectile, such as a bullet. The ballistic pendulum is a kind of “transformer,” exchanging the high speed of a light object (the bullet) for the low speed of a massive object (the block). When a bullet is fired into the block, its momentum is transferred to the block. The bullet’s momentum can be determined from the amplitude of the pendulum swing.
When the bullet is embedding itself in the block, it occurs so quickly that the block does not move appreciably. The supporting strings remain nearly vertical, so negligible external horizontal force acts on the bullet–block system, and the horizontal component of momentum is conserved. However, mechanical energy is not conserved during this stage because a nonconservative force does work (the force of friction between bullet and block).
In the second stage, the bullet and block move together. The only forces acting on this system are gravity (a conservative force) and string tensions (which do not work). Thus, as the block swings, mechanical energy is conserved. Momentum is not conserved during this stage because there is a net external force (the forces of gravity and string tension don’t cancel when the strings are inclined).
Equations governing the ballistic pendulum
In the first stage momentum is conserved and therefore:
where v is the initial velocity of the projectile of mass mP. v’ is the velocity of the block and embedded projectile (both of mass mP + mB) just after the collision, before they have moved significantly.
In the second stage mechanical energy is conserved. We choose y = 0 when the pendulum hangs vertically and then y = h when the block and embedded projectile system reach maximum height. The system swings up and comes to rest for an instant at a height y, where its kinetic energy is zero, and the potential energy is (mP + mB)gh. Thus we write the law of conservation of energy:
which is the initial velocity of the projectile and our final result.
When we use some realistic numbers:
- mP = 5 g
- mB = 2 kg
- h = 3 cm
- v = ?
then we have:
Conservation of Energy in Chemical Reactions
Example: Combustion of Hydrogen
Consider the combustion of hydrogen in air. In a flame of pure hydrogen gas burning in the air, the hydrogen (H2) reacts with oxygen (O2) to form water (H2O) and releases energy.
Energetically, the process can be considered to require the energy to dissociate the H2 and O2, but then the bonding of the H2O returns the system to a bound state with negative potential. It is more negative than the bound states of the reactants, and the formation of the two water molecules is, therefore, an exothermic reaction, which releases 5.7 eV of energy.
2H2(g) + O2(g) → 2H2O(g)
The balance of energy before and after the reaction can be illustrated schematically with the state in which all atoms are free taken as the reference for energy.
Example: Law of Conservation of Mass-Energy – Mass-Energy Equivalence
Example: Conservation of Energy in Nuclear Reactions
The DT fusion reaction of deuterium and tritium is particularly interesting because of its potential to provide future energy. Calculate the reaction Q-value.
3T (d, n) 4He
The atom masses of the reactants and products are:
m(3T) = 3.0160 amu
m(2D) = 2.0141 amu
m(1n) = 1.0087 amu
m(4He) = 4.0026 amu
Using the mass-energy equivalence, we get the Q-value of this reaction as:
Q = {(3.0160+2.0141) [amu] – (1.0087+4.0026) [amu]} x 931.481 [MeV/amu]
= 0.0188 x 931.481 = 17.5 MeV
A high-energy photon (gamma ray) can, under certain conditions, eject a neutron from a nucleus. It occurs when its energy exceeds the binding energy of the neutron in the nucleus. Most nuclei have binding energies above 6 MeV, above the energy of most gamma rays from fission. On the other hand, few nuclei with sufficiently low binding energy are of practical interest. These are 2D, 9Be, 6Li, 7Li, and 13C. As can be seen from the table, the lowest threshold have 9Be with 1.666 MeV and 2D with 2.226 MeV.
threshold energies.
In case of deuterium, neutrons can be produced by the interaction of gamma rays (with a minimum energy of 2.22 MeV) with deuterium:
The reaction Q-value is calculated below:
The atom masses of the reactant and products are:
m(2D) = 2.01363 amu
m(1n) = 1.00866 amu
m(1H) = 1.00728 amu
Using the mass-energy equivalence, we get the Q-value of this reaction as:
Q = {2.01363 [amu] – (1.00866+1.00728) [amu]} x 931.481 [MeV/amu]
= -0.00231 x 931.481 = -2.15 MeV
