The Energy Levels of Molecules
The Energy Levels of Molecules
The phenomena of chemistry cannot be understood thoroughly without a firm grasp of the principle concept of quantum mechanics. When talking about energy levels, quantisation has to be considered, and this quantisation is the basic assumption of Quantum theory. This is when all the energy levels are quantised meaning that they can have only discrete values. This quantisation is defined through a quantum number, which usually have integers or half integers as values. These quantum numbers can only be fixed values whereas a classical particle can have any value when changing from one energy level to another. The molecule normally accepts one quantum of energy and the size of this quantum must be exactly the same as the difference between these two levels. Therefore this statement describes us the quantisation of energy. An element can only be present in certain energy levels corresponding to various quantum states (as shown below), but the energy of an electron cannot be found between any shown levels. Energy quantisation is universal because it holds for all kinds of systems like atoms, nuclei, molecules or electrons in solids.
The principle energy levels of electrons
Shell
Principle Quantum Number
Period (horizontal row)
Maximum Number of Electrons = 2n2
Energy sublevels
K
2
S
L
2
2
8
S p
M
3
3 - 4
8
S p d
N
4
4 - 6
32
S p d f
O
5
5 - 7
32 (2n2 = 50)
S p d f
P
6
6 - 7
8 (2n2 = 72)
S p d *
Q
7
7
* *Man-Made
* Elements
S *
Kinetic Energy
There are three different forms of Kinetic Energy in molecules, as follows:
> Translational Energy - This is when single diatomic molecules move at a certain velocity in a straight line.
> Rotational Energy - This is when single diatomic molecules can rotate around their centre of gravity.
> Vibrational Energy - This is when single diatomic molecules at the end of the bonds can move relative to each other. There are approx. 1012 oscillations per second i.e. the frequency of vibration is 1012 s-1.
The above are classified as forms of Kinetic Energy because they are forms of energy
due to their movement. Each of the above energy levels are quantised, but the molecule can change its speed, it's rotational frequency or vibrational frequency to other allowed values and therefore the change in the energy stored in this mode. The molecule also has the equivalent Potential energy i.e. the energy due to the arrangement of the electrons in the molecule, therefore the following energy must be taken into account, Electronic Energy.
The sum of these four types of energies must be equivalent to the total energy in the molecule, as follows:
ETotal = ETranslational + ...
This is a preview of the whole essay
due to their movement. Each of the above energy levels are quantised, but the molecule can change its speed, it's rotational frequency or vibrational frequency to other allowed values and therefore the change in the energy stored in this mode. The molecule also has the equivalent Potential energy i.e. the energy due to the arrangement of the electrons in the molecule, therefore the following energy must be taken into account, Electronic Energy.
The sum of these four types of energies must be equivalent to the total energy in the molecule, as follows:
ETotal = ETranslational + ERotational + E Vibrational + EElectronic
The value for each of these energies can change as time increases, but for a moment in time they can have single fixed values. For a mole of gas the total energy is called the Internal Energy of the System (U / KJ mol-1). This is the sum of each of the individual values of ETotal for each of the 6.0223 ? 1023 molecules in the system..
Translational Energy
According to Quantum Theory the nth translational energy could be calculated by the following:
En = n2 h 2 Where n = Quantum number ( 1, 2, 3, 4 etc. and not in
8 m l2 between values and no values less than 1)
h = Planck's constant (6.6256 ? 10-34 J s)
m = mass of particle (kg)
l = size of the container (cm)
From the above equation the quantum number specifies certain physical properties of the system, in the above equation n is only specifying the energy of the particle through that equation. From the equation you can define that the lowest energy of a particle is when n = 1. A molecule can never have a zero energy value, therefore shows that molecules never stop.
Rotational Energy
As the above we can represent how to calculate the Rotational energy by an equation, as follows:
EJ = BJ (J + 1) Where B = Roatational Constant (KJ mol-1)
J = Rotational Quantum number (0, 1, 2, 3, 4 etc. and
not in between values, an no values less than 0)
B is the only unknown in this equation, but can be calculated by the following equation:
B = h2 / (2?)2 Where h = Planck's constant (6.6256 ? 10-34 J s)
2 I I = Moment of inertia (Kg m2)
From this equation the only unknown is I (moment of inertia / kg m2), but this can also be calculated by the following:
I = m1m2 ? (r)2 Where m = mass (kg)
m1 +m2 r = bond length (m)
Now the Rotational energy can be calculated. Note that in this case the lowest energy state corresponds to the lowest rotational quantum number, which in this case is zero so therefore the molecule can have zero rotational energy.
Vibrational Energy
The Vibrational energy can also be calculated from the following equation:
EV = (V + 1/2)h wosc Where V = Vibrational quantum number (0, 1, 2, 3, 4 etc. and not
In between values, and no values less than 0)
wosc = Oscillation frequency of molecule
Where wosc = 2 ?? Where ? = frequency (Hz)
? = 1 k Where k = constant
2? µ µ = reduced mass = m1m2
m1 + m2
wosc could also be calculated by the following providing that the values are known:
wosc = k Where k = constant
µ µ = reduced mass (as shown previously)
Therefore when a molecule moves from Level 0 to Level 1:
For level 0 : EV=0 = 1/2 h wosc
This value is commonly known as the vibrational zero point. But when the vibrational quantum number equals 1, the corresponding energy is:
For Level 1 : EV=1 = 3/2 h wosc
The change in energy would therefore be as follows:
?E = 3/2 h wosc - 1/2 h wosc
= h wosc
The energy therefore required from the photon absorbed is as follows , which is the frequency of radiation:
E = h? Where h = Planck's constant (6.6256 ? 10-34 J s)
? = frequency (Hz)
Therefore from the above (change in the energy from level 0 to 1) and the frequency of radiation the following can be deduced:
h? = 3/2 h wosc - 1/2 h wosc
h? = h wosc divide both sides by h (Planck's constant)
Therefore wosc (oscillating frequency of a molecule) can be calculated as being equivalent to ?, (frequency / Hz) as shown from above which can be calculated:
? = wosc
Electronic Energy
For electronic energy levels there is no simple equations to give its value. But we can calculate it by using Quantum Theory, but even this requires separate calculations for different molecules.
Electronic energy levels are limited to two electrons with paired spins per level, whereas molecular electronic states can have different numbers of molecules and can have different configurations.
If we put in numbers for each of the energies i.e. for translational, rotational, vibrational and electronic we would be able to determine the following:
Eelectronic ? E vibrational ?? Erotational ?? Etranslational ?? = more greater than
This shows that the electronic energy has the greatest value and the translational energy has the lowest energy value.
Other modes in which molecules can store energy are, for example spin in molecules can store energy. Also nuclei of some elements can store energy, but the value of these energies is so small in the absence of magnetic fields; as a result we can ignore them.
Now we can deduce that Quantum Theory can tell us what the different contributions to the total energy of a single molecule would be i.e. in terms of translational, rotational, vibrational and also electronic. But when molecules are in a fixed volume they can collide with each other and also collide with the sides of the container, and during these collisions a quantum of energy can be transferred from one molecule to another. When this happens the two molecules that collide will move from one energy state to another i.e. excitation occurs, but they eventually return to their normal ground state. This will cause an increase in energy by one molecule, which will be the same as the energy lost by the second molecule. Therefore the energy of the two molecules is conserved. From this we can deduce that at a fixed temperature in an isolated system, the total internal energy (U) of the system is constant.
To calculate the internal energy of a system all we have to do is find out how many molecules are in a certain energy level, because at a fixed temperature the number of molecules in any one energy state will be constant. Therefore to calculate the internal energy (U) we have to multiply the number of molecules by the energy value of that energy level.
U = n1?E1 + n2?E2 + n3?E3 etc. (J) Where n = number of molecules in different
Energy levels.
E (J) = the corresponding energy levels.
The only unknown from the above will be the number of molecules, but this can be calculated by the Boltzmanns equation. If we consider energy states 1 and 2 (i.e. E1 & E2)/ J molecule-1, the Boltzmanns equation shows that at a temperature the number of molecules in the two states, n1 and n2 would be given by the following:
n2 = e - (E2 - E1) / kT Boltzmanns Equation
n1
Where n = number of molecules
E = the energy value of the energy level (J)
K = Boltzmanns constant (1.3805 ? 10-23JK-1)
T = Temperature (K)
This equation shows the exponential dependency explicitly. It represents that the spread of the number of molecules to high energies, increasing with temperature. The energy kT is the dividing line between states that have significant number of molecules and those that have not.
The Boltzmanns constant could be calculated as follows:
K = R Where R = Gas constant (8.3145 JK-1mol-1)
NA NA = Avagadros constant (6.0223 ? 1023)
Just to add to this as we are talking about the Boltzmanns constant, the internal energy for one atom or a molecule where the rotational levels are widely spaced at a particular temperature (or a low temperature) can be calculated as follows:
U = 3/2 KT Where U = Internal energy of the molecule (J)
K = Boltzmanns constant (1.3805 ?10-23 JK-1)
T = Temperature (k)
We can say that the Bolzmanns constant can be defined as the amount of energy required to raise the temperature of an atom by one kelvin.
Mudassar Azam Dr. H. Morris Rm 207
07/12/2003
