The objectives of this investigation were to study how molecular theory and the various calculations associated with it could assist with predicting certain properties in basic molecular structures, such as carbon monoxide (CO) and carbon dioxide (CO2). These predictions were then compared with literature values in order to assess their reliability. With the use of a rotational-vibrational infrared spectrum for CO2 and CO, an in-depth analysis of their respective molecular structures was produced.
1. Molecular Shapes:
In order to determine the molecular shape of a compound, the distribution of electron density within each atom which makes up the compound has to be analysed. Although CO2 and CO are two very distinct molecules, they are both made up of carbon and oxygen atoms. Carbon is an element found in group 4, hence it contains 4 valence electrons, which means that at ground state its electronic configuration is the following: 1s2 2s2 2p2. Oxygen differs since it is part of group 6, which means that with two additional electrons in its 2p atomic orbitals its electronic configuration is defined as 1s2 2s2 2p4.
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By knowing the electronic configurations of the constituent atoms, reasonable schematics of both compounds could be drawn as Lewis structures as seen in Figures 1 and 2 below.
Figure 1: Lewis structure of CO Figure 2: Lewis structure of CO2
Using VSEPR theory, the occupancy (i.e. regions of electron density) around the carbon atom in CO2, which is the centre atom, is two, based on the fact that each double bond accounts for one region of electron density. Each double bond in CO2 prevents any rotation around the bond between the carbon and oxygen atoms, therefore causing the C-O-C angle to be 180°, hence why CO2 is a linear molecule. If a triple bond is also considered as being one region of electron density, then the molecular shape of CO is also linear, again because of the restricted rotation about the bond.
When bonding occurs in any compound, there has to be an interaction between symmetry matching and energy matching atomic orbitals of the bonding atoms. The wave-particle duality property of electrons allows them to behave as both waves and particles. This theory, introduced by Louis de Broglie in 1924, suggested that all matter has wave-like properties. This was such that the wavelength of the wave, λ (m), associated with a particle was linked to the momentum of this particle, p (kg ms-1) by the following de Broglie relationship :
Where h (6.62608 - 10-34 Js) is defined as Planck's constant.
The three dimensional motion of an electron orbiting around an atom is described by a wavefunction, ψ, which gives the probability amplitude of finding the electron at a location in space and time . These wavefunctions are known as atomic orbitals, and they are the waves associated to the electrons of an atom. If the motion of the electron is restricted to one dimension, the behaviour of the wavefunction is determined by the following time-independent Schrödinger equation:
Where Ä§ = h/2π, m is the mass of the electron (kg), x is the spatial x-coordinate of the electron, V is the potential energy (J), and E is the total energy of the particle.
When atoms interact, this is due to their atomic orbitals interfering in two possible ways; constructively or destructively. In constructive interference, there is an increase in electron density because the two wave functions of like sign interact, leading to a bonding action. In destructive interference, there is a decrease in electron density in this region because two wave functions of opposite signs interact, leading to an anti-bonding action. When the orbitals do not interact (i.e. no bonding interaction), no bond is formed because the overall net effect in electron density is zero, such as the electron density from the lone pairs of electrons represented as red lines on both Fig. 1 and Fig. 2.
Another approach to finding out the shape of CO2 and CO would be the Molecular Orbital theory. This theory involves the interactions of atomic orbitals to form molecular orbitals, whereby following Aufbau principle (i.e. electrons are filled from the lowest energy orbital to the highest energy orbital) , the valence electrons from each atom are used to fill these molecular orbitals.
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In CO2 and CO, the electron density within the 2s atomic orbital in each oxygen atom is found to experience greater attraction from the nucleus in comparison to the electron density in the 2s atomic orbital of the carbon atom. This is the case as a result of a greater effective nuclear charge in the oxygen atom than in the carbon atom, meaning that the electrons in oxygen are found closer to the nucleus. For that reason, the 2s orbital in oxygen is lower in energy than the 2s orbital for carbon for both compounds; therefore the extent to which these two orbitals overlap is considered to be negligible, hence leading to a non bonding interaction.
The hybridisation concept, i.e. the mixing of the atomic orbitals, has to be used for both CO2 and CO in order to explain the bonding in these compounds. The 2s and one of the 2p atomics orbitals of carbon (in CO2 and CO) have to mix to form a sp hybrid, which would explain the linear shape of these two molecules. Constructive interference between two s atomic orbitals causes a σ bonding molecular orbital to form, whereas destructive interference leads to the formation of a σ anti-bonding molecular orbital. The interactions of two p orbitals create π bonding and π anti-bonding molecular orbitals for constructive and destructive interference respectively. All this information allows molecular orbital diagrams (MOs) for both CO2 and CO to be drawn below.
The bond order of any diatomic molecule can be found using a molecular orbital diagram, since it is defined as the difference between the number of electrons in the bonding levels (in Fig. 3: σ2 and π1) and the number of electrons in the anti-bonding levels (in Fig. 3: none) divided by two. This gives a bond order of 3 for carbon monoxide, agreeing with the Lewis model which predicted a triple bond between the carbon and oxygen atoms. In Figure 3 below, σ1 and σ3 represent non-bonding molecular orbitals that correspond to the lone pairs of electrons on the oxygen and carbon atoms respectively.
Figure 3: MO diagram for CO
In the case of carbon dioxide, the molecular orbital diagram is slightly more complex because CO2 is a triatomic molecule. Without taking into account the use of Ligand Group Orbitals (LGOs), a simplified MO diagram, which consider a localised cloud of electron density around each double bond , can be drawn, such as in Figure 4 below.
Figure 4: MO diagram for CO2
In the molecular diagram of Figure 4 above, the bond order around each C-O bond is found to be 2, confirming the existence of double bonds as suggested in Fig. 2. In this diagram, σ1 and σ3 also represent non-bonding molecular orbitals, which correspond to the two lone pairs of electrons on each of the two oxygen atoms.
2. Rotational-vibrational properties by Infrared spectra analysis:
In infrared (IR) spectroscopy, photons of wavelength between 2μ - 20 μ from the infrared part of the electromagnetic spectrum (EM spectrum) interact with matter. These interactions, involving the transfer of energy from the infrared photons to the matter being studied, are only possible because the radiation emitted by the photons is matching in frequency with the transition of the compound (i.e. principle of frequency matching resonance) . The photons in this region of the EM spectrum have low energies, and therefore they can only bring about vibrational and rotational excitations of covalently bonded atoms. For analysis of a compound molecular structure, an infrared spectrophotometer can then be used to produce an absorption spectrum unique to this particular compound.
A net change in the dipole moment of the compound (i.e. , where μ is the dipole moment and r is the coordinate with respect to position) caused by both vibrations and rotations within this compound, allows the latter to absorb infrared photons of electromagnetic radiation. This radiation consists of an oscillating magnetic field perpendicular to an alternating electrical field which interacts with the fluctuations in the dipole moment of the molecule. If the frequency of the compound matches perfectly the frequency of the infrared photons, then electromagnetic radiation will be absorbed by the compound .
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Vibrational and rotational energies of the molecule both contribute to the high resolution gas phase IR spectrum produced for that particular molecule. By solving Schrodinger's equation for a diatomic compound experiencing vibrations, the following solution is found using the harmonic oscillator approximation:
Where Ev is the energy associated with molecular vibrations (J), v = 0, 1, 2, ... is the vibrational quantum number, and ν is the fundamental vibration frequency (s-1). This frequency allows the vibrational energy of the molecule to be related to its molecular structure through the following relationship:
Where k is the force constant (N m-1), and is the reduced/effective mass of the diatomic molecule (kg).
In a vibrational IR spectrum, the wavenumber is often given instead of the vibrational frequency. The conversion between the two values can easily be made because:
Where is the vibration wavenumber in cm-1.
In terms of rotations, Equation (2) can also be solved to find the rotational energy of a diatomic molecule (using the rigid linear rotor approximation), as shown by the following solution:
Here EJ is the energy associated with molecular rotations (J), c is the speed of light in a vacuum (2.99792558 - 108 m s-1), B is the rotational constant (cm-1), J = 0, 1, 2, ... is the quantum number associated with rotations, and I is the moment of inertia for the molecule (kg m2) given by:
Where m is the mass of one of the atoms in the molecule (kg), and r is the bond length (m).
The rotational constant, B, is related to the bond length of a diatomic molecule, which again allows the molecular structure of the compound to be connected to its energy. This is observed by the relationship below:
The presence of both quantum numbers v and J, in Equation (3) and Equation (6) respectively, implies that during rotations and vibrations, the energy of the system is restricted to a discrete number of values, i.e. it is quantised. The difference in energy for a transition between two consecutive energy levels, EJ+1 and EJ is:
This energy difference is related to the spacing between two adjacent peaks on a rotation-vibration spectrum. On this spectrum, the frequency of the absorbed radiation increases from the left hand side region to the right hand side region. The heights of the peaks represent the intensity of these particular absorbed radiations, which in turn reflects the population of the initial rotational energy levels . The rotational-vibrational spectrum of CO below shows all these features.
Figure 5: Vibrational-rotational infrared spectrum for carbon monoxide 
The peaks shown in Fig. 5 above also reflect the fact that energy is quantised because there are a discrete number of peaks corresponding to a discrete number of absorbed electromagnetic radiations.
If one takes for example a homonuclear diatomic molecule, as the molecule spins/rotates this causes the bond between the two nuclei to stretch (i.e. similar to the stretching of a spring), therefore implying that the bond length increases with increasing frequency of rotations. This is reflected on the spectrum since it can be seen that the spacing between two successive peaks increases from the left hand side of the spectrum to the right hand side.
Transitions from one energy state to another also depend on quantum mechanical selection rules, as not all transitions are permitted. The absorption of a photon of IR radiation causes a simultaneous change in both the vibration and the rotational quantum numbers of a diatomic molecule as shown in Figure 6 below. Rotational transitions are limited to ΔJ = ±1 and vibrational transitions are restricted to Δv = ±1. The P branch, the Q branch and the R branch corresponds to rotational transitions of type ΔJ = +1, ΔJ = 0 and ΔJ = -1 respectively.
J = 3
J = 2
J = 1
J = 0
v = 1
J = 3
J = 2
J = 1
J = 0
v = 0
Figure 6: Schematic of the changes in vibrational and rotational quantum numbers due to electronic transitions.
On Fig. 6 below, the blue and the red arrows represent the P-branch and R-branch transitions respectively. The green arrow represents the forbidden Q-branch transition, which appears as an absent peak on the spectrum in Fig. 5.
In the spectrum provided for carbon monoxide, a large number of peaks were measured using a 30 cm ruler from both the P and R branches to find the mean spacing, allowing the rotational constant for this compound to be estimated accurately. The accuracy of the ruler used was ± 0.5 mm; therefore this had to be taken into account for the final value for B. Moreover, the scale of the spectrum also had to be considered, since 4.3 cm represented 50 cm-1, so 1 m on the graph corresponded to. From this, the following calculations were performed for both branches.
For 20 peaks, the length measured was 68.5 ± 0.5 mm. 20 peaks where chosen because the spacing between each of these was relatively similar enough for an average to be taken. Beyond 20 peaks, the spacing started to become wider, so no more than 20 peaks were selected.
Therefore the average spacing was:
For 21 peaks, the length measured was 56.5 ± 0.5 mm. 21 peaks were taken for the same reasons as for the P branch.
So the mean spacing was:
However to find the average rotational constant, the propagation formula has to be used to consider the uncertainties introduced in taking measurements, which means that:
From this average rotational constant, the bond length of the triple bond in CO can be determined as these two parameters are related, since the moment of inertia in Equation (8) can be substituted in Equation (7) to find the bond length, r.
In carbon monoxide, the reduced mass is used as a simplification of the problem so that only a single-body problem has to be considered.
The reduced mass of CO was found to be: ,
where u = 1.66054 - 10-27 is the atomic mass unit in kg.
Since the moment of inertia had to be found before finding the bond length, Equation (8) had to be rearranged, so that the value for I could be substituted into Equation (7) to give:
The literature value found for the bond length in carbon monoxide was found to be equal to 113 pm, therefore there is a percentage error of about 0.7 % between the experimental value of 114 pm and the theoretical value of 113 pm. This percentage error is a relatively small (i.e. less than 10%), therefore the estimated value of 114 pm can be considered to be reliable.
The vibrations of the carbon monoxide molecule can be modelled as two atoms experiencing vibrations relative to each other with the bond between them acting like a spring. Since the oxygen atom is the heaviest atom, it can be considered to be fixed, leaving carbon to experience simple harmonic motion. The carbon atom therefore experiences a resistive force proportional to its displacement, each time it 'bounces' back towards the oxygen atom. This force is related to the stiffness of the triple bond between the two atoms, i.e. the spring constant - k.
However the vibrational frequency which corresponded to the peaks obtained on the spectrum was not known, but only the vibrational wavenumber. Given that the two parameters are related by Equation (5), the vibrational frequency could be computed into a rearranged form of Equation (4) to determine the value of k. The vibrational wavenumber could be read off the spectrum, as it corresponded to the missing peak in the Q branch. This value was estimated to be approximately 2144 cm-1 (nearest cm-1) by again taking into consideration the scaling of the spectrum, i.e. 4.3 cm for every 50 cm-1.
As a result of this, the value for the force constant could be estimated:
Using this value, the energy of this compound at the ground state, i.e. zero-point vibrational energy (v = 0), could be calculated by substituting Equation (5) into Equation (3) as shown below:
As vibrational molar energies are usually found to be within an order of magnitude of 104 J mol-1, the answer above can be considered to be reliable.
The procedure described to find the rotational constant in carbon monoxide was repeated to find the same constant in carbon dioxide. The spectrum being different to the carbon dioxide spectrum, the scale differed as well, as 2.2 cm corresponded to 10 cm-1 on the spectrum.
For 22 peaks, the length measured was 85.5 ± 0.5 mm.
Therefore the average spacing was:
For 22 peaks, the length measured was 59.5 ± 0.5 mm.
So the mean spacing was:
This means that the estimated rotational constant lies in between 0.533 cm-1 and 0.252 cm-1. As the literature value of 0.390 cm-1 lies within that range of values, the estimated Brot (rotational constant, B) can be considered to be reliable.
Vibrational motion in a molecule, such as CO2, can be subdivided into normal vibrational modes. The greater the number of atoms in a molecule, the more normal modes the molecule will have. Carbon dioxide is a linear molecule consisting of 3 different atoms; therefore it has 3 degrees of freedom (i.e. N atoms give N degrees of freedom). For a linear molecule, the number of possible vibrational modes is 3N - 5, where N is the number of degrees of freedom in the molecule. Carbon dioxide has 3 atoms; consequently it has 4 possible vibrational modes. These four normal modes correspond to four different types of vibrations which are the symmetric stretch, asymmetric stretching and two types of bending.
In the symmetric stretch, the carbon atom is vibrating around a fixed point, whilst the two neighbouring oxygen atoms are vibrating and moving towards it. This motion occurs due to the double bonds being stretched inwards to the same extent on both sides of the carbon atom, as represented in Figure 7. The same type of stretch would take place if the double bonds were stretching outwards.
Figure 7: Schematic of symmetric stretch
(1 = 1333 cm-1)
In asymmetric stretching, one of the double bonds is being stretched inwards as the carbon atom and the oxygen atom, which are part of this bond, are brought closer together. At the same time, the other oxygen atom is vibrating in the other direction as the double bond linking it to the carbon atom is being stretched outwards. This is shown in Figure 8 below.
Figure 8: Schematic of asymmetric stretch
(3 = 2349 cm-1)
When it comes to bending, there are two distinct modes representing two different types of motions in the carbon dioxide molecule. Both modes are equivalent as the second bending mode (c.f. Fig. 10) is identical to the first bending mode (c.f. Fig. 9) if the molecule is rotated through an angle of 90°. This entails the same energy requirements for both distinct motions, hence why both bending modes, which occur at ν2 = 667 cm-1, are said to be two-fold degenerate. The bending modes are represented in Figures 9 and 10 below.
In Figure 9, the arrows represent the bending vibrational mode which takes place on the vertical axis.
Figure 9: Schematic representation of bending mode 1 (2 = 667 cm-1)
In Figure 10, the arrows represent a different motion taking place on the horizontal axis, whereby the carbon atom is going into the plane and the two oxygen atoms are coming out of the plane.
Figure10: Schematic diagram of bending mode 2 (2 = 667 cm-1)
Infrared spectroscopy detects a change in the magnitude of the dipole moment and therefore for the CO2 molecule it would only detect the asymmetric stretch and both types of bending. In symmetric stretch, there is no change in dipole moment due to symmetry, so detecting it using this type of spectroscopy is not possible.
Raman spectroscopy involves detecting a change in the polarisability of the molecule, i.e. , where α is the polarisability and r is the coordinate relative to motion. This method of spectroscopy allows the symmetric stretch in CO2 to be detected due to the change in the electron density of the molecule. When the molecule experiences bending, the actual shape and density of the electron cloud in the CO2 molecule does not change, therefore this mode is not detected by Raman spectroscopy. Asymmetric stretching is slightly more complex; only Raman spectrometers of high resolutions are able to detect such changes in the polarisability of the molecule.
From the average rotational constant worked out above, the bond length could be calculated. Instead of using the reduced mass in this case, for CO2 the centre of mass was the carbon atom, therefore the moment of inertia for this molecule was easily calculated because at the position of the carbon atom, the distance from the centre of mass was zero. This can be shown in the formula below using Equation (7).
By rearranging this fomula obtained from Equation (7), the bond length could therefore be found.
LITERATURE VALUE COMPARISON
The bond length in CO2 is greater in magnitude in comparison to the bond length in CO since. This is because carbon monoxide has a triple bond (bond order of 3) whereas carbon dioxide has a double bond (bond order of 2), and a triple bond is shorter than a double bond because there is a greater pull of the electrons by the atoms sharing that bond.
From this bond length value and following the method used in carbon monoxide, the force constant for the double bond in CO2 could be found.
To compare the force constant of both CO2 and CO, the same vibrational mode has to be considered for both molecules. Carbon monoxide is a linear molecule with 2 atoms, hence 2 degrees of freedom. Therefore it only has one possible normal mode for vibrations, which is symmetric stretching. The wavenumber to be used for CO2 in order to work its force constant has to be 1 = 1333 cm-1.
From there the value of k could be deduced from Equation (4) the following way:
If a comparison is made between carbon monoxide and carbon dioxide, it is clear that as 1856 Nm-1 > 1675 Nm-1. The bond in CO was expected to be greater than the bond in CO2 from the Lewis structures in Fig. 1 and 2 because CO has a C-O triple bond which is much 'stiffer' than a C-O double bond. As the force constant is a measure of stiffness, a stiffer bond will have a greater k value.
3. Heat capacity
The heat capacity of a compound is the ability of a compound to store heat for a given increase in temperature. The change in the internal energy of this compound with respect to temperature at constant volume is called the heat capacity of this compound at constant volume, Cv. The change in the enthalpy of this compound with respect to temperature at constant pressure is Cp, i.e. the heat capacity at constant pressure.. For an ideal gas, Cp and Cv are related by the following equation:
Where Cp is the heat capacity of the ideal gas at constant pressure (J K-1), Cv is the heat capacity of the same gas at constant volume, and R is the gas constant (8.31447 J K-1 mol-1).
This gas constant is defined by the product of two other constants as:
With NA = 6.02214 - 1023 being Avogadro's number (i.e. the number of molecules in 1 mol of a substance), and kB = 1.38065 - 1023 being Boltzmann's constant, which is a constant describing the relationship between the temperature and the kinetic energy of the molecules in an ideal gas.
The total internal energy of a compound can be calculated using the Equipartition theorem which states that for any compound, the mean energy for each type of motion is equal to, where T is the absolute temperature in Kelvin. According to this Equipartition principle, the total internal energy of a compound is equally split by each degree of freedom. The number of degrees of freedom that a compound possesses depends on its ability to translate, to rotate around its centre of mass and to vibrate . If each type of motion is analysed separately, the average contribution, towards the total internal energy of the compound, for each different mode of motion can be found. For instance, translations and rotations can occur in three dimensions (x, y and z directions and θ, ø, and φ directions respectively), therefore these directions account for 3 degrees of freedom in total for translation and rotations separately, with each direction having a contribution to the total energy. In terms of vibrations, it's slightly different because vibrations store both potential energy and kinetic energy; therefore the contribution due to this type of motion is greater, i.e. . As Cv is the derivative of the total internal energy, U, with respect to temperature, the heat capacity of a compound can be easily calculated using this theorem.
Rotational and vibrational modes only become active at specific temperatures θR and θV respectively whereby:
Below these activation temperatures, the only mode active is translational motion.
The Equipartition theorem is only valid at high temperatures and does not account for the quantisation of energy. In terms of quantised states, the number of energy levels that molecules can occupy is limited, for that reason the total internal energy of each molecule is limited as well. The Boltzmann distribution takes into account the fact that not all molecules are occupying the same energy levels but the molecules are distributed in a wide range of energy levels. This distribution shows that the population of a state i decreases exponentially as the energy of any level relative to that state increases. Using the Boltzmann distribution, the heat capacity contribution for both rotations and vibrations can be determined by the following formulae:
Using the spectroscopic data provided for CO2, its Cv can be plotted as a function of temperature for further analysis on heat capacity.
Figure 11: Graph showing the heat capacity of carbon dioxide as a function of temperature (in logarithmic scale).
The graph in Fig. 11 shows the activation temperatures for both the rotational and vibrational modes. The graph can be separated into three parts corresponding to the relative contributions from the translational, rotational and vibrational types of motion. The contribution due to translations corresponds to the section where and the contribution due to rotations corresponds to the part where. Both sections appear as flat lines because these contributions do not vary with temperature as opposed to the vibrational contribution. The latter contribution is represented by an
Beyond 5000K, the CO2 molecule dissociates as the temperature would be so high that the bonds between the atoms would break, hence the sudden drop in the heat capacity value beyond this temperature on Fig. 11. Before dissociation there was N number of particles, whereas after there was 3N number of particles. Monoatomic particles do not vibrate or rotate as they are spherical and possess no bonds, therefore one would assume that for the dissociated particles in CO2 the degree of freedom would be 3 but this is not the case because. The degree of freedom for a compound does not change after dissociation, therefore it should remain 9, and hence why the specific heat capacity drops to.
The calculated values for the Cv of CO2 can be compared against literature values for a range of temperatures by converting Cv into Cp using Equation (10), assuming CO2 behaves as an ideal gas. These sets of values were then graphically compared (c.f. Fig. 12 below) to evaluate the reliability of the use of equations (13) and (14) to calculate the contribution of heat capacity for vibrational and rotational modes.
In Fig. 12 below, the values obtained from Rogers and Mayhew are found to be directly on the smooth curve representing the calculated data. This means that the results obtained from the equations mentioned above perfectly agree with experimental data, therefore Equation (13) and Equation (14) are a reliable way of calculating the heat capacity contribution of various types of motion in a compound. Furthermore the use of Equation (10) signifies the behaviour of carbon dioxide is similar to that of an ideal gas because the calculated Cp values which were based on this assumption, completely matched the experimental Cp values.
Figure 12: Graph of Cp of carbon dioxide as a function of temperature for calculated data and literature values from Mayhew and Rogers 
Figure 14: Graph showing the viscosity of carbon dioxide as a function of temperature for theoretical values and experimental data