1.3. Transformación de la energía a partir la biomasa forestal
1.3.3. Tipos de gasificadores
Vibrational spectroscopy of molecules is studied at infra-red wavelengths and is the subject of many texts (e.g. [HI]). Vibrational spectroscopy concerns the motions o f nuclei relative to one another within a molecule, the equilibrium positions of the vibrations being determined by a balance of forces:
i) the repulsive forces between the positively charged nuclei
ii) the repulsive forces between the inner shell electrons surrounding each nucleus iii) the attractive forces between the nuclei and the electrons throughout the molecule.
The energy o f the molecular system is a minimum when the forces are balanced and the equilibrium intemuclear distances are then defined as bond lengths.
To describe the motion of nuclei within a molecule we may choose the Cartesian coordinate system. If a molecule contains N nuclei, the positions and motions o f them can be described using 3N coordinate values. However, translations along the axes and rotations about the axes do not
affect the vibrational motion of a molecule, hence information on them is not required to describe the vibrations. Thus for an N-atomic molecule there are 3N-6 (3N-5 for linear molecules which have only two distinguishable rotational modes) vibrational degrees o f freedom, i.e. the molecule has 3N-6 (3N-5) fundamental vibrational frequencies, or fundamental modes o f vibration.
A simple diatomic molecule is linear by default and has only one fundamental mode of vibration; stretching and contracting of the bond. To a first approximation the bond can be likened to a spring obeying Hooke's law; in this case the potential energy curve is parabolic having the form:
[J] (1.3.7)
where k is the force constant, r is the intemuclear separation and r^q is the equilibrium separation. This model of a diatomic molecule is called the simple harmonic oscillator model and the oscillation frequency is given by:
U(r) = '/a k (r - r^q)
(13.8)
where p is the reduced mass of the system and c is the speed of light in vacuo. Using this potential function to solve the Schrodinger equation it is found that the allowed vibrational energies of the molecule are quantised and given by:
Evib = h CÛOSC (o + '/a) [J] (1.3.9)
where h is Planck's constant and u is the vibrational quantum number (u=0, 1, 2, ...). This shows that the molecule can never have zero vibrational energy (when u=0, Evib='/2hc0osc’. the zero point energy), i.e. the nuclei can never be completely at rest relative to each other. The energy
levels thus obtained for the simple harmonic oscillator model are shown in figure 1 3 .1.
U(r)
u-0
Figure 1.3.1. Vibrational energy levels of the diatomic simple harmonic oscillator.
Further use of the Schrodinger equation yields the selection rule for the simple harmonic oscillator model: Ao= ±1.
However, real molecules do not execute simple harmonic motion: the potential in which the nuclei of a diatomic molecule are bound can typically be described by the Morse function:
U(r) = Deq [1 -exp{o(req-r)}f [JJ (1.3.10)
where Dgq is the dissociation energy (the depth of the potential well) and a is a constant for the molecule. A system vibrating in such a potential is called an anharmonic oscillator (figure 1.3.2). Replacing the harmonic oscillator potential function with equation 1.3.10 in the Schrodinger equation gives the allowed vibrational energy levels for the anharmonic oscillator:
Evib = h Cùe (u + Vz) - h ©e Xe (U + Vzf [J] (1.3.11)
where is the anharmonicity constant and ©e is the equilihrium oscillation frequency.
Comparing the results for the harmonic and anharmonic oscillators (equations 1.3.9 and 1.3.11) shows that:
®osc = 03e {1 -Xe(u + ‘/2)} [s'^j (1.3.12)
Thus the anharmonic oscillator behaves like the harmonic oscillator but with an oscillation frequency which decreases as v increases (figure 1.3.2).
U(r)
Zero-point energy
Figure 1.3.2. Energy levels of the anharmonic oscillator.
The selection rule for transitions in the anharmonic oscillator model becomes: Au=±l, ±2, ±3, ... although the probability of a transition decreases rapidly as Au increases. The frequency of a transition between vibrational levels u=0 and u -1 is known as the fundamental frequency, whilst those corresponding to u=0 and u'>l are known as overtones.
The case of the diatomic molecule is simple in that it can only support one vibrational mode, and the intemuclear potential depends on a single variable (r, the intemuclear separation). However, the potential functions of polyatomic molecules depend on greater numbers o f variables (3N-6 for non-linear molecules, 3N-5 for linear) and can be mathematically complicated. These functions can be visualised by considering just two o f the degrees of freedom (by freezing the other vibrational modes) and considering those two modes only. The potential energy function describing the interaction of the two modes can then be represented as a two-dimensional surface, allowing the effects o f multiple vibrational modes to be considered simultaneously for the molecule. The allowed energy levels of such molecules are given by:
E ^ ,= h 2 < o ,(U i+ i) + h 2 Ê x „ (U i+ i)(U j+ i)
[J]
(1.3.13)
i=l
j^i i=l
where (Oi is the equilibrium frequency (or zero-order frequency) and Ui the vibrational quantum
number of the i^ vibrational mode, n is the number of vibrational modes o f the molecule (3N-6 or
3N-5), and are the anharmonicity constants (corresponding to cOe^e for diatomic molecules); this approximation assumes higher order terms in
Oi
are negligible.For a simple triatomic molecule such as ozone (N=3, n=3N-6=3), this equation reduces to:
Evib = hcoi ( o i + Yi) + h(ù2 (U2 + Yz) + h©3 (0 3 + Yz) [J] (1.3.14) + h x ii ( u i + YzŸ + 11x22(02 + Y z f + I1X33 (0 3 + Y z f
+ h x i 2 ( O i + Yz) { V 2 + Yz) + h X ] 3 ( U ] + * / 2 ) ( U 3 + Yz) + h X23 ( v > 2 + % ) ( l ) 3 + /4)
Hence it can be seen that the anharmonicity constants
Xi^^j
are a measure o f the interactions between the different vibrational modes. In practice, it is common to measure the vibrational energy with respect to the zero-point energy (i.e.01=1)2=1)3=0,
the lowest vibrational state):The vibrational energy above the zero-point for a triatomic molecule can then be represented as (neglecting higher order terms in u j:
E(i)i,U2,ü3) - E(0,0,0) = h (coi^O] +© 2^02 +© 3°03 [J] (1.3.16)
+ + X22Ü2^ + %33 U3^ + Xi2U]U2 + X13U1U3 + X23O2O3 } where:
[s']
(1.3.17)
2 j ^ i
The observed fundamental frequencies (vi) corresponding to transitions from Oi=0 to Oi=l with all other oj=0 are hence:
Vi = ©i° + Xii [s'^] (1.3.18)
The quantities vi, © ©i and Xÿ can thus be derived from experimental data.
The vibrational selection rule for polyatomic molecules is the same as that for diatomic molecules. In addition to the fundamental transitions mentioned above, infra-red absorption spectra of polyatomic molecules may also exhibit overtone bands (e.g. E(0,0,0) -> E(2,0,0) for a triatomic molecule), combination bands (e.g. E(0,0,0) E(2,l,0)) and difference bands (e.g. E(0,0,0) -> E(2,0,-l)). However, the fundamental bands are usually observed to be the most intense. It is also found that as Aoi becomes large (>10) it can change by non-integer amounts, i.e. the approximations used in the quantum mechanical solution begin to break down.
A vibrating molecule may also be rotating, in accordance with the Bom-Oppenheimer approximation (equation 1.3.1). The two motions interact with each other, and it may be shown that the selection rules for the combined motions are the same as those for each separately, i.e. Au=±l, ±2, ±3, ... and AJ=±1. This results in fine structure due to rotational transitions on the vibrational bands observed in infra-red spectra. This structure (sometimes referred to as
rovibrational structure) is often quite complicated, although it can be analysed to yield much information about the molecule; very high resolution (<1 meV) spectrometers are required to resolve these fine structures. Such equipment was not used in the work presented in this thesis, hence a rigorous description of rovibrational transitions is not required here; a thorough account of this type of spectroscopy can be found in [Bl]. The important result of the combined motions is that the width o f observed vibrational peaks may have a limit imposed by rotational transitions.
Vibrational modes that do not lead to a change in permanent electric dipole moment o f the molecule are not observed by infra-red spectroscopy, e.g. the stretching modes o f homonuclear diatomic molecules. However, such modes may be observable using Raman-spectroscopy which involves the inelastic scattering of radiation by the molecular target, i.e. the Raman effect. In classical terms, this requires that the polarisibility o f the molecule (a) must be changed by a nuclear motion such that a dipole moment is induced allowing an interaction with radiation. If a molecular state is Raman-active, photons of incident energy Am will scatter inelastically with final energy Am’, where A(m-m’) is the energy required to excite the molecule. The scattered light intensity is usually very small and decreases as the change in photon energy increases, so in practice a bright hght source is used (e.g. a laser beam) and the energy change kept small (<1 eV). A rovibrational mode of a molecule may be infra-red and/or Raman active, or inactive to both. A detailed discussion o f Raman spectroscopy is not relevant to this thesis hence will not be given; a full description can be found in [Bl]. Raman active modes may also be observed using electron impact spectroscopy (section 1.5.3).