reduced mass of polyatomic molecules

R r 2 Legal. 0000100491 00000 n 0000011477 00000 n Have questions or comments? Molecules with the same atoms in different arrangements are called isomers. Then substituting above gives a new Lagrangian. For diatomic molecules, we define the reduced mass \(\mu_{AB}\) by: \[ \mu_{AB}=\dfrac{m_A\, m_B}{m_A+m_B} \label{5.2.1}\] Reduced mass is the representation of a two-body system as a single-body one. Equation \ref{freq} is used to predict the respective vibrational frequencies of these two molecules. Legal. 1 {\displaystyle \mu } Anything that consists of one atom is called atomic, for example helium or other noble gasses. The LibreTexts libraries are Powered by MindTouch® and are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. Reduced mass is the representation of a two-body system as a single-body one. The science of molecules is called molecular chemistry or molecular physics, depending on the focus. For diatomic molecules, we define the reduced mass \(\mu_{AB}\) by: \[ \mu_{AB}=\dfrac{m_A\, m_B}{m_A+m_B} \label{5.2.1}\]. The reduced mass is always less than or equal to the mass of each body: "Reduced mass" may also refer more generally to an algebraic term of the form[citation needed]. We cannot use it, for example, to describe vibrations of diatomic molecules, where quantum effects are important. 2 wavefunction. The reduced mass is frequently denoted by As discussed previously, the Schrödinger equation for the angular motion of a rigid (i.e., having fixed bond length \(R\)) diatomic molecule is, \[\dfrac{\hbar^2}{2 μ} \left[ \dfrac{1}{R^2 \sin θ} \dfrac{∂}{∂θ} \left(\sin θ \dfrac{∂}{∂θ} \right) + \dfrac{1}{R^2 \sin^2 θ} \dfrac{∂^2}{∂φ^2} \right] |ψ \rangle = E | ψ \rangle\], \[ \dfrac{L^2}{2 μ R^2 } | ψ \rangle = E | ψ\rangle \], The Hamiltonian in this problem contains only the kinetic energy of rotation; no potential energy is present because the molecule is undergoing unhindered "free rotation". 5.2: The Equation for a Harmonic-Oscillator Model of a Diatomic Molecule Contains the Reduced Mass of the Molecule, [ "article:topic", "reduced mass", "showtoc:no", "license:ccbyncsa", "transcluded:yes", "source-chem-92374" ], 5.1: A Harmonic Oscillator Obeys Hooke's Law, 5.3: The Harmonic Oscillator is an Approximation. mass and spring model? However, as with most quantum modules (and in contrast to the classical harmonic oscillator), the energies are quantized in terms of a quantum number (\(v\) in this case): \[ \begin{align} E_v &= \hbar \left(\sqrt {\dfrac {k}{\mu}} \right) \left(v + \dfrac {1}{2} \right) \\[4pt] &= h \nu \left(v+\dfrac {1}{2} \right) \end{align} \], with the natural vibrational frequency of the system given as, \[\nu = \dfrac{1}{2 \pi}\sqrt {\dfrac {k}{\mu}} \label{freq}\]. . However, high resolution IR spectroscopy can easily distinguish the vibrations between these two molecules. If the unique rotational axis has a greater inertia than the degenerate axes the molecule is called an oblate symmetrical top (Figure \(\PageIndex{1}\)). 0000032654 00000 n F In the case of the gravitational potential energy, we find that the position of the first body with respect to the second is governed by the same differential equation as the position of a body with the reduced mass orbiting a body with a mass equal to the sum of the two masses, because, Consider the electron (mass me) and proton (mass mp) in the hydrogen atom. : If This simplifies the description of the system to one force (since reduced to a point-mass is quite common, and simple estimates provide a satisfactory description of system’sdynamics. m 0000032861 00000 n R and atomic masses (reduced mass). r + ( For this problem, we need the exact mass of the \(\ce{^1H}\), \(\ce{^35Cl}\), and \(\ce{^37Cl}\) isotopes. or when converted into kg is \(1.6291 \times 10^{-27}\,kg\). The eigenfunctions \(|J, M,K>\) are the same rotation matrix functions as arise for the spherical-top case. + 12 such that they are co-linear, the two distances CC BY-SA 3.0. https://en.wikipedia.org/wiki/Properties_of_water For example, carbohydrates have the same ratio (carbon: hydrogen: oxygen = 1:2:1) and thus the same empirical formula, but have different total numbers of atoms in the molecule. Caution: Do Not Use Atomic Weights to Calculate Reduced Masses. {\displaystyle m_{1}} ) 0000002803 00000 n 0000002350 00000 n The rotational energy in Equation \(\ref{Ediatomic}\) can be expressed in terms of the moment of inertia \(I\), \[I =\sum_i m_i R_i^2 \label{Idiatomic}\]. One problem with this classical formulation is that it is not general. The reduced mass is always less than or equal to the mass of each body: and has the reciprocal additive property: which by re-arrangement is equivalent to half of the harmonic mean. {\displaystyle m_{i}} m 0000007271 00000 n Polyatomic molecules are electrically neutral groups of three or more atoms held together by covalent bonds. In H-X, particle X is stationary, and particle H R 0000007985 00000 n times that of H-H. r This require the formulation for Schrödinger Equation using Equation \ref{potential}. contributes 100% of the energy to the vibration. a R In a collision with a coefficient of restitution e, the change in kinetic energy can be written as. The relative abundances are reported for the ions formed and some general patterns for negative-ion formation have been dedu particle contributes is proportional to its intensity in the wavefunction. This approach greatly simplifies many calculations and problems. 1 H-X, where X has a very large mass, compared to that of the H, the effective m the problem is, to what extent does each atom contribute to the mass, and to Alternatively, a Lagrangian description of the two-body problem gives a Lagrangian of, where 0000003693 00000 n In physics, the reduced mass is the "effective" inertial mass appearing in the two-body problem of Newtonian mechanics.It is a quantity which allows the two-body problem to be solved as if it were a one-body problem.Note, however, that the mass determining the gravitational force is not reduced. This holds for a rotation around the center of mass. 2 μ l 1 0000008595 00000 n It is oblate if, 13.9: Normal Modes in Polyatomic Molecules, Telluride Schools on Theoretical Chemistry. To analyze the motion of the electron, a one-body problem, the reduced mass replaces the electron mass, and the proton mass becomes the sum of the two masses. Central to this model is the formulation of the quadratic potential energy, \[V(x) \approx \dfrac {1}{2} kx^2 \label{potential}\]. The LibreTexts libraries are Powered by MindTouch® and are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. 0000003935 00000 n For this reason, vibrational spectra (IR and Raman) can provide detailed structural information. m l The potential energy V is a function as it is only dependent on the absolute distance between the particles. Determine the reduced mass of the two body system of a proton and electron with \(m_{proton} = 1.6727 \times 10^{-27}\, kg\) and \(m_{electron} = 9.110 \times 10^{-31}\, kg\)). 1 Exercise \(\PageIndex{1}\) will demonstrate that this "isotope effect" is not always a small effect. CC BY-SA 3.0. https://en.wikipedia.org/wiki/Allotrope 2 . This idea is used to set up the Schrödinger equation for the hydrogen atom. 2 A molecular vibration is a periodic motion of the atoms of a molecule relative to each other, such that the center of mass of the molecule remains unchanged.

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