Lecture 32: Theoretical Kinetics & Reaction Dynamics
Collision Theory we need to find the collision rate and the probability of reaction (many parameters), and integrate their product. Consider: D 2 + H DH + D
clas. vib.
General reaction rate: - total rat
Lecture 31: Phenomenological Kinetics
Stationary-state approximation used to simplify the solutions of complex reactions
B, B C, . Consider again the consecutive reactions: A When the intermediate is slowly generated, but it decays fast: [B] < [A] and [C
Lecture 30: Phenomenological Kinetics
Temperature dependence of the rate coefficient Reactions need less than the (H2) dissociation energy: H2 + D HD + H (9.3% of it) k(T) ~ P(EEa) molecules need to gain the activation energy Ea to react; Arrhenius equati
Lecture 29: Phenomenological Kinetics
We will study chemical reaction rates. The systems are out of equilibrium and often evolve fast (they require special experimental techniques). In principle, we can calculate the reaction rates by combining quantum an
Lecture 28: Statistical Thermodynamics
Vibrational contributions to thermodynamic functions Consider diatomic molecule with the energies of vibrational states:
- at room T, one state is OK If we neglect the anharmonicity, we can sum the power series:
The
Lecture 27: Statistical Thermodynamics
Rotational contributions to thermodynamic functions Consider first linear molecules and neglect the rot-vib coupling. If we also neglect distortion (non-linear) terms, we can find the rotational molecular partition f
Lecture 26: Statistical Thermodynamics
We found that in the canonical ensemble all the functions used in thermodynamics can be obtained from the partition function Z. For practical use of these rules, we need to clarify, how Z is related to the molecular
Lecture 25: Statistical Thermodynamics
Ensembles of identical systems (each in one microstate) allow for calculation of average values of thermodynamic properties. The microcanonical ensemble (MCE) consists of a set of M systems each characterized by N, V
Lecture 33: Theoretical Kinetics & Reaction Dynamics
Monte Carlo integration methods they can be used to approximately evaluate integrals and other expressions (convenient for finding R, if PAB is known).
We can evaluate <f(x)> by summing over a large and
Lecture 34: Theoretical Kinetics & Reaction Dynamics
Simple transition-state theory We can minimize the flux FStotal and get closer to R by maximizing the total energy E needed for the passage of the AB-pair (reactants) over the dividing surface. This can
Lecture 24: The Kinetic Theory of Gases
The collision frequency: We transform it to CMS coordinates:
Since:
we obtain the probability product in the form:
The collision frequency integrated over the CMS velocities is:
After integrating it also over the an
Lecture 23: The Kinetic Theory of Gases
The Maxwell-Boltzmann Distribution of Molecular Speeds (ideal gas so far) Last time, starting from the microcanonical ensemble, we have derived the Boltzmann distribution of (distinguishable) particles across variou
Lecture 22: The Kinetic Theory of Gases
Microstates: The combined state of all the particles in any system define the microstate that this system populates. Assumption: - in equilibrium, microstates of the same energy are populated with the same probabili
Lecture 10: Simple Quantum Mechanical Systems
Quantum mechanical angular momentum The Hamiltonian for 2D motion of a particle is:
(particle on a ring no azimuthal component)
r
Analogously, the Hamiltonian for relative motion of 2 particles in 3D is:
(part
Lecture 9: Simple Quantum Mechanical Systems
We separate the translational, rotational and vibrational motions. Rotational motion we use center of mass and relative coordinates
The original coordinates & velocities in terms of the CMS & relative coordinat
Lecture 8: Simple Quantum Mechanical Systems
Superposition of states Let us consider a superposition of energy eigenstates:
(this is also a possible solution of TSE)
- get norm. const. of this new state
- normalized state - wf (but not eigenstate of H)
By
Lecture 7: Simple Quantum Mechanical Systems
We will study quantized translational, rotational and vibrational motions (in molecules). Translational motion we solve the SE for a free particle: V(x,y,z,t)=0
The time-dependent SE (TSE) can be solved by the
. Possible steps leading to the time-dependent Schrodinger equation:
Lecture 6: Introduction to Quantum Mechanics
The solution of the classical wave equation in a box of length l is a superposition of plane waves (wavector) We assume that de Broglie's mat
Lecture 5: Introduction to Quantum Mechanics
The Wave Nature of Matter: The Compton experiment (1922) Energy conservation: p, p' assumed incident and scattered momenta of the photon Compton assumed that
from the right sides above
X-ray photon
Momentum co
Lecture 4: Introduction to Quantum Mechanics
Blackbody Radiation The electromagnetic theory was completed by J. C. Maxwell in 1861. Accelerated charges are the sources of electromagnetic waves.
(Far from the sources, they are planar waves often coming in