probability

Examples of probability in the following topics:

• Indeterminacy and Probability Distribution Maps

  • Quantum indeterminacy can be quantitatively characterized by a probability distribution on the set of outcomes for measurements of an observable.
  • Quantum mechanics provides a recipe for calculating this probability distribution.
  • This is called a probability distribution map, a statistical representation of the probable locations of electrons as they exist in an atom.
  • With repeated measurements, a pattern of probability emerges.
  • The clouds of probability do not look like nor do they correspond to classical orbits.

• Particle in a Box

  • The probability of finding the particle at the edge of the box is 0 because the potential there is infinite; the cosine term, therefore, can be removed to leave only the sine term.
  • The size or amplitude of the wave function at any point determines the probability of finding the particle at that location, as given by the equation:
  • Finally, the unknown constant may be found by normalizing the wavefunction so that the total probability density of finding the particle in the system is 1.
  • When treated as a probability density, the square of the wave function (Ψ2) describes the probability of finding the particle at a given point and at a given time.

• Microstates and Entropy

  • For a given set of macroscopic variables, the entropy measures the degree to which the probability of the system is spread out over different possible microstates.
  • The more such microstates, the greater is the probability of the system being in the corresponding macrostate.
  • Even though it is beyond human comprehension to compare numbers that seem to verge on infinity, the thermal energy contained in actual physical systems manages to discover the largest of these quantities with no difficulty at all, quickly settling in to the most probable macrostate for a given set of conditions.

• Single Covalent Bonds

  • An atomic orbital is defined as the probability of finding an electron in an area around an atom's nucleus.
  • Higher intensity of the red color indicates a greater probability of the bonding electrons being localized between the nuclei.

• Reactions of Ethers

  • Indeed, with the exception of the alkanes, cycloalkanes and fluorocarbons, ethers are probably the least reactive, common class of organic compounds.
  • The 2º-alkyl group in example #3 is probably cleaved by an SN2 mechanism, but the SN1 alternative cannot be ruled out.

• Distribution of Molecular Speeds and Collision Frequency

  • By understanding the nature of the particle movement, however, we can predict the probability that a particle will have a certain velocity at a given temperature.
  • If we assume that all velocity states are equally probable, higher velocity states are favorable because there are greater in quantity.
  • The peak of the curve represents the most probable velocity among a collection of gas particles.

• Combustion

  • The alkanes and cycloalkanes, with the exception of cyclopropane, are probably the least chemically reactive class of organic compounds.
  • The pyrolytic conversion of β-pinene to myrcene probably takes place by an initial rupture of the 1:6 bond, giving an allylic 3º-diradical, followed immediately by breaking of the 5:7 bond.

• The Favorskii Rearrangement

  • In this case cleavage of the cyclopropanol at the more substituted α-carbon probably reflects the inductive effect of the THPO substituent.

• Addition to Carbonyl Double Bonds

  • Nucleophilic addition reactions to aldehydes and ketones are probably the earliest and most actively studied examples of stereoselectivity.

• Background and Properties

  • The relatively high boiling points of equivalent 3º-amides and nitriles are probably due to the high polarity of these functions.