Physical Chemistry 3 Reviewer PDF

**Physical Chemistry 3 Reviewer** I. **Origins of Quantum Mechanics** 1. A **black body** is an object capable of emitting and absorbing all wavelengths of radiation without favoring any wavelength. 2. At each temperature ***T*** there is a wavelength, ***λ~max~***, at which the intens...

**Physical Chemistry 3 Reviewer** I. **Origins of Quantum Mechanics** 1. A **black body** is an object capable of emitting and absorbing all wavelengths of radiation without favoring any wavelength. 2. At each temperature ***T*** there is a wavelength, ***λ~max~***, at which the intensity of the radiation is a maximum, with ***T*** and ***λ~max~*** related by the empirical **Wien's law: *(***[**λ**]{.math.inline}***~max~ T = constant)***. The constant is found to have the value 2.9 mm K. 3. The **energy density**, ***E(T)***, is the total energy inside the container divided by its volume. Empirically, the energy density is found to vary as ***T^4^***, an observation expressed by the **Stefan--Boltzmann law:** ***\[E(T) = constant x T^4^\],*** with the constant found to have the value 7.567 × 10^−16^ J m^−3^ K^−4^. 4. An electromagnetic field of a given frequency can take up energy only in discrete amounts. 5. Atomic and molecular spectra show that atoms and molecules can take up energy only in discrete amounts. 6. The limitation of energies to discrete values is called **energy quantization**. On this basis Planck was able to derive an expression for the energy spectral density which is now called the **Planck distribution**. The currently accepted value is h = 6.626 × 10^−34^ J s. 7. If the energy of an atom or molecule decreases by ***ΔE***, and this energy is carried away as radiation, the frequency of the radiation [**ν**]{.math.inline} and the change in energy are related by the **Bohr frequency condition: (**[**Δ**]{.math.inline}**E = h**[**ν**]{.math.inline}**)** 8. The **photoelectric effect** ***(E~k~ = h***[**ν−** **Φ)** ]{.math.inline}establishes the view that electromagnetic radiation, regarded in classical physics as wave-like, consists of particles (photons). 9. The diffraction of electrons establishes the view that electrons, regarded in classical physics as particles, are wavelike with a wavelength given by the **de Broglie relation *(***[**λ**]{.math.inline} ***= h/p).*** 10. **Wave--particle duality** is the recognition that the concepts of particle and wave blend together. II. **Wavefunctions** 1. A **wavefunction** is a mathematical function that contains all the dynamical information about a system. 2. The ***Schrödinger equation*** is a second-order differential equation used to calculate the wavefunction of a system. 3. According to the **Born interpretation**, the probability density at a point is proportional to the square of the wavefunction at that point. 4. A **node** is a point where a wavefunction passes through zero. 5. A wavefunction is **normalized** ***(***[**∫ψ** **\*** **ψdτ** **=** **1)**]{.math.inline} if the integral over all space of its square modulus is equal to 1. 6. A wavefunction must be single-valued, continuous, not infinite over a finite region of space, and (except in special cases) have a continuous slope. 7. The quantization of energy stems from the constraints that an acceptable wavefunction must satisfy. 8. For a one-dimensional system, the probability ***P*** of finding the particle between ***x~1~*** and ***x~2~*** is given by  III. **Operators and Observables** 1. The Schrödinger equation is an **eigenvalue equation.** 2. The Schrödinger equation written as in A screenshot of a computer Description automatically generated is an **eigenvalue equation**, an equation of the form (operator)(function) = (constant factor) x (same function). In an eigenvalue equation, the action of the operator on the function generates the same function, multiplied by a constant. 3. An **operator** carries out a mathematical operation on a function, in general transforming it into a new function. 4. The **Hamiltonian operator** is the operator corresponding to the total energy of the system, the sum of the kinetic and potential energies. 5. The wavefunction corresponding to a specific energy is an **eigenfunction** of the Hamiltonian operator. 6. Two different functions are **orthogonal** if the integral (over all space) of their product is zero.  7. **Hermitian operators** have real eigenvalues and orthogonal eigenfunctions. 8. A **Hermitian operator** is one for which the following relation is true: A screenshot of a computer Description automatically generated ***d***[**τ**]{.math.inline} implies integration over the full range of all relevant spatial variables. 9. **Observables** are represented by Hermitian operators. 10. **Orthonormal** functions are sets of functions that are both normalized and mutually orthogonal. 11. When the system is not described by a single eigenfunction of an operator, it may be expressed as a **superposition** of such eigenfunctions. 12. The mean value of a series of observations is given by the **expectation value** of the corresponding operator.  13. The **Heisenberg** **uncertainty principle** A screenshot of a computer Description automatically generated restricts the precision with which complementary observables may be specified and measured simultaneously. 14. **Complementary observables** are observables for which the corresponding operators do not commute. 15. The different outcomes of the effect of applying ***Ωˆ~1~*** and ***Ωˆ~2~*** in a different order are expressed by introducing the **commutator** of the two operators, which is defined as