Planck s relationship

Planck s relationship E = hv lets us convert Equation 5.38 into a relation between energy and measurement time. 

The precessional motion of the magnetic moment around Bq occurs with angular frequency wq, called the Larmorfrequency, whose units are radians per second (rad s ). As Bq increases, so does the angular frequency that is, coq cx Bq, as is demonstrated in Appendix 1. The constant of proportionality between o>o and Bq is the gyromagnetic ratio 7, so that wq = Bq. The natural precession frequency can be expressed as linear frequency in Planck s relationship AE = Hvq or angular frequency in Planck s relationship AE = h(x)Q (coq = 2 rrvo). In this way, the energy difference between the spin states is related to the Larmor frequency by the formula... 

Planck s relationship is used to convert the frequency to energy ... 

The amount of energy associated with a quantum of radiation is proportional to the frequency E = hv, where h is Planck s constant. This is known as Planck s relationship and can be rewritten as E = h x c/X since c = v x X (the wave equation). A wide variety of problems can be solved using Planck s relationship, the wave equation and the Avogadro constant. 

Example 4. For a given lattice, a relationship is to be found between the lattice resistivity and temperature usiag the foUowiag variables mean free path F, the mass of electron Af, particle density A/, charge Planck s constant Boltzmann constant temperature 9, velocity and resistivity p. Suppose that length /, mass m time /, charge and temperature T are chosen as the reference dimensions. The dimensional matrix D of the variables is given by (eq. 55) ... 

DeBroglie, in the mid 1920s, proposed the idea that particles could be treated as waves by the relationship, X = h/mv. This equation related the mass (m) and velocity (v) of a particle to its wavelength (X) by using Planck s constant (h). 

A similar relationship is also derived by the absolute reaction rate theory, which is used almost exclusively in considering, and understanding, the kinetics of reactions in solution. The activated complex in the transition state is reached by reactants in the initial state as the highest point of the most favorable reaction path on the potential energy surface. The activated complex Xms in equilibrium with the reactants A and B, and the rate of the reaction V is the product of the equilibrium concentration of X and the specific rate at which it decomposes. The latter can be shown to be equal to kT/h, where k is Boltzmannn s constant and h is Planck s constant ... 

This tells us that the energy, E, carried by a photon is related to its frequency. The relationship between energy and frequency is = hf, where h is Planck s constant and has the value 6-63 x 10" J s. Using this relationship, the energy is calculated in joules (J). In chemistry, energy values are normally expressed in units of kj mol. To convert from J to kj the number value must be divided by 1000. To obtain mol in units of kJ mol, the relationship changes to E = Lhf, where L is Avogadro s constant. L has the value 6-02 x 10 mol. ... 

The critical energy required for a photon to remove an electron is then hvc = q>, where Vc is the critical frequency of the photon and h is Planck s constant. When the frequency of the incident radiation is less than the critical frequency, electrons will not be ejected. Similarly, when the wavelength of the incident radiation is greater than the critical wavelength, Xc, electrons will also not be emitted (recall that V = c/X, where c is the speed of light in vacuum). This relationship between... 

If the electron is now brought from the first into the second orbit, then an amount of energy must be expended, equivalent to E2 — Ev If, on the other hand, the electron moves from state 2 to state 1, then an amount of energy E2 — Ex is set free, which, according to Bohr, is released as monochromatic radiation of a frequency given by the relationship v = E2 — Ej) jh, in which h is Planck s constant. 

The relationship between the energy of a transition and the frequency is given by AL" = hv or = he/A or Af = Acv where h is Planck s constant. The energy of a particular transition is, therefore, proportional to the frequency or wavenumber, and inversely proportional to the wavelength. 

The only non-classical feature of this relationship is the appearance of Planck s constant. The uncertainty itself is well-known to occur in any classical system described in wave formalism. The reason why it never became an issue in classical physics is because h is too small to cause measurable effects in macroscopic systems. 

It is significant that in both cases Planck s constant appears in the specification of the dynamic variables of angular momentum and energy, associated with wave motion. The curious relationship between mass and energy that involves the velocity of a wave, seems to imply that the motion of mass points also has some wavelike quality. Only because Planck s constant is almost vanishingly small, dynamic variables of macroscopic systems appear to be continuous. However, when dealing with atomic or sub-atomic systems... 

The weird properties that came to be associated with quantum systems, because of the probability doctrine, obscured the simple mathematical relationship that exists between classical and quantum mechanics. The lenghthy discussion of this aspect may be of less interest to chemical readers, but it may dispel the myth that a revolution in scientific thinking occured in 1925. Actually there is no break between classical and non-classical systems apart from the relative importance of Planck s action constant in macroscopic and microscopic systems respectively. Along with this argument goes the realization that even in classical mechanics, as in optics, there is a wave-like aspect associated with all forms of motion, which becomes more apparent, at the expense of particle behaviour, in the microscopic domain. 

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