Boltzmann constant relations

The temperature T of a system is related to the mean kinetic energy of all atoms N via Eq. (36), where kg is the Boltzmann constant and the average of the squared velocities of atom i. 

Constant relating wave number and energy per molecule k = Boltzmann constant... 

The quantities n, V, and (3 /m) T are thus the first five (velocity) moments of the distribution function. In the above equation, k is the Boltzmann constant the definition of temperature relates the kinetic energy associated with the random motion of the particles to kT for each degree of freedom. If an equation of state is derived using this equilibrium distribution function, by determining the pressure in the gas (see Section 1.11), then this kinetic theory definition of the temperature is seen to be the absolute temperature that appears in the ideal gas law. 

The hydrodynamic radius reflects the effect of coil size on polymer transport properties and can be determined from the sedimentation or diffusion coefficients at infinite dilution from the relation Rh = kBT/6itri5D (D = translational diffusion coefficient extrapolated to zero concentration, kB = Boltzmann constant, T = absolute temperature and r s = solvent viscosity). 

The effect of surface potential on interfacial ionic concentration is given by a Boltzmann distribution relating the total solution concentration to that at the interface. For charged, amphiphilic species, binding constants replace these ionic concentrations, and the expression... 

The Eyring activated-complex (or transition-state) treatment relates the observed rate constant k to multiplied by the frequency factor k TIh, where k is the Boltzmann constant, T is the absolute temperature, and h is Planck s constant ... 

In statistical mechanics entropy of a system with respect to a particular state is related to the probability, W, of a system being in that state ie.,S = kBltiW + b where is a constant and /cb is the Boltzmann constant. Hence, entropy represents the degree of disorder within a sys-... 

Here, ks is the Boltzmann constant (1.38 x 10-23 J/K), T is the absolute temperature (300 K at room temperature), B is the bandwidth of measurement [typically about 1000 Hz for direct current (dc) measurement], /o is the resonant frequency of the cantilever, and Q is the quality factor of the resonance, which is related to damping. It is clear from Eq. (12.8) that lower spring constant, K, produces higher thermal noise. This thermal motion can be used as an excitation technique for resonance frequency mode of operation. 

In equation (5.5), k is the Boltzmann constant (1.38066 x 10 JK ), which is related to the (thermal) energy of one molecule, and V is the number of all accessible states of the molecule. Generally, in physical chemistry, we consider not a single molecule (or particle), but rather one mole, i.e., 6 x 10 molecules (or particles). Thus equation (5.5) becomes... 

A fundamental theorem of classical mechanics called the equipartition theorem (which we shall not derive here) states that the average energy of each degree of freedom of a molecule in a sample at a temperature T is equal to kT. In this simple expression, k is the Boltzmann constant, a fundamental constant with the value 1.380 66 X 10-21 J-K l. The Boltzmann constant is related to the gas constant by R = NAk, where NA is the Avogadro constant. The equipartition theorem is a result from classical mechanics, so we can use it for translational and rotational motion of molecules at room temperature and above, where quantization is unimportant, but we cannot use it safely for vibrational motion, except at high temperatures. The following remarks therefore apply only to translational and rotational motion. 

Diffusion of small solute particles (atoms, molecules) in a dense liquid of larger particles is an important but ill-understood problem of condensed matter physics and chemistry. In this case one does not expect the Stokes-Einstein (SE) relation between the diffusion coefficient D of the tagged particle of radius R and the viscosity r/s of the medium to be valid. Indeed, experiments [83, 112-115] have repeatedly shown that in this limit SE relation (with slip boundary condition) significantly underestimates the diffusion coefficient. The conventional SE relation is D = C keT/Rr]s, where k T is the Boltzmann constant times the absolute temperature and C is a numerical constant determined by the hydrodynamic boundary condition. To explain the enhanced diffusion, sometimes an empirical modification of the SE relation of the form... 

In this equation, the diffusion coefficient D is related to air viscosity r A and particle diameter dp, with k being the Boltzmann constant and T the absolute temperature. It is clear from this description that diffusion is a rather slow deposition mechanism compared with impaction and sedimentation processes because it depends on the thermal velocity of the particles and not on airflow. It is the primary transport mechanism for small particles and is important when the transport distance becomes small, as in the deep lung. Efficiency of this deposition mechanism can be increased significantly by breath-holding because a portion of the ultrafine particles that are not deposited will be exhaled by the patient. 

The use of the Stokes-Einstein equation (2) relating the diffusion coefficient (D) of a spherical solute molecule to its radius (r), the viscosity of the medium (tj) and the Boltzmann constant (k) permits the rate coefficient ( en) to be expressed in (3) in terms of the viscosity of the medium. In this derivation, the... 

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