Examples of Boltzmann's constant in the following topics:
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- The third law of thermodynamics states that the entropy of a system approaches a constant value as the temperature approaches absolute zero.
- The third law of thermodynamics states that the entropy of a system approaches a constant value as the temperature approaches zero.
- Entropy is related to the number of possible microstates according to $S = k_Bln(\Omega)$, where S is the entropy of the system, kB is Boltzmann's constant, and Ω is the number of microstates (e.g. possible configurations of atoms).
- The constant value (not necessarily zero) is called the residual entropy of the system.
- Mathematically, the absolute entropy of any system at zero temperature is the natural log of the number of ground states times Boltzmann's constant kB.
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- The kinetic theory of gases describes a gas as a large number of small particles (atoms and molecules) in constant, random motion.
- The kinetic theory of gases describes a gas as a large number of small particles (atoms or molecules), all of which are in constant, random motion.
- The distribution of the speeds (which determine the translational kinetic energies) of the particles in a classical ideal gas is called the Maxwell-Boltzmann distribution.
- (k: Boltzmann's constant).
- (R: ideal gas constant, n: number of moles of gas) from a microscopic theory.
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- while Charles' law states that volume of a gas is proportional to the absolute temperature T of the gas at constant pressure
- where C is a constant which is directly proportional to the amount of gas, n (representing the number of moles).
- The proportionality factor is the universal gas constant, R, i.e.
- where k is Boltzmann's constant and N is the number of molecules.
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- An isothermal process is a change of a system in which the temperature remains constant: ΔT = 0.
- An isothermal process is a change of a system in which the temperature remains constant: ΔT = 0.
- According to the ideal gas law, the value of the constant is NkT, where N is the number of molecules of gas and k is Boltzmann's constant.
- This means that $p = {N k T \over V} = {\text{Constant} \over V}$ holds.
- (This equation is derived in our Atom on "Constant Pressure" under kinetic theory.
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- For any given liquid with constant density throughout, pressure increases with increasing depth.
- For many liquids, the density can be assumed to be nearly constant throughout the volume of the liquid and, for virtually all practical applications, so can the acceleration due to gravity (g = 9.81 m/s2).
- Thus the force contributing to the pressure of a gas within the medium is not a continuous distribution as for liquids and the barometric equation given in must be utilized to determine the pressure exerted by the gas at a certain depth (or height) within the gas (p0 is the pressure at h = 0, M is the mass of a single molecule of gas, g is the acceleration due to gravity, k is the Boltzmann constant, T is the temperature of the gas, and h is the height or depth within the gas).
- The force contributing to the pressure of a gas within the medium is not a continuous distribution as for liquids and the barometric equation given in this figure must be utilized to determine the pressure exerted by the gas at a certain depth (or height) within the gas (p0 is the pressure at h = 0, M is the mass of a single molecule of gas, g is the acceleration due to gravity, k is the Boltzmann constant, T is the temperature of the gas, and h is the height or depth within the gas)
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- where k is the Boltzmann's constant.
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- The actual number of atoms or molecules in one mole is called Avogadro's constant (NA), in recognition of Italian scientist Amedeo Avogadro .
- The value of Avogadro's constant, NA , has been found to equal 6.02×1023 mol−1.
- Avogadro's constant is a scaling factor between macroscopic and microscopic (atomic scale) observations of nature.
- As such, it provides the relation between other physical constants and properties.
- For example, it establishes a relationship between the gas constant R and the Boltzmann constant k,
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- The equation combines the concepts of activation energy and the Boltzmann distribution law into one of the most important relationships in physical chemistry:
- In this equation, k is the rate constant, T is the absolute temperature, Ea is the activation energy, A is the pre-exponential factor, and R is the universal gas constant.
- This means that high temperatures and low activation energies favor larger rate constants, and therefore these conditions will speed up a reaction.
- Recall that the exponential part of the Arrhenius equation ($e^{\frac{-E_a}{RT}}$) expresses the fraction of reactant molecules that possess enough kinetic energy to react, as governed by the Maxwell-Boltzmann distribution.
- Therefore, A represents the maximum possible rate constant; it is what the rate constant would be if every collision between any pair of molecules resulted in a chemical reaction.
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- The Maxwell-Boltzmann Distribution describes the average molecular speeds for a collection of gas particles at a given temperature.
- According to the Kinetic Molecular Theory, all gaseous particles are in constant random motion at temperatures above absolute zero.
- In the above formula, R is the gas constant, T is absolute temperature, and Mm is the molar mass of the gas particles in kg/mol.
- Given the constant changes, it is difficult to gauge the particles' velocities at any given time.
- This results in an asymmetric curve, known as the Maxwell-Boltzmann distribution.
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- A gas of many molecules has a predictable distribution of molecular speeds, known as the Maxwell-Boltzmann distribution.
- Maxwell-Boltzmann distribution is a probability distribution.
- The Maxwell-Boltzmann distribution of molecular speeds in an ideal gas.
- The Maxwell-Boltzmann distribution is shifted to higher speeds and is broadened at higher temperatures.
- Describe the shape and temperature dependence of the Maxwell-Boltzmann distribution curve