Boltzmann Constant K Links Entropy With Probability
For an isolated system of particles in equilibrium, entropy S is the natural logarithm of W , the number of possible micro-states for the given macroscopic properties :
S=k ln W
Thus Boltzmann constant k has the dimensions Energy/Temperature.
Thus Boltzmann constant k bridges macroscopic physics with microscopic physics .
Role In Boltzmann Factors
More generally, systems in equilibrium at temperature T have probability Pi of occupying a state i with energy E weighted by the corresponding Boltzmann factor:
- P , \propto }\right)}},}
where Z is the partition function. Again, it is the energy-like quantity kT that takes central importance.
Consequences of this include the Arrhenius equation in chemical kinetics.
What Does K Stand For Physics
We compiled queries of the k abbreviation in Physics in search engines. The most frequently asked k acronym questions for Physics were selected and included on the site.
We thought you asked a similar k question to the search engine to find the meaning of the k full form in Physics, and we are sure that the following Physics k query list will catch your attention.
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How To Calculate The Spring Constant
There are two simple approaches you can use to calculate the spring constant, using either Hookes law, alongside some data about the strength of the restoring force and the displacement of the spring from its equilibrium position, or using the elastic potential energy equation alongside figures for the work done in extending the spring and the displacement of the spring.
Using Hookes law is the simplest approach to finding the value of the spring constant, and you can even obtain the data yourself through a simple setup where you hang a known mass from a spring and record the extension of the spring. Ignoring the minus sign in Hookes law and dividing by the displacement, x, gives:
Using the elastic potential energy formula is a similarly straightforward process, but it doesnt lend itself as well to a simple experiment. However, if you know the elastic potential energy and the displacement, you can calculate it using:
In any case youll end up with a value with units of N/m.
Elastic Potential Energy Equation
The concept of elastic potential energy, introduced alongside the spring constant earlier in the article, is very useful if you want to learn to calculate k using other data. The equation for elastic potential energy relates the displacement, x, and the spring constant, k, to the elastic potential PEel, and it takes the same basic form as the equation for kinetic energy:
As a form of energy, the units of elastic potential energy are joules .
The elastic potential energy is equal to the work done , and you can easily calculate it based on the distance the spring has been stretched if you know the spring constant for the spring. Similarly, you can re-arrange this equation to find the spring constant if you know the work done in stretching the spring and how much the spring was extended.
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Distinguishing Temperature Heat And Internal Energy
Using the kinetic theory, a clear distinction between these three properties can be made.
- Temperature is related to the kinetic energies of the molecules of a material. It is the average kinetic energy of individual molecules.
- Internal energy refers to the total energy of all the molecules within the object. It is an extensive property, therefore when two equal-mass hot ingots of steel may have the same temperature, but two of them have twice as much internal energy as one does.
- Finally, heat is the amount of energy flowing from one body to another spontaneously due to their temperature difference.
It must be added, when a temperature difference does exist heat flows spontaneously from the warmer system to the colder system. Thus, if a 5 kg cube of steel at 100°C is placed in contact with a 500 kg cube of steel at 20°C, heat flows from the cube at 300°C to the cube at 20°C even though the internal energy of the 20°C cube is much greater because there is so much more of it.
A particularly important concept is thermodynamic equilibrium. In general, when two objects are brought into thermal contact, heat will flow between them until they come into equilibrium with each other.
Internal energymicroscopic scaleUU
U = Upot;+ Ukin
The microscopic potential energy,;Upot, involves the;chemical bonds;between the atoms that make up the molecules, binding forces in the nucleus and also the physical force fields within the system .
Greek Letters Used In Mathematics Science And Engineering
Greek letters are used in mathematics, science, engineering, and other areas where mathematical notation is used as symbols for constants, special functions, and also conventionally for variables representing certain quantities. In these contexts, the capital letters and the small letters represent distinct and unrelated entities. Those Greek letters which have the same form as Latin letters are rarely used: capital A, B, E, Z, H, I, K, M, N, O, P, T, Y, X. Small , and are also rarely used, since they closely resemble the Latin letters i, o and u. Sometimes font variants of Greek letters are used as distinct symbols in mathematics, in particular for / and /. The archaic letter digamma is sometimes used.
The Bayer designation naming scheme for stars typically uses the first Greek letter, , for the brightest star in each constellation, and runs through the alphabet before switching to Latin letters.
In mathematical finance, the Greeks are the variables denoted by Greek letters used to describe the risk of certain investments.
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What Does The Brackets Around And Represent And What Does It Do
In this context, the brackets means #concentration#
It is tempting to assume that #HO^-# and #H_3O^+# are actual species in aqueous species. As far as anyone knows these species are CLUSTERS of water molecules LESS or PLUS a proton
And so or or somethingwe use the #HO^/H_3O^+# as a label of conveniencethe acidium species is a water cluster associated with an extra proton, and the hydroxide species is a water cluster LESS a proton. And if you play rugby think of #H^+# as the ball in a maul.
But certainly, when we do acid-base titrations, we can find the equivalence of acids and bases, straightforwardly and quantitatively, using very simple equipment, calibrated burettes, and standard indicatos.
And we use the brackets to designate a concentration term: #=10^# under standard conditions. And for simplicity, when we see ##
Role In The Statistical Definition Of Entropy
In statistical mechanics, the entropyS of an isolated system at thermodynamic equilibrium is defined as the natural logarithm of W, the number of distinct microscopic states available to the system given the macroscopic constraints :
- K . } \over T}=\approx 8.61733034\times 10^\ \mathrm .}
At room temperature 300;K , VT is approximately 25.85;mV. and at the standard state temperature of 298.15;K , it is approximately 25.69;mV. The thermal voltage is also important in plasmas and electrolyte solutions ; in both cases it provides a measure of how much the spatial distribution of electrons or ions is affected by a boundary held at a fixed voltage.
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Physical Meaning Of Wave Vector
Recently I have been studying the Rayleigh-Jeans derivation, and for this problem we need the three dimensional wave equation for electromagnetic radiation:
$$ \frac}^2} + \frac}^2} + \frac}^2} = \frac\frac}^2} $$
For our blackbody radiation problem, we need to define a field, E, that is zero at the boundaries of the cavity it is contained within. From a simple perspective, I think that we can use a sinusoidal functional form with no phase shift :
$$ \vec=\vec\sin $$$$ \vec = \vec\sin $$
My notation here is that k is the wave vector and Em is the magnitude and direction of the wave. Plugging this into the wave equation gives us the following:
$$ k_x^2 +k_y^2 + k_z^2 = \frac $$
Which implies that the magnitude of the wave vector is equal to the wave number, k = w/c. In addition, we should have corresponding variables for the modes of the standing wave:
$$ n_x, n_y, n_z $$and “wavelengths”:$$ \lambda_x, \lambda_y, \lambda_z $$
We know that there is physically only one wavelength, lambda, so what is the meaning of the x, y and z components? I did some math on this earlier today and if the following is true:
$$ n^2 = n_x^2+n_y^2+n_z^2 $$$$ k^2 = k_x^2+k_y^2+k_z^2 $$
I found that:
$$ \lambda^2 \ne \lambda_x^2 + \lambda_y^2 + \lambda_z^2 $$
I feel more comfortable with the notion of nx, ny and nz, because they are the number of “half” wavelengths that can fit into the respect x, y and z directions.
In addition, what do the various components of the wave vector, k, mean?
Example Evaporation Of Water At Atmospheric Pressure
Calculate heat required to evaporate;1 kg;of water at the atmospheric pressure and at the temperature of 100°C.
Since these parameters corresponds to the saturated liquid state, only latent heat of vaporization of 1 kg of water is required. From steam tables, the latent heat of vaporization is L = 2257 kJ/kg. Therefore the heat required is equal to:
H = 2257 kJ
Note that the initial specific enthalpy h1;= 419 kJ/kg, whereas the final specific enthalpy will be h2;= 2676 kJ/kg. The specific enthalpy of low-pressure dry steam is very similar to the specific enthalpy of high-pressure dry steam, despite the fact they have different temperatures.
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Conservation Of Mechanical Energy
A mass hanging from the ceiling will have a kinetic energy equal to zero. Ifthe cord breaks, the mass will rapidly increase its kinetic energy. Thiskinetic energy was somehow stored in the mass when it was hanging from theceiling: the energy was hidden, but has the potential to reappear as kineticenergy. The stored energy is called potential energy. Conservation ofenergy tells us that the total energy of the system is conserved, and inthis case, the sum of kinetic and potential energy must be constant. Thismeans that every change in the kinetic energy of a system must beaccompanied by an equal but opposite change in the potential energy:
U + K = 0
E = U + K = constant
The work-energy theorem discussed in Chapter 7 relates the amount ofwork W to the change in the kinetic energy of the system
W = K
The change in the potential energy of the system can now be related tothe amount of work done on the system
U = – K = – W
which will be the definition of the potential energy. The unit ofpotential energy is the Joule .
The potential energy U can be obtained from the applied force F
where U is the potential energy of the system at itschosen reference configuration. It turns out that only changes in thepotential energy are important, and we are free to assign the arbitraryvalue of zero to the potential energy of the system when it is in its referenceconfiguration.
Sometimes, the potential energy function U is known. The forceresponsible for this potential can then be obtained
The Spring Constant: Car Suspension Problem
A 1800-kg car has a suspension system that cannot be allowed to exceed 0.1 m of compression. What spring constant does the suspension need to have?
This problem might appear different to the previous examples, but ultimately the process of calculating the spring constant, k, is exactly the same. The only additional step is translating the mass of the car into a weight on each wheel. You know that the force due to the weight of the car is given by F = mg, where g = 9.81 m/s2, the acceleration due to gravity on Earth, so you can adjust the Hookes law formula as follows:
However, only one quarter of the total mass of the car is resting on any wheel, so the mass per spring is 1800 kg / 4 = 450 kg.
Now you simply have to input the known values and solve to find the strength of the springs needed, noting that the maximum compression, 0.1 m is the value for x youll need to use:
This could also be expressed as 44.145 kN/m, where kN means kilonewton or thousands of newtons.
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;;extensive Variables And Eulers Thermodynamic Equation
Here’s a fancy example. Suppose we have a thermodynamic system with acertain set of extensive variables, such as volume , entropy, and the amount of each chemical component . Also suppose the energy can be expressed as a function ofthose variables.
It is more elegant to lump all the extensive variables into a vectorX with components Xi for all i.
We introduce the general notion of homogeneous function asfollows: If we have a function with the property:
|∂ Xi | Xj≠i|
There are conventional names for the partial derivatives on the LHS:temperature, pressure, and chemical potentials. Note that these areintensive quantities . Using these names, weget:
|T S − P V + |
which is called Euler’s thermodynamic equation. It is a consequence ofthe fact that the extensive variables are extensive. It imposes aconstraint, which means that not all of the variables are independent.
Note: Nothing is ever perfectly extensive. There arealways boundary terms that don’t scale the same way as thebulk terms. However, for big-enough systems, the boundaryterms can be neglected, and this scaling analysis is anexcellent approximation.
If we take the exterior derivative of equation 14 weobtain:
|S dT − V dP + |
The Gibbs-Duhem equation has many other practical applications.
The Base Dissociation Constant
Historically, the equilibrium constant Kb for a base has been defined as the association constant for protonation of the base, B, to form the conjugate acid, HB+.
B + H_2O \leftrightharpoons HB^+ + OH^-
As with any equilibrium constant for a reversible reaction, the expression for Kb takes the following form:
K_ = \frac
Kb is related to Ka for the conjugate acid. Recall that in water, the concentration of the hydroxide ion, , is related to the concentration of the hydrogen ion by the autoionization constant of water:
4.3/5KaKbKa and KbKa and Kbabout it here
Solve the equation for Kb by dividing the Kw by the Ka. You then obtain the equation Kb = Kw / Ka. Put the values from the problem into the equation. For example, for the chloride ion, Kb = 1.0 x 10^-14 / 1.0 x 10^6.
Beside above, what are the KA and KB equations? Ka and Kb values measure how well an acid or base dissociates. Higher values of Ka or Kb mean higher strength. General Ka expressions take the form Ka = / . General Kb expressions take the form Kb = / .
Keeping this in consideration, what is KA and KB in chemistry?
Ka is the acid dissociation constant. pKa is simply the -log of this constant. Similarly, Kb is the base dissociation constant, while pKb is the -log of the constant. The acid and base dissociation constants are usually expressed in terms of moles per liter .
What is the Ka in chemistry?
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Dr Michael De Podesta National Physical Laboratory Uk
Arguably todays best measurement has been performed by Dr Michael de Podesta MBE CPhys MInstP, Principal Research Scientist at the National Physical Laboratory NPL in Teddington, UK, who has kindly discussed his measurements and todays status of the work on the system of SI units and its redefinition with me, and has greatly assisted in the preparation of this article. Dr Podestas measurements of Boltzmanns constant have been published in:Michael de Podesta et al. A low-uncertainty measurement of the Boltzmann constant, Metrologia 50 354-376.
Over the last 10 years there was intense effort in Europe and the USA to build rebuild the SI unit system. In particular NIST , NPL , several French institutions and Italian institutions, as well as the German PTB have undertaken this effort.
Force: Colliding With A Car
In a situation where car B collides with car C, we have different force considerations. Assuming that car B and car C are complete mirrors of each other , they would collide with each other going at precisely the same speed but in opposite directions. From conservation of momentum, we know that they must both come to rest. The mass is the same, therefore, the force experienced by car B and car C is identical, and also identical to that acting on the car in case A in the previous example.
This explains the force of the collision, but there is a second part of the question: the energy within the collision.
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;; K Times Greater Than
The phrase “… k times greater than …” is so problematic thatit deserves detailed discussion.
Consider the following scenario: VA denotes the volume of objectA, while VB denotes the volume of object B. We are given thatVA = 12 liters and VB = 36 liters.
The first three of the following statements are goodterminology, but then things go to pot:
- 1.    VB is three times as great as VA.
- 2.    B has 3 times the volume of A.
- 3.    VBis 300% of VA.
- 4.    VB is 200% greater than VA.
- 5.    VB is greater than VA by a factor of3.
- 6.    VB is three times greater than VA.
- 7.    VB is two times greater than VA.
- 8.    B is three times as big as A.
A different problem crops up in statement 8. Does thestatement mean that every dimension of B is bigger by a factor of 3, or does it only mean that the volume isbigger by a factor of 3? When you are scaling some property, you needto be specific about which property you are scaling.