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Determination of Number of Particles in I-th State for Bose-Einstein Statistics Calculator

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Indistinguishable Particles Distinguishable Particles

✖Number of Degenerate States can be defined as the number of energy states that have the same energy.ⓘ Number of Degenerate States [g]			+10% -10%
✖Lagrange's Undetermined Multiplier 'α' is denoted by μ/kT, Where μ= chemical potential; k= Boltzmann constant; T= temperature.ⓘ Lagrange's Undetermined Multiplier 'α' [α]			+10% -10%
✖Lagrange's Undetermined Multiplier 'β' is denoted by 1/kT. Where, k= Boltzmann constant, T= temperature.ⓘ Lagrange's Undetermined Multiplier 'β' [β]			+10% -10%
✖Energy of i-th State is defined as the total quantity of energy present in a particular energy state.ⓘ Energy of i-th State [ε_i]			+10% -10%

✖Number of particles in i-th State can be defined as the total number of particles present in a particular energy state.ⓘ Determination of Number of Particles in I-th State for Bose-Einstein Statistics [n_i]

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Determination of Number of Particles in I-th State for Bose-Einstein Statistics Solution

STEP 0: Pre-Calculation Summary

Formula Used

Number of particles in i-th State = Number of Degenerate States/(exp(Lagrange's Undetermined Multiplier 'α'+Lagrange's Undetermined Multiplier 'β'*Energy of i-th State)-1)
n_i = g/(exp(α+β*ε_i)-1)
This formula uses 1 Functions, 5 Variables

Functions Used

exp - n an exponential function, the value of the function changes by a constant factor for every unit change in the independent variable., exp(Number)

Variables Used

Number of particles in i-th State - Number of particles in i-th State can be defined as the total number of particles present in a particular energy state.
Number of Degenerate States - Number of Degenerate States can be defined as the number of energy states that have the same energy.
Lagrange's Undetermined Multiplier 'α' - Lagrange's Undetermined Multiplier 'α' is denoted by μ/kT, Where μ= chemical potential; k= Boltzmann constant; T= temperature.
Lagrange's Undetermined Multiplier 'β' - (Measured in Joule) - Lagrange's Undetermined Multiplier 'β' is denoted by 1/kT. Where, k= Boltzmann constant, T= temperature.
Energy of i-th State - (Measured in Joule) - Energy of i-th State is defined as the total quantity of energy present in a particular energy state.

STEP 1: Convert Input(s) to Base Unit

Number of Degenerate States: 3 --> No Conversion Required
Lagrange's Undetermined Multiplier 'α': 5.0324 --> No Conversion Required
Lagrange's Undetermined Multiplier 'β': 0.00012 Joule --> 0.00012 Joule No Conversion Required
Energy of i-th State: 28786 Joule --> 28786 Joule No Conversion Required

STEP 2: Evaluate Formula

Substituting Input Values in Formula

n_i = g/(exp(α+β*ε_i)-1) --> 3/(exp(5.0324+0.00012*28786)-1)

Evaluating ... ...

n_i = 0.000618692918280003

STEP 3: Convert Result to Output's Unit

0.000618692918280003 --> No Conversion Required

FINAL ANSWER

0.000618692918280003 ≈ 0.000619 <-- Number of particles in i-th State

(Calculation completed in 00.020 seconds)

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Determination of Number of Particles in I-th State for Bose-Einstein Statistics Formula

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What is Statistical Thermodynamics?

Statistical thermodynamics is a theory that uses molecular properties to predict the behavior of macroscopic quantities of compounds. While the origins of statistical thermodynamics predate the development of quantum mechanics, the modern development of statistical thermodynamics assumes that the quantized energy levels associated with a particular system are known. From these energy-level data, a temperature-dependent quantity called the partition function can be calculated. From the partition function, all of the thermodynamic properties of the system can be calculated. Statistical thermodynamics has also been applied to the general problem of predicting reaction rates. This application is called transition state theory or the theory of absolute reaction rates.

How to Calculate Determination of Number of Particles in I-th State for Bose-Einstein Statistics?

Determination of Number of Particles in I-th State for Bose-Einstein Statistics calculator uses Number of particles in i-th State = Number of Degenerate States/(exp(Lagrange's Undetermined Multiplier 'α'+Lagrange's Undetermined Multiplier 'β'*Energy of i-th State)-1) to calculate the Number of particles in i-th State, The Determination of Number of Particles in I-th State for Bose-Einstein Statistics formula is defined as the number of indistinguishable boson particles that can be present in a particular energy state. Number of particles in i-th State is denoted by n_i symbol.

How to calculate Determination of Number of Particles in I-th State for Bose-Einstein Statistics using this online calculator? To use this online calculator for Determination of Number of Particles in I-th State for Bose-Einstein Statistics, enter Number of Degenerate States (g), Lagrange's Undetermined Multiplier 'α' (α), Lagrange's Undetermined Multiplier 'β' (β) & Energy of i-th State (ε_i) and hit the calculate button. Here is how the Determination of Number of Particles in I-th State for Bose-Einstein Statistics calculation can be explained with given input values -> 0.000619 = 3/(exp(5.0324+0.00012*28786)-1).

FAQ

What is Determination of Number of Particles in I-th State for Bose-Einstein Statistics?

The Determination of Number of Particles in I-th State for Bose-Einstein Statistics formula is defined as the number of indistinguishable boson particles that can be present in a particular energy state and is represented as n_i = g/(exp(α+β*ε_i)-1) or Number of particles in i-th State = Number of Degenerate States/(exp(Lagrange's Undetermined Multiplier 'α'+Lagrange's Undetermined Multiplier 'β'*Energy of i-th State)-1). Number of Degenerate States can be defined as the number of energy states that have the same energy, Lagrange's Undetermined Multiplier 'α' is denoted by μ/kT, Where μ= chemical potential; k= Boltzmann constant; T= temperature, Lagrange's Undetermined Multiplier 'β' is denoted by 1/kT. Where, k= Boltzmann constant, T= temperature & Energy of i-th State is defined as the total quantity of energy present in a particular energy state.

How to calculate Determination of Number of Particles in I-th State for Bose-Einstein Statistics?

The Determination of Number of Particles in I-th State for Bose-Einstein Statistics formula is defined as the number of indistinguishable boson particles that can be present in a particular energy state is calculated using Number of particles in i-th State = Number of Degenerate States/(exp(Lagrange's Undetermined Multiplier 'α'+Lagrange's Undetermined Multiplier 'β'*Energy of i-th State)-1). To calculate Determination of Number of Particles in I-th State for Bose-Einstein Statistics, you need Number of Degenerate States (g), Lagrange's Undetermined Multiplier 'α' (α), Lagrange's Undetermined Multiplier 'β' (β) & Energy of i-th State (ε_i). With our tool, you need to enter the respective value for Number of Degenerate States, Lagrange's Undetermined Multiplier 'α', Lagrange's Undetermined Multiplier 'β' & Energy of i-th State and hit the calculate button. You can also select the units (if any) for Input(s) and the Output as well.

How many ways are there to calculate Number of particles in i-th State?

In this formula, Number of particles in i-th State uses Number of Degenerate States, Lagrange's Undetermined Multiplier 'α', Lagrange's Undetermined Multiplier 'β' & Energy of i-th State. We can use 1 other way(s) to calculate the same, which is/are as follows -

Number of particles in i-th State = Number of Degenerate States/(exp(Lagrange's Undetermined Multiplier 'α'+Lagrange's Undetermined Multiplier 'β'*Energy of i-th State)+1)

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