Maximum Vibrational Quantum Number Solution

STEP 0: Pre-Calculation Summary
Formula Used
Max Vibrational Number = (Vibrational Wavenumber/(2*Anharmonicity Constant*Vibrational Wavenumber))-1/2
vmax = (ω'/(2*xe*ω'))-1/2
This formula uses 3 Variables
Variables Used
Max Vibrational Number - Max Vibrational Number is the maximum scalar quantum value that defines the energy state of a harmonic or approximately harmonic vibrating diatomic molecule.
Vibrational Wavenumber - (Measured in Diopter) - Vibrational Wavenumber is simply the harmonic vibrational frequency or energy expressed in units of cm inverse.
Anharmonicity Constant - Anharmonicity Constant is the deviation of a system from being a harmonic oscillator which is related to the vibrational energy levels of diatomic molecule.
STEP 1: Convert Input(s) to Base Unit
Vibrational Wavenumber: 15 1 per Meter --> 15 Diopter (Check conversion here)
Anharmonicity Constant: 0.24 --> No Conversion Required
STEP 2: Evaluate Formula
Substituting Input Values in Formula
vmax = (ω'/(2*xe*ω'))-1/2 --> (15/(2*0.24*15))-1/2
Evaluating ... ...
vmax = 1.58333333333333
STEP 3: Convert Result to Output's Unit
1.58333333333333 --> No Conversion Required
FINAL ANSWER
1.58333333333333 1.583333 <-- Max Vibrational Number
(Calculation completed in 00.020 seconds)

Credits

Created by Akshada Kulkarni
National Institute of Information Technology (NIIT), Neemrana
Akshada Kulkarni has created this Calculator and 500+ more calculators!
Verified by Pragati Jaju
College Of Engineering (COEP), Pune
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22 Vibrational Spectroscopy Calculators

Maximum Vibrational Number using Anharmonicity Constant
Go Max Vibrational Number = ((Vibrational Wavenumber)^2)/(4*Vibrational Wavenumber*Vibrational Energy*Anharmonicity Constant)
Vibrational Quantum Number using Rotational Constant
Go Vibrational Quantum Number = ((Rotational Constant vib-Rotational Constant Equilibrium)/Anharmonic Potential Constant)-1/2
Rotational Constant Related to Equilibrium
Go Rotational Constant Equilibrium = Rotational Constant vib-(Anharmonic Potential Constant*(Vibrational Quantum Number+1/2))
Rotational Constant for Vibrational State
Go Rotational Constant vib = Rotational Constant Equilibrium+(Anharmonic Potential Constant*(Vibrational Quantum Number+1/2))
Anharmonic Potential Constant
Go Anharmonic Potential Constant = (Rotational Constant vib-Rotational Constant Equilibrium)/(Vibrational Quantum Number+1/2)
Maximum Vibrational Quantum Number
Go Max Vibrational Number = (Vibrational Wavenumber/(2*Anharmonicity Constant*Vibrational Wavenumber))-1/2
Anharmonicity Constant given Fundamental Frequency
Go Anharmonicity Constant = (Vibration Frequency-Fundamental Frequency)/(2*Vibration Frequency)
Vibrational Quantum Number using Vibrational Frequency
Go Vibrational Quantum Number = (Vibrational Energy/([hP]*Vibrational Frequency))-1/2
Vibrational Quantum Number using Vibrational Wavenumber
Go Vibrational Quantum Number = (Vibrational Energy/[hP]*Vibrational Wavenumber)-1/2
Anharmonicity Constant given Second Overtone Frequency
Go Anharmonicity Constant = 1/4*(1-(Second Overtone Frequency/(3*Vibrational Frequency)))
Anharmonicity Constant given First Overtone Frequency
Go Anharmonicity Constant = 1/3*(1-(First Overtone Frequency/(2*Vibrational Frequency)))
Energy Difference between Two Vibrational States
Go Change in Energy = Equilibrium Vibrational Frequency*(1-(2*Anharmonicity Constant))
Vibrational Frequency given Second Overtone Frequency
Go Vibrational Frequency = Second Overtone Frequency/3*(1-(4*Anharmonicity Constant))
Second Overtone Frequency
Go Second Overtone Frequency = (3*Vibrational Frequency)*(1-4*Anharmonicity Constant)
First Overtone Frequency
Go First Overtone Frequency = (2*Vibrational Frequency)*(1-3*Anharmonicity Constant)
Vibrational Frequency given First Overtone Frequency
Go Vibrational Frequency = First Overtone Frequency/2*(1-3*Anharmonicity Constant)
Vibrational Frequency given Fundamental Frequency
Go Vibrational Frequency = Fundamental Frequency/(1-2*Anharmonicity Constant)
Fundamental Frequency of Vibrational Transitions
Go Fundamental Frequency = Vibrational Frequency*(1-2*Anharmonicity Constant)
Vibrational Degree of Freedom for Nonlinear Molecules
Go Vibrational Degree Nonlinear = (3*Number of Atoms)-6
Vibrational Degree of Freedom for Linear Molecules
Go Vibrational Degree Linear = (3*Number of Atoms)-5
Total Degree of Freedom for Nonlinear Molecules
Go Degree of Freedom Non Linear = 3*Number of Atoms
Total Degree of Freedom for Linear Molecules
Go Degree of Freedom Linear = 3*Number of Atoms

21 Important Calculators of Vibrational Spectroscopy Calculators

Maximum Vibrational Number using Anharmonicity Constant
Go Max Vibrational Number = ((Vibrational Wavenumber)^2)/(4*Vibrational Wavenumber*Vibrational Energy*Anharmonicity Constant)
Vibrational Quantum Number using Rotational Constant
Go Vibrational Quantum Number = ((Rotational Constant vib-Rotational Constant Equilibrium)/Anharmonic Potential Constant)-1/2
Rotational Constant Related to Equilibrium
Go Rotational Constant Equilibrium = Rotational Constant vib-(Anharmonic Potential Constant*(Vibrational Quantum Number+1/2))
Rotational Constant for Vibrational State
Go Rotational Constant vib = Rotational Constant Equilibrium+(Anharmonic Potential Constant*(Vibrational Quantum Number+1/2))
Anharmonic Potential Constant
Go Anharmonic Potential Constant = (Rotational Constant vib-Rotational Constant Equilibrium)/(Vibrational Quantum Number+1/2)
Maximum Vibrational Quantum Number
Go Max Vibrational Number = (Vibrational Wavenumber/(2*Anharmonicity Constant*Vibrational Wavenumber))-1/2
Anharmonicity Constant given Fundamental Frequency
Go Anharmonicity Constant = (Vibration Frequency-Fundamental Frequency)/(2*Vibration Frequency)
Vibrational Quantum Number using Vibrational Frequency
Go Vibrational Quantum Number = (Vibrational Energy/([hP]*Vibrational Frequency))-1/2
Vibrational Quantum Number using Vibrational Wavenumber
Go Vibrational Quantum Number = (Vibrational Energy/[hP]*Vibrational Wavenumber)-1/2
Anharmonicity Constant given Second Overtone Frequency
Go Anharmonicity Constant = 1/4*(1-(Second Overtone Frequency/(3*Vibrational Frequency)))
Anharmonicity Constant given First Overtone Frequency
Go Anharmonicity Constant = 1/3*(1-(First Overtone Frequency/(2*Vibrational Frequency)))
Vibrational Frequency given Second Overtone Frequency
Go Vibrational Frequency = Second Overtone Frequency/3*(1-(4*Anharmonicity Constant))
Second Overtone Frequency
Go Second Overtone Frequency = (3*Vibrational Frequency)*(1-4*Anharmonicity Constant)
First Overtone Frequency
Go First Overtone Frequency = (2*Vibrational Frequency)*(1-3*Anharmonicity Constant)
Vibrational Frequency given First Overtone Frequency
Go Vibrational Frequency = First Overtone Frequency/2*(1-3*Anharmonicity Constant)
Vibrational Frequency given Fundamental Frequency
Go Vibrational Frequency = Fundamental Frequency/(1-2*Anharmonicity Constant)
Fundamental Frequency of Vibrational Transitions
Go Fundamental Frequency = Vibrational Frequency*(1-2*Anharmonicity Constant)
Vibrational Degree of Freedom for Nonlinear Molecules
Go Vibrational Degree Nonlinear = (3*Number of Atoms)-6
Vibrational Degree of Freedom for Linear Molecules
Go Vibrational Degree Linear = (3*Number of Atoms)-5
Total Degree of Freedom for Nonlinear Molecules
Go Degree of Freedom Non Linear = 3*Number of Atoms
Total Degree of Freedom for Linear Molecules
Go Degree of Freedom Linear = 3*Number of Atoms

Maximum Vibrational Quantum Number Formula

Max Vibrational Number = (Vibrational Wavenumber/(2*Anharmonicity Constant*Vibrational Wavenumber))-1/2
vmax = (ω'/(2*xe*ω'))-1/2

What is Dissociation energy?

The term dissociation energy may be appreciated by reference to potential energy internuclear distance curves. At about 0 K all molecules have no rotational energy but are merely vibrating with their zero-point energy. Thus, diatomic molecules are in the v = 0 vibrational level. The energy required to separate the stable molecule A - B initially in the v = 0 level into two unexcited atoms A and B, that is: A - B → A+B is known as the dissociation energy (D).

How to Calculate Maximum Vibrational Quantum Number?

Maximum Vibrational Quantum Number calculator uses Max Vibrational Number = (Vibrational Wavenumber/(2*Anharmonicity Constant*Vibrational Wavenumber))-1/2 to calculate the Max Vibrational Number, The Maximum vibrational quantum number formula is defined as the maximum scalar quantum value that defines the energy state of a harmonic or approximately harmonic vibrating diatomic molecule. Max Vibrational Number is denoted by vmax symbol.

How to calculate Maximum Vibrational Quantum Number using this online calculator? To use this online calculator for Maximum Vibrational Quantum Number, enter Vibrational Wavenumber (ω') & Anharmonicity Constant (xe) and hit the calculate button. Here is how the Maximum Vibrational Quantum Number calculation can be explained with given input values -> 1.583333 = (15/(2*0.24*15))-1/2.

FAQ

What is Maximum Vibrational Quantum Number?
The Maximum vibrational quantum number formula is defined as the maximum scalar quantum value that defines the energy state of a harmonic or approximately harmonic vibrating diatomic molecule and is represented as vmax = (ω'/(2*xe*ω'))-1/2 or Max Vibrational Number = (Vibrational Wavenumber/(2*Anharmonicity Constant*Vibrational Wavenumber))-1/2. Vibrational Wavenumber is simply the harmonic vibrational frequency or energy expressed in units of cm inverse & Anharmonicity Constant is the deviation of a system from being a harmonic oscillator which is related to the vibrational energy levels of diatomic molecule.
How to calculate Maximum Vibrational Quantum Number?
The Maximum vibrational quantum number formula is defined as the maximum scalar quantum value that defines the energy state of a harmonic or approximately harmonic vibrating diatomic molecule is calculated using Max Vibrational Number = (Vibrational Wavenumber/(2*Anharmonicity Constant*Vibrational Wavenumber))-1/2. To calculate Maximum Vibrational Quantum Number, you need Vibrational Wavenumber (ω') & Anharmonicity Constant (xe). With our tool, you need to enter the respective value for Vibrational Wavenumber & Anharmonicity Constant 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 Max Vibrational Number?
In this formula, Max Vibrational Number uses Vibrational Wavenumber & Anharmonicity Constant. We can use 2 other way(s) to calculate the same, which is/are as follows -
  • Max Vibrational Number = ((Vibrational Wavenumber)^2)/(4*Vibrational Wavenumber*Vibrational Energy*Anharmonicity Constant)
  • Max Vibrational Number = ((Vibrational Wavenumber)^2)/(4*Vibrational Wavenumber*Vibrational Energy*Anharmonicity Constant)
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