Consider the following three transitions in a hydrogen atom: (A) ni=5, nf=2 ; (B) ni=7, nf=2 ;(C

Consider the following three transitions in a hydrogen atom:...

Key Concepts

Quantum Energy Levels

In quantum mechanics, electrons in atoms can only occupy discrete energy levels, and the transitions between these levels involve the absorption or emission of specific energy quanta. This concept is fundamental to understanding atomic spectra, as the energy difference between levels determines the properties of the emitted or absorbed photon.

Hydrogen Atomic Transitions

In the hydrogen atom, the energy levels are quantized and transitions occur when an electron moves from one level to a lower or higher level. These transitions are responsible for the characteristic spectral lines, with the energy difference between the initial and final states dictating the wavelength and frequency of the emitted or absorbed light.

Rydberg Formula

The Rydberg formula provides a mathematical relationship to calculate the wavelengths of the spectral lines in hydrogen and hydrogen-like species. It connects the inverse of the wavelength to the differences between the inverses of the squares of the principal quantum numbers, making it an essential tool for predicting and analyzing spectral transitions.

Photon Energy and Frequency Relationship

The energy of a photon is directly proportional to its frequency and inversely proportional to its wavelength, as expressed by the equation E = hf. This relationship is important because it means that transitions with larger energy differences will produce photons with higher frequencies and shorter wavelengths.

Inverse Relationship between Wavelength and Frequency

Wavelength and frequency are inversely related in electromagnetic radiation; as the wavelength decreases, the frequency increases, and vice versa. This concept is critical when ranking transitions based on the characteristics of the emitted photons since it allows one to determine the order of transitions by comparing energy differences.

Transcript

00:01 In this problem three transitions of hydrogen atom are given so in the first part we want to rank transitions in order of increasing wavelength of the emitted photon so we have the relation for wavelength of radiation emitted by a transition is lambda is equal to at c upon delta here h is a belongs constant c is speed of light so h and the c are constants so we have wavelength is inversely proportional to energy so well length of radiation is inversely proportional to changing change in the energies of two energy levels as we have delta e that is changing energies that is equal to energy of final state minus energy of initial set and we also have the formula for energy of any state is en is equal to minus 13 .6 electron volt upon n square now in the first transition we have initial state is 5 and final state is 2 so we have changing energy of the two states in this transition is equal to ef minus ei here ef is ef is e2 minus e initial is e5 now putting the value of of e2 and e5 here we have e2 is minus 13 .6 electron volt upon n sugar here and is two minus e5 is minus 13 .6 electron volt upon n square here and is five now by taking minus 13 .6 electron volt common from both the terms so we have left 1 upon 4 in the first term and 1 upon 25 in the second term so minus 13 .6 electron volt into lcm of 4 and 25 is 4 into 25 so we have 25 minus 4 that equal to minus 13 .6 electron volt into 21 upon 100 that is equal to minus 2 .856 electron volt so the absolute value will be equal to 2 .856 electron volt so the absolute value will be equal to 2 .856 electron volt so this is the energy in transition a now in transition b we have initial state is 7 and final state is 2 so energy of the emitted photon in this transition is equal to absolute value of ef minus ei here final state is e 7 e 2 minus initial state is e 7 now by putting the value of e 2 and e 7 here we have e 2 equal to minus 13 .6 electron volt upon n square here and is 2 and minus e 7 is minus e 7 is minus 13 .6 electron volt upon n square is 7 square so we have energy of photon in this transition is equal to minus 13 .6 electron volt into 1 upon 4 minus 1 upon 49 which is equal to minus…

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