Why Do Different Elements Have Unique Spectral Lines?

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Different elements exhibit unique spectral lines due to the distinct energy levels of their electrons, which are influenced by the number of protons in the nucleus, represented by the variable Z. When electrons transition between energy levels, they emit photons at specific frequencies that correspond to these energy differences. The variation in the number of protons affects the electrostatic forces acting on the electrons, resulting in different energy level spacings for each element. Consequently, an electron jumping from a high orbital to a lower one will release energy at a frequency unique to that element. This fundamental principle explains why each element has a characteristic spectral signature.
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why do different spectral lines appear for the excitation of different elements? I know that electrons jump to lower energy levels gives off photons of distinct frequencies but why characterizes them to be a certain frequency for each element? why doesn't an electron in a high orbital (say n=2) give off the same energy when they jump to the n=1 energy level for all elements? why is there a difference in frequency depending on the element?
 
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There is a Z in the formula for the energy of a given level, like n=1 an n=2. Z is the number of protons in the nucleus. So the frequency depends on the element.
 
Thread 'Variable mass system : water sprayed into a moving container'

Thread 'Variable mass system : water sprayed into a moving container'

Starting with the mass considerations #m(t)# is mass of water #M_{c}# mass of container and #M(t)# mass of total system $$M(t) = M_{C} + m(t)$$ $$\Rightarrow \frac{dM(t)}{dt} = \frac{dm(t)}{dt}$$ $$P_i = Mv + u \, dm$$ $$P_f = (M + dm)(v + dv)$$ $$\Delta P = M \, dv + (v - u) \, dm$$ $$F = \frac{dP}{dt} = M \frac{dv}{dt} + (v - u) \frac{dm}{dt}$$ $$F = u \frac{dm}{dt} = \rho A u^2$$ from conservation of momentum , the cannon recoils with the same force which it applies. $$\quad \frac{dm}{dt}...
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