How does the quantization of energy reconcile the apparent contradiction in the Planck equation E = hv, where energy peaks at a specific frequency rather than continually increasing?

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This is a tricky question, and there are several levels to it.

First of all, the Planck equation is intended to describe the energy of individual particles at rest in a vacuum (the same as E = mc^2 describes the correlation between a particle's mass and its resting energy). The total energy can change and there are many—more complicated—equations to calculate those values.

A group of particles will behave in a more advanced way, and this can disrupt the equation significantly, because there is not an exact- and equal energy between each particle.

Second of all, the Planck equation is the result of energy observation, not energy input. It is used primarily to calculate the total energy of a particle that is giving off measurable light/radiation. What you are asking is the opposite—higher and lower frequency input give different amounts of energy to a body, and that is not a linear relationship. There is a specific, harmonic frequency for every particle or item, and the closer you get to that frequency the more efficiently the energy will transfer into it.

When thinking about quantized energy units, the different frequencies are essentially different sized packages, or quanta. A particle is much more efficient at taking in energy closer to its harmonic frequency: this is comparable to handing an individual a stack of papers that is taller than their head—they won't be able to carry all of it at once, so much of it gets cast off. This is why the energy you input at higher frequencies doesn't translate properly to the energy of a particle.

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