A nanoparticle containing $6$ atoms can be modeled approximately as an Einstein solid of $18$ independent oscillators. The evenly spaced energy levels of each oscillator are $5{10}^{-21}J$ apart. Use $k=1\mathrm{.}4{10}^{-23}\frac{J}{K}.$

Part $1$

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$($a$)$ When the nanoparticle's energy is in the range $55{10}^{-21}J$ to $65{10}^{-21}J,$ what is the approximate temperature? $($In order to keep precision for calculating the heat capacity, give the result to the nearest tenth of a degree.$)$

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Part $2$

$($b$)$ When the nanoparticle's energy is in the range $85{10}^{-21}J$ to $95{10}^{-21}J,$ what is the approximate temperature? $($In order to keep precision for calculating the heat capacity, give the result to the nearest tenth of a degree.$)$

$$ K

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$40\%$ score reduction after attempt $3$

Part $3$

$($c$)$ When the nanoparticle's energy is in the range $55{10}^{-21}J$ to $95{10}^{-21}J,$ what is the approximate heat capacity per atom?