For nanocrystalline materials, the band gap energy of a sample is dependent on t

Question

For nanocrystalline materials, the band gap energy of a sample is dependent on the particle size of the nanocrystals. (See parts F & G of the Chemical Foundations section.) The energy is directly proportional to 1/size2.

Suppose you measure the band gap energy of several nanocrystalline samples that each have a known particle size and produce the following plot of energy vs. 1/size2, with a best-fit line for the data. ?

For a nanocrystalline sample containing particles of unknown size, you measure a band gap energy of 3.75×10-19 J. Use the plot and the equation of the best-fit line to calculate the particle size (in nm) of this unknown sample. Be sure to show your calculation. (Note: “E” in the plot represents “×10^”.)

Explanation / Answer

It is given that equation of the best fit line is:

y = (4.29 E-19) x + (3.04 E-19)

y represents the Energy in J

and x is (1/size^2) where size is in nanometer.

We have to find the size when energy (y) = 3.75 E -19

Putting the value of y in the equation of straight line, we get

3.75 E-19 = 4.29 E-19 x + 3.04 E-19

=> x = 0.1655 = (1 / Size^2)

=> Size = 2.46 nm

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