Very small semiconductor crystals, composed of approximately 1000 to 10,000 atoms, are called quantum dots. Quantum dots made of the semiconductor CdSe are now being used in electronic reader and tablet displays because they emit light efficiently and in multiple colors, depending on dot size. The density of CdSe is \(5.82 \mathrm{~g} / \mathrm{cm}^{3}\) (a) What is the mass of one \(2.5-\mathrm{nm}\) CdSe quantum dot? (b) CdSe quantum dots that are \(2.5 \mathrm{nm}\) in diameter emit blue light upon stimulation. Assuming that the dot is a perfect sphere and that the empty space in the dot can be neglected, calculate how many Cd atoms are in one quantum dot of this size. (c) What is the mass of one \(6.5-\mathrm{nm}\) CdSe quantum dot? (d) CdSe quantum dots that are \(6.5 \mathrm{nm}\) in diameter emit red light upon stimulation. Assuming that the dot is a perfect sphere, calculate how many Cd atoms are in one quantum dot of this size. (e) If you wanted to make one \(6.5-\mathrm{nm}\) dot from multiple \(2.5-\mathrm{nm}\) dots, how many \(2.5-\mathrm{nm}\) dots would you need, and how many CdSe formula units would be left over, if any?

Step by step solution

01

Calculate the mass of a 2.5 nm CdSe quantum dot

: To find the mass of a 2.5 nm quantum dot, we first need to find the volume of the sphere and then use the density of CdSe to calculate the mass. The formula for the volume of a sphere is: \[ V = \frac{4}{3} \pi r^3\] where V is the volume, and r is the radius of the sphere. Given that the diameter of the quantum dot is 2.5 nm, its radius r would be: \(r = \dfrac{2.5}{2} = 1.25 nm\) Now, convert the radius from nm to cm: \(r = 1.25 nm \times 10^{-7} cm/nm = 1.25 \times 10^{-7} cm\) Now, find the volume of the sphere: \[ V = \frac{4}{3} \pi (1.25 \times 10^{-7})^3 cm^3\] Finally, use the density of CdSe to calculate the mass of the quantum dot: \[m = \rho × V\] where m is the mass, \(\rho\) is the density of CdSe (5.82 g/cm³). Now, we can calculate the mass of the 2.5 nm CdSe quantum dot.

02

Calculate the number of Cd atoms in a 2.5 nm CdSe quantum dot

: To find the number of Cd atoms in a 2.5 nm quantum dot, we'll use Avogadro's number and the molar mass of CdSe compound. First, find the number of moles of CdSe in the quantum dot: \[n = \frac{m}{MM}\] where n is the number of moles, m is the mass of the CdSe quantum dot (calculated in step 1), and MM is the molar mass of CdSe. Then, use the number of moles and Avogadro's number (6.022 x 10²³) to find the total number of formula units in the quantum dot. Now, knowing that there is one Cd atom in each CdSe formula unit, we have found the number of Cd atoms in the 2.5 nm CdSe quantum dot.

03

Calculate the mass of a 6.5 nm CdSe quantum dot

: Repeat the process in step 1 with the new diameter of the CdSe quantum dot (6.5 nm): Given that the diameter of the quantum dot is 6.5 nm, its radius r would be: \(r = \dfrac{6.5}{2} = 3.25 nm\) Now, convert the radius from nm to cm: \(r = 3.25 nm \times 10^{-7} cm/nm = 3.25 \times 10^{-7} cm\) Now, find the volume of the sphere: \[ V = \frac{4}{3} \pi (3.25 \times 10^{-7})^3 cm^3\] Finally, use the density of CdSe to calculate the mass of the 6.5 nm quantum dot: Now, we can calculate the mass of the 6.5 nm CdSe quantum dot.

04

Calculate the number of Cd atoms in a 6.5 nm CdSe quantum dot

: Repeat the process in step 2 with the mass of the 6.5 nm CdSe quantum dot (calculated in step 3).

05

Determine the number of 2.5 nm quantum dots needed and the remaining CdSe formula units when making a 6.5 nm quantum dot

: First, we need to determine the volume ratio of a 6.5 nm CdSe quantum dot to a 2.5 nm CdSe quantum dot. This is because volume ratios will equal the number of smaller quantum dots needed to make a bigger one. Now, we have the volume ratio. We can determine how many 2.5 nm quantum dots are needed to make a 6.5 nm quantum dot by rounding down the volume ratio. After that, we can find the remaining volume that is not used from the 2.5 nm quantum dots when making the 6.5 nm quantum dot. Using this remaining volume, we can convert it back to mass using the density of CdSe. Then, find the number of moles and formula units in this remaining mass, similar to the processes in steps 1 and 2.

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