Very small semiconductor crystals, composed of approximately 1000 to \(10,000\) atoms, are called quantum dots. Quantum dots made of the semiconductor Case 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 -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-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 -nm dot from multiple \(2.5-\)nm dots, how many 2.5 -nm dots would you need, and how many CdSe formula units would be left over, if any?

Step by step solution

01

Calculate the volume of a 2.5-nm quantum dot

First, calculate the volume of a 2.5-nm quantum dot, assuming it is a perfect sphere with a radius of 1.25 nm. Use the formula for the volume of a sphere: Volume = \( \frac{4}{3}\pi r^3 \) where r is the radius. Convert the radius from nm to cm to match the units in the density value: 1.25 nm = 1.25 x 10^{-7} cm Volume = \( \frac{4}{3}\pi (1.25 \times 10^{-7})^3 \) = \( 8.18 \times 10^{-21} cm^3 \)

02

Calculate the mass of a 2.5-nm quantum dot

Use the density formula to calculate the mass of a 2.5-nm quantum dot: Density = \( \frac{Mass}{Volume}\) Mass = Density x Volume Mass = \( 5.82 \frac{g}{cm^3} \times 8.18 \times 10^{-21} cm^3 \) = \( 4.76 \times 10^{-20} g \)

03

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

To find the number of Cd atoms in the quantum dot, first find the number of CdSe formula units in the dot. Number of CdSe formula units = \( \frac{Mass_{CdSe}}{Molar\_mass_{CdSe}} \times Avogadro's\_Number \) Molar mass of CdSe = 112.41 g/mol (Cd) + 78.96 g/mol (Se) = 191.37 g/mol Number of CdSe formula units = \( \frac{4.76 \times 10^{-20} g}{191.37 \frac{g}{mol}} \times 6.022 \times 10^{23} formula\_units/mol \) = 1.51 x 10^3 formula units. Since there is one Cd atom in one CdSe formula unit, there are 1510 Cd atoms in one 2.5-nm quantum dot.

04

Calculate the mass of a 6.5-nm quantum dot

First, calculate the volume of a 6.5-nm quantum dot, assuming it is a perfect sphere with a radius of 3.25 nm. Radius = 3.25 nm = 3.25 x 10^{-7} cm Volume = \( \frac{4}{3}\pi (3.25 \times 10^{-7})^3 \) = \( 1.44 \times 10^{-19} cm^3 \) Now, use the density formula to calculate the mass: Mass = Density x Volume Mass = \( 5.82 \frac{g}{cm^3} \times 1.44 \times 10^{-19} cm^3 \) = \( 8.38 \times 10^{-19} g \)

05

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

Find the number of CdSe formula units in the 6.5-nm quantum dot: Number of CdSe formula units = \( \frac{8.38 \times 10^{-19} g}{191.37 \frac{g}{mol}} \times 6.022 \times 10^{23} formula\_units/mol \) = 2.65 x 10^4 formula units. There are 26,500 Cd atoms in one 6.5-nm quantum dot.

06

Calculate the number of 2.5-nm quantum dots needed and the leftover CdSe formula units

First, determine how many times more CdSe formula units there are in a 6.5-nm quantum dot than in a 2.5-nm quantum dot: 26,500 (6.5-nm dot) ÷ 1510 (2.5-nm dot) = 17.55 Since you cannot have a fraction of a quantum dot, you would need 18 of the 2.5-nm quantum dots to create one 6.5-nm quantum dot. Now, calculate the leftover CdSe formula units when using 18 of the 2.5-nm quantum dots to make a 6.5-nm quantum dot: Number of CdSe formula units in 18 of the 2.5-nm quantum dots: 18 × 1510 = 27,180 Leftover CdSe formula units: 27,180 - 26,500 = 680. There would be 680 leftover CdSe formula units when creating a 6.5-nm quantum dot from 18 of the 2.5-nm quantum dots.

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