Question

A solid circular cone made of pure platinum \((d=\) \(\left.21.45 \mathrm{~g} / \mathrm{cm}^{3}\right)\) has a diameter of \(2.75 \mathrm{~cm}\) and a height of 3.00 inches. (a) How many moles of platinum are there in the cone? (b) How many electrons are there in the cone?

Short answer

Answer: There are approximately 1.435 moles of platinum and 6.735 × 10²⁵ electrons in the cone.

Step-by-step solution

Convert height to centimeters

Since the diameter is given in centimeters, we need to convert the height from inches to centimeters. 1 inch is equal to 2.54 cm. Height in cm = 3.00 inches × 2.54 cm/inch ≈ 7.62 cm

Find the volume of the cone

The formula to find the volume (V) of a cone is: V = (1/3)πr²h, where r is the radius, and h is the height. The diameter is given as 2.75 cm, so the radius is half of that: r = 2.75/2 = 1.375 cm Volume = (1/3)π(1.375 cm)²(7.62 cm) ≈ 13.051 cm³

Find the mass of the cone

Given the density (d) of pure platinum is 21.45 g/cm³, we can find the mass (m) of the platinum cone using the formula: m = d × V Mass = 21.45 g/cm³ × 13.051 cm³ ≈ 280.060 g

Calculate the number of moles

To find the number of moles (n) of platinum, use the formula n = m / M, where M is the molar mass of platinum. The molar mass of platinum is approximately 195.08 g/mol. Number of Moles = 280.060 g / 195.08 g/mol ≈ 1.435 mol (a) There are approximately 1.435 moles of platinum in the cone.

Calculate the number of electrons

Using Avogadro's constant (6.022 × 10²³ particles/mol), multiply the number of moles by the Avogadro's constant to find the number of platinum atoms. Number of Platinum Atoms = 1.435 mol × 6.022 × 10²³ atoms/mol ≈ 8.635 × 10²³ atoms Each platinum atom contains 78 electrons. To find the total number of electrons in all the platinum atoms, multiply the number of platinum atoms by the number of electrons per atom. Number of Electrons = 8.635 × 10²³ atoms × 78 electrons/atom ≈ 6.735 × 10²⁵ electrons (b)There are approximately 6.735 × 10²⁵ electrons in the cone.