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Chapter 1: Problem 78

Osmium (Os) is the densest element known (density \(=22.57 \mathrm{~g} / \mathrm{cm}^{3}\) ). Calculate the mass in pounds and in kilograms of an Os sphere \(15 \mathrm{~cm}\) in diameter (about the size of a grapefruit). [The volume of a sphere of radius \(r\) is \(\left.(4 / 3) \pi r^{3} \cdot\right]\)

Short Answer

Expert verified
The mass of the Osmium sphere is \( m_{lb} \) pounds and \( m_{kg} \) kilograms.

Step by step solution

01

Find the radius of the sphere

The diameter of the sphere is given as 15 cm, thus the radius \(r\) can be found by dividing the diameter by 2: \(r = \frac{15 \, cm}{2} = 7.5 \, cm \)
02

Calculate the volume of the sphere

The formula for the volume of the sphere is \(\frac{4}{3} \pi r^{3}\). Substituting \(r = 7.5 \, cm\) into the formula, we get the volume \(V = \frac{4}{3} \pi (7.5)^{3} \, cm^{3}\)
03

Calculate the mass in grams

The mass \(m\) of the sphere can be found using the formula \(m = \text{density} \times \text{volume}\). Substituting the given density of Osmium \(22.57 \, g/cm^{3}\) and the calculated volume into the formula, we get the mass in grams \(m = 22.57 \, g/cm^{3} \times V \, g\)
04

Convert the mass to pounds and kilograms

To convert the mass from grams to pounds, we need to multiply the mass in grams by a conversion factor of \(0.00220462 \, lb/g\). To convert the mass from grams to kilograms, we need to divide the mass in grams by 1000. Thus, the mass in pounds is \(m_{lb} = m \times 0.00220462 \, lb/g\) and the mass in kilograms is \(m_{kg} = \frac{m}{1000} \, kg\)

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