Chapter 2: Q157CP (page 88)

In a sample of any metal, spherical atoms pack closely together, but the space between them means that the density of the sample is less than that of the atoms themselves. Iridium (Ir) is one of the densest elements: 22.56 g/cm³. The atomic mass of Ir is 192.22 amu, and the mass of the nucleus is 192.18 amu. Determine the density (in g/cm³) of (a) an Ir atom and (b) an Ir nucleus. (c) How many Ir atoms placed in a row would extend 1.00 cm [radius of Ir atom = 1.36 Å; the radius of Ir nucleus = 1.5 femtometers (fm); V of a sphere =4.3πr³ ]?

Short Answer

Expert verified

The density of an Ir atom is $30.40 g / {cm}^{3}$
The density of an Ir nucleus is $2.26 \times 10^{16} g / {cm}^{3}$
The number of Ir atoms is $3.68 \times 10^{7} lr atoms$

Step by step solution

Calculating the density of an Ir atom

The volume of the atom is calculated using the given formula,

$V = \frac{4}{3} π r^{3} = \frac{4}{3} π {(1.36 \dot{\dot{A} \times \frac{1 \times 10^{‐ 10} m}{1 \dot{A}}} \times \frac{100 c m}{1 m})}^{3} V = 1.05 \times 10^{‐ 23} {cm}^{3}$

On solving for density,

$d = \frac{mass}{volume} = \frac{192.22 amu}{1.05 \times 10^{‐ 23} {cm}^{3}} \times \frac{1 g}{6.022 \times 10^{23} amu} d = 30.40 g / {cm}^{3}$

Calculating the density of an Ir nucleus

On solving the volume of the atom,

$V = \frac{4}{3} {πr}^{3} = \frac{4}{3} π {(1.5 fm \times \frac{1 \times 10^{‐ 15} m}{1 fm} \times \frac{100 cm}{1 m})}^{3} V = 1.41 \times 10^{‐ 38} {cm}^{3}$

On solving for density,

$d = \frac{192.18 amu}{1.41 \times 10^{‐ 38} {cm}^{3}} \times \frac{1 g}{6.022 \times 10^{23} amu} d = 2.26 \times 10^{16} g / {cm}^{3}$

Calculating the number of Ir atoms

For finding the diameter,

$diameter = 2 r = 2 (1.36 \dot{A)} = 2.72 \dot{A} 2.72 \dot{A} = 2.72 \dot{A} \times \frac{1 \times 10^{‐} m}{1 \dot{A}} \times \frac{100 cm}{1 m} = 2.72 \times 10^{‐ 8} cm$

Ir atoms in 1 cm would be

$\frac{1}{2.7 \times 10^{‐ 8}} = 3.68 \times 10^{7} lr atoms$

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Short Answer

Step by step solution

Calculating the density of an Ir atom

Calculating the density of an Ir nucleus

Calculating the number of Ir atoms

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