### Estimating the Atomic Radius of a Hypothetical MetalA hypothetical metal, which consists of a single crystal and is composed of one type of chemical element, has an FCC (Face-Centered Cubic) crystal structure. Given the following data:- **Density of the metal**: \(10 \, \text{g/cm}^3\)- **Atomic weight/mass of the metal**: \(100 \, \text{g/mol}\)**Objective**: Estimate the atomic radius of the metal.**Note**: Avogadro’s number \(N_A\) is \(6.022 \times 10^{23} \, \text{atoms/mol}\).#### Explanation of FCC Crystal StructureIn an FCC crystal structure:- Atoms are located at each of the corners and the centers of all the cube faces of the unit cell.- There are 4 atoms per unit cell (8 corners * 1/8 atom per corner + 6 faces * 1/2 atom per face).#### Calculating Molar Volume:1. **Density Equation**:\[ \text{Density} = \frac{\text{Mass}}{\text{Volume}} \]Hence, the volume occupied by one mole of the hypothetical metal can be calculated by:\[ \text{Volume} = \frac{\text{Mass}}{\text{Density}} = \frac{100 \, \text{g}}{10 \, \text{g/cm}^3} = 10 \, \text{cm}^3 \]2. **Volume of Metal Per Atom**:\[ \text{Volume per atom} = \frac{\text{Volume per mole}}{N_A} = \frac{10 \, \text{cm}^3}{6.022 \times 10^{23} \, \text{atoms/mol}} = 1.66 \times 10^{-23} \, \text{cm}^3/\text{atom} \]3. **Relation between Volume and Atomic Radius in FCC Structure**:The volume of the unit cell \( V_{\text{cell}} \) can also be related to the atomic radius \( r \) by:\[a^3 = \text{Volume of the unit cell}, \quad \text{where} \ a = 2\sqrt{2}r\]Since there are 4 atoms

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A hypothetical metal, which is a single crystal and made up of one chemical element, has an FCC crystal structure. Given the density of the metal is 10 g/cm³ and the atomic weight/mass of the metal is 100 g/mol, estimate the atomic radius of the metal. Note: Avogadro's number Na = 6.022×10²3 atoms/mol %3D

A hypothetical metal, which is a single crystal and made up of one chemical element, has an FCC crystal structure. Given the density of the metal is 10 g/cm³ and the atomic weight/mass of the metal is 100 g/mol, estimate the atomic radius of the metal. Note: Avogadro's number Na = 6.022×10²3 atoms/mol %3D

Chemistry

10th Edition

ISBN:9781305957404

Author:Steven S. Zumdahl, Susan A. Zumdahl, Donald J. DeCoste

Publisher:Steven S. Zumdahl, Susan A. Zumdahl, Donald J. DeCoste

Chapter1: Chemical Foundations

Section: Chapter Questions

Problem 1RQ: Define and explain the differences between the following terms. a. law and theory b. theory and...

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