### Estimating the Atomic Radius of a Hypothetical Metal**Problem Statement:**A hypothetical metal, which is a single crystal and made up of one chemical element, has an FCC (Face-Centered Cubic) 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.**Given Data:**1. Density ($\rho$) of the metal: 10 g/cm³2. Atomic weight/mass (M) of the metal: 100 g/mol3. Avogadro's number ($N_A$): $6.022 \times 10^{23}$ atoms/mol**Solution:**To estimate the atomic radius, we use the following steps:1. **Determine the Number of Atoms per cm³:** The density equation for atoms in a unit volume of the metal can be expressed as: \[\rho = \frac{n \cdot M}{V\cdot N_A}\] Where $n$ is the number of atoms per unit cell and $V$ is the volume of the unit cell. Rearranging for $V$: \[V = \frac{n \cdot M}{\rho \cdot N_A}\] For an FCC structure, the number of atoms per unit cell $n$ is 4.2. **Calculate the Volume of the Unit Cell:** \[V = \frac{4 \cdot 100 \text{ g/mol}}{10 \text{ g/cm}^3 \cdot 6.022 \times 10^{23} \text{ atoms/mol}}\]3. **Convert the Volume to Appropriate Units:** Calculate $V$ to find the unit cell volume in cm³ and subsequently convert it to cubic angstroms (Å³) if necessary for precision in determining the atomic radius.4. **Determine the Atomic Radius:** The volume of a cubic unit cell $V$ can also be related to the lattice parameter $a$ (where $V = a^3$). For an FCC lattice, the relationship between the lattice parameter $a$ and the atomic radius $R$ is: \[a = \frac{4R}{\sqrt{2}}\]

FCC structure means face centered cubic arrangement, here total 4 atoms used to present in a

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

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

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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