Question

A gold atom has a radius of \(145 \mathrm{pm} .\) If you could string gold atoms like beads on a thread, how many atoms would you need to have a necklace \(36 \mathrm{cm}\) long?

Short answer

Approximately 1,241,379,310 gold atoms are needed.

Step-by-step solution

Convert Length Units

First, we need to convert the length of the necklace from centimeters to picometers since the radius of the gold atom is given in picometers. There are \(1 imes 10^{12}\) picometers in a meter, so \(36\) cm equals \(36 imes 10^{-2}\) meters. We then convert this to picometers: \(36 imes 10^{-2} imes 10^{12} = 3.6 \times 10^{11}\) pm.

Diameter of a Gold Atom

Each gold atom is a sphere, and for the atoms to line up like a necklace, we consider the diameter. Since the radius of a gold atom is \(145\) pm, the diameter is \(2 \times 145 = 290\) pm.

Calculate Number of Atoms

To find how many atoms are needed to reach the total length of the necklace, divide the total length in picometers by the diameter of one gold atom. Therefore, the number of atoms is \(\frac{3.6 \times 10^{11} ext{ pm}}{290 ext{ pm}}\).

Perform Division

Carrying out the division \(\frac{3.6 \times 10^{11}}{290}\) gives approximately \(1.24 \times 10^9\), which means approximately \(1,241,379,310\) gold atoms are needed.

Key concepts

Gold Atom

A gold atom is the basic structural unit of gold, which is a metal known for its attractive color and luster. Gold atoms contain protons, neutrons, and electrons, similar to other atoms, with a nucleus composed of protons and neutrons surrounded by electron clouds. Gold is unique due to its heavy atomic weight and distinct properties. - Gold is a transition metal and is found in group 11 of the periodic table. - It has an atomic number of 79, meaning there are 79 protons in the nucleus. - The heavy nature of gold results from its high density, making it very compact at the atomic level. The exciting characteristics of gold atoms contribute to their widespread use in jewelry, electronics, and even as a monetary standard.

Atomic Radius

The atomic radius is a measure of the size of an atom, typically represented by the distance from the nucleus to the outer shell of electrons. The radius can vary depending on the atom's environment. - For a gold atom, the atomic radius is given as 145 picometers (pm). - The radius indicates how far the electron cloud extends around the nucleus. - When discussing atomic structures, it is essential to consider the radius along with other factors like atomic bonding and electron configuration. When visualizing how gold atoms link together, as in the problem of creating a necklace, it is the atomic radius that helps determine distances between atoms.

Unit Conversion

Unit conversion is a crucial step in solving chemistry problems, especially when measurements are given in different units. In this exercise, converting centimeters to picometers is essential for solving the problem correctly.- To convert from centimeters to meters, multiply by 0.01 (since 1 cm = 0.01 m).- Then, convert meters to picometers by multiplying by \(10^{12}\) (since 1 m = \(10^{12}\) pm).Using these conversions, you can convert any length measure to match the unit of atomic measurements, ensuring accuracy when calculating quantities involving atomic dimensions.

Chemistry Problem Solving

Effective chemistry problem-solving involves a systematic approach to breaking down problems into manageable steps. In the case of determining how many gold atoms fit on a necklace:1. **Understand the Problem**: Identify what the question is asking for and what data you have. Here, it's necessary to determine the number of atoms, knowing the necklace's length and the atomic radius.
2. **Convert Units Well**: Ensure all measurements are in compatible units. This step uses unit conversion, as seen in changing centimeters to picometers.
3. **Apply Relevant Formulas**: Calculate the diameter when dealing with atomic radii since atoms align with diameters touching. Use \(\text{Diameter} = 2 \times \text{Radius}\).
4. **Perform Calculation**: Divide the total distance by one atomic diameter to find the number of atoms.Repeated practice and a clear problem-solving methodology are pivotal to mastering chemistry problems like this example.