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Chapter 10: Problem 20

What is the mass in grams of each of the following? \begin{equation}\begin{array}{l}{\text { a. } 6.02 \times 10^{24} \text { atoms Bi }} \\ {\text { b. } 1.00 \times 10^{24} \text { atoms Mn }} \\\ {\text { c. } 3.40 \times 10^{22} \text { atoms He }} \\ {\text { d. } 1.50 \times 10^{15} \text { atoms } \mathrm{N}} \\ {\text { e. } 1.50 \times 10^{15} \text { atoms } U}\end{array}\end{equation}

Short Answer

Expert verified
(a) 2090 g Bi, (b) 91.18 g Mn, (c) 0.226 g He, (d) \(3.49 \times 10^{-8}\) g N, (e) \(5.92 \times 10^{-7}\) g U.

Step by step solution

01

Calculate Moles

To find the mass, we first need to calculate the number of moles for each element by using Avogadro's number, which is approximately \( 6.022 \times 10^{23} \) atoms/mol. For example, to find moles of Bi, use the formula \( \text{moles} = \frac{\text{number of atoms}}{6.022 \times 10^{23}} \). Repeat for each element.
02

Bi - Moles and Mass

Number of atoms of Bi = \( 6.02 \times 10^{24} \). Moles of Bi = \( \frac{6.02 \times 10^{24}}{6.022 \times 10^{23}} \approx 10 \) mol. Molar mass of Bi (Bismuth) is 209.0 g/mol. Mass = moles \( \times \) molar mass = \( 10 \times 209.0 = 2090 \) g.
03

Mn - Moles and Mass

Number of atoms of Mn = \( 1.00 \times 10^{24} \). Moles of Mn = \( \frac{1.00 \times 10^{24}}{6.022 \times 10^{23}} \approx 1.66 \) mol. Molar mass of Mn (Manganese) is 54.94 g/mol. Mass = moles \( \times \) molar mass = \( 1.66 \times 54.94 \approx 91.18 \) g.
04

He - Moles and Mass

Number of atoms of He = \( 3.40 \times 10^{22} \). Moles of He = \( \frac{3.40 \times 10^{22}}{6.022 \times 10^{23}} \approx 0.0564 \) mol. Molar mass of He (Helium) is 4.00 g/mol. Mass = moles \( \times \) molar mass = \( 0.0564 \times 4.00 \approx 0.226 \) g.
05

N - Moles and Mass

Number of atoms of N = \( 1.50 \times 10^{15} \). Moles of N = \( \frac{1.50 \times 10^{15}}{6.022 \times 10^{23}} \approx 2.49 \times 10^{-9} \) mol. Molar mass of N (Nitrogen) is 14.01 g/mol. Mass = moles \( \times \) molar mass = \( 2.49 \times 10^{-9} \times 14.01 \approx 3.49 \times 10^{-8} \) g.
06

U - Moles and Mass

Number of atoms of U = \( 1.50 \times 10^{15} \). Moles of U = \( \frac{1.50 \times 10^{15}}{6.022 \times 10^{23}} \approx 2.49 \times 10^{-9} \) mol. Molar mass of U (Uranium) is 238.03 g/mol. Mass = moles \( \times \) molar mass = \( 2.49 \times 10^{-9} \times 238.03 \approx 5.92 \times 10^{-7} \) g.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Avogadro's Number
Let's dive into the concept of Avogadro's number, a cornerstone in chemistry that links atoms to grams. This number is often denoted as \( 6.022 \, \times \, 10^{23} \) atoms/mol. Think of it as a bridge connecting the atomic world鈥攖iny, invisible particles鈥攖o amounts we can see and measure. For instance, when we say a mole of any element, it means it contains \( 6.022 \, \times \, 10^{23} \) atoms of that element. Here鈥檚 how it helps you:
  • Converting atoms to moles: Divide the number of atoms by Avogadro's number to find how many moles you have.
  • Understanding scale: Although elements vary in size, each mole contains exactly the same number of atoms.
This magical number helps chemists predict the amount of product in reactions based on the number of moles. By knowing how many moles of an element you have, you can find its mass, and this is essential in laboratory settings.
Molar Mass
Molar mass is a term you'll encounter frequently in chemistry. It refers to the mass of one mole of a substance, usually expressed in grams per mole (g/mol). Each element has a unique molar mass based on the mass of its atoms.

To calculate the molar mass:
  • Check the periodic table: Each element鈥檚 atomic weight equals its molar mass.
  • Units: It's always in g/mol, allowing you to easily convert from moles to grams.
Molar mass is critical because it allows you to calculate the mass of a given number of atoms. For example, Bismuth (Bi) has a molar mass of 209.0 g/mol, meaning if you have 1 mole of Bismuth, its mass will be 209.0 grams. Knowing the molar mass helps you balance chemical equations and predicts the yield of reactions.
Mass in Grams
Converting moles to mass in grams is a straightforward step using the molar mass. Once you know the number of moles you have, just multiply it by the molar mass of the element to get its mass in grams.

For example, if you have \( 1.66 \) moles of Manganese (Mn) with a molar mass of \( 54.94 \, \text{g/mol} \):
  • Calculate mass: \( \text{mass} = \text{moles} \times \text{molar mass} = 1.66 \, \times \, 54.94 \approx 91.18 \, \text{g} \).
By using this method, you can determine the mass of any quantity of an element if you know the moles and its molar mass. This ability to calculate mass in grams is crucial for preparing solutions, conducting reactions, and various laboratory applications.
Chemical Elements
Chemical elements are the building blocks of matter, classified in the periodic table. Each element has its unique properties and atomic arrangement, which define its chemical behavior.

There are several things to remember about elements:
  • Unique identity: Every element has a specific number of protons defining its identity (atomic number).
  • Variety of states: Elements can exist in various states鈥攕olid, liquid, or gas鈥攄epending on environmental conditions.
  • Role in chemistry: Understanding elements is crucial for predicting how substances react and combine.
For instance, understanding elements like Helium (He), Nitrogen (N), and Uranium (U) helps explain their roles in chemistry. Each has distinct characteristics, indicated by differences in parameters like molar mass and atomic structure. When solving problems in chemistry, knowledge of these elements and their properties allows for accurate computations and predictions.

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