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Chapter 19: Problem 61

Find the mass in kilograms of 3.23 1024 atoms of aluminum, which has a molar mass of 26.98 g/mol.

Short Answer

Expert verified
The mass is 0.144643 kg.

Step by step solution

01

Understand the given information

We are given 3.23 脳 10^{24} atoms of aluminum and need to convert them to mass in kilograms. The molar mass of aluminum is 26.98 g/mol.
02

Convert atoms to moles

First, we convert the number of atoms to moles using Avogadro's number, which is 6.022 脳 10^{23} atoms/mol. Calculate the number of moles using the formula: \[ ext{Number of moles} = \frac{3.23 imes 10^{24} ext{ atoms}}{6.022 imes 10^{23} ext{ atoms/mol}} \]This will give us the number of moles of aluminum.
03

Calculate the number of moles

Calculate the moles of aluminum: \[ ext{Number of moles} = \frac{3.23 imes 10^{24} }{6.022 imes 10^{23}} = 5.36 ext{ mol} \]
04

Convert moles to grams

Use the molar mass of aluminum to convert moles of aluminum to grams. The molar mass is 26.98 g/mol. Calculate the mass in grams: \[ ext{Mass (g)} = 5.36 ext{ mol} imes 26.98 ext{ g/mol} \]This multiplication gives the mass in grams.
05

Calculate the mass in grams

Calculate the mass in grams:\[ ext{Mass (g)} = 5.36 imes 26.98 = 144.643 ext{ g} \]
06

Convert grams to kilograms

Finally, convert the mass in grams to kilograms by dividing by 1000:\[ ext{Mass (kg)} = \frac{144.643}{1000} = 0.144643 ext{ kg} \]

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Key Concepts

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

Avogadro's Number
Understanding Avogadro's Number is essential in chemistry. It is the constant used to define the number of particles, usually atoms or molecules, in one mole of a substance. Avogadro's Number is
  • approximately 6.022 脳 10^{23} particles/mol
This massive number helps us bridge the world of atoms, which are incredibly tiny, with macroscopic quantities of material that we can easily measure.
When dealing with atoms of a substance, converting their count to moles using Avogadro's Number facilitates operations and comparisons in chemical calculations.
Molar Mass
Molar mass is a fundamental concept that links the mass of a substance to the amount of substance in moles. It is defined as the mass of one mole of a given substance, measured in grams per mole (g/mol).
In the context of aluminum, as seen in the exercise, its molar mass is 26.98 g/mol. This value tells us that one mole of aluminum atoms has a mass of 26.98 grams.
The molar mass derived from the atomic or molecular mass unit allows us to easily transition from the microscopic to the macroscopic world in chemical analysis.
Unit Conversion
Unit conversion is a vital process in chemistry that ensures consistency and accuracy in calculations. In this exercise, converting between units of atoms, moles, grams, and kilograms was crucial. Here鈥檚 how it鈥檚 done step-by-step:
  • First, convert atoms to moles using Avogadro's Number.
  • Next, convert moles to grams using the molar mass.
  • Finally, convert grams to kilograms, as
    • 1 kilogram = 1000 grams
Navigating these conversions accurately helps in translating calculations into meaningful, real-world measurements.
Atoms to Moles
Converting atoms to moles is a crucial calculation that helps us work with amounts of a substance on a manageable scale. Starting with a given number of atoms, such as the 3.23 脳 10^{24} atoms of aluminum, we can determine the amount in moles by dividing the number of atoms by Avogadro's Number.
This is often the first step in a series of conversions needed for practical applications, such as finding the mass of a substance. This process underscores the importance of Avogadro's constant, ensuring quantifiable and precise chemical calculations.

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Most popular questions from this chapter

Air that initially occupies 0.280 m3 at a gauge pressure of 103.0 kPa is expanded isothermally to a pressure of 101.3 kPa and then cooled at constant pressure until it reaches its initial volume. It then returns to its initial pressure in a constant-volume process. Compute the net work done by the air. (Gauge pressure is the difference between the actual pressure and atmospheric pressure.)

Suppose 2.80 mol of an ideal diatomic gas, with molecular rotation but not oscillation, experienced a temperature increase of 45.0 K under constant-pressure conditions.What are (a) the energy transferred as heat Q, (b) the change Eint in internal energy of the gas, (c) the work W done by the gas, and (d) the change K in the total translational kinetic energy of the gas?

Opening champagne. In a bottle of champagne, the pocket of gas (primarily carbon dioxide) between the liquid and the cork is at pressure of \(p_{i}=4.00 \mathrm{~atm}\). When the cork is pulled from the bottle, the gas undergoes an adiabatic expansion until its pressure matches the ambient air pressure of \(1.00\) atm. Assume that the ratio of the molar specific heats is \(y=\frac{4}{3}\). If the gas has initial temperature \(T_{i}=5.00^{\circ} \mathrm{C}\), what is its temperature at the end of the adiabatic expansion?

We give 90 J as heat to a diatomic gas, which then expands at constant pressure.The gas molecules rotate but do not oscillate. By how much does the internal energy of the gas increase?

The volume of an ideal gas is adiabatically reduced from 350 L to 130 L. The initial pressure and temperature are 2.00 atm and 380 K.The final pressure is 8.00 atm. (a) Is the gas monatomic, diatomic, or polyatomic? (b) What is the final temperature? (c) How many moles are in the gas?
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