Problem 33 - Chemistry The Central Science In Si Units

Question

What is the mass, in kilograms, of an Avogadro's number of people, if the average mass of a person is \(160 \mathrm{lb}\) ? How does this compare with the mass of Earth, \(5.98 \times 10^{24} \mathrm{~kg}\) ?

Step-by-Step Solution

Answer

An Avogadro's number of people has a mass of \(4.372 \times 10^{25} \text{ kg}\) which is about 7.31 times the mass of Earth.

Step 1: Convert Pounds to Kilograms

First, we need to convert the average mass of a person from pounds to kilograms. We know that 1 pound is approximately equal to 0.453592 kilograms. Therefore, to convert 160 pounds to kilograms, we use the formula:\[160 \text{ lb} \times 0.453592 \text{ kg/lb} = 72.57472 \text{ kg}.\]

Step 2: Determine Mass of Avogadro's Number of People

Avogadro's number is approximately \(6.022 \times 10^{23}\). To find the total mass of Avogadro's number of people, we multiply the mass of one person (in kilograms) by Avogadro's number:\[72.57472 \text{ kg/person} \times 6.022 \times 10^{23} \text{ persons}.\] Solving this gives \[4.372 \times 10^{25} \text{ kg}.\]

Step 3: Compare Mass of People and Earth

The mass of Earth is given as \(5.98 \times 10^{24} \text{ kg}\). We compare this with the total mass of the people calculated:\[4.372 \times 10^{25} \text{ kg} \div 5.98 \times 10^{24} \text{ kg} \] which approximates to 7.31. This shows that the mass of Avogadro's number of people is about 7.31 times the mass of Earth.

Key Concepts

Mass Conversion: Pounds to Kilograms

When converting units of mass, a common task is changing from pounds to kilograms. This is often necessary because the metric system, where kilograms are standard, is widely used in scientific contexts. To convert pounds to kilograms, we need to know the conversion factor between these two units.

One pound is approximately equal to 0.453592 kilograms. By multiplying the number of pounds by this factor, you can easily convert to kilograms. For instance, a person weighing 160 pounds would weigh about 72.57472 kilograms. This step involves a straightforward multiplication:

• 160 pounds × 0.453592 kg/pound = 72.57472 kg

This conversion is crucial when dealing with large quantities, as it ensures consistency with other scientific measurements, which often use the metric system.

Understanding Avogadro's Constant

Avogadro's constant is a fundamental number in chemistry and physics that represents the number of constituent particles, usually atoms or molecules, that are contained in one mole of a substance. It’s named after the scientist Amedeo Avogadro and is approximately equal to \(6.022 \times 10^{23}\).

This number is crucial for calculations involving large quantities of small entities. For example, when calculating the total mass of an Avogadro's number of people, each weighing 72.57472 kg, you use this constant to multiply the mass of one person by \(6.022 \times 10^{23}\). This results in a massive number, illustrating how Avogadro's constant functions to connect macroscopic measurements with a vast number of microscopic entities.

Comparison with Earth's Mass

In order to appreciate the enormity of Avogadro's number, it helps to compare the resultant mass to something tangible, like Earth's mass.

Earth's mass is approximately \(5.98 \times 10^{24} \text{ kg}\). When calculating the mass of an Avogadro's number of people, it results in \(4.372 \times 10^{25} \text{ kg}\).

• By comparing the two, you find that the mass of these people is about 7.31 times the mass of Earth.

This powerful comparison provides a tangible sense of scale, highlighting just how incredibly large Avogadro's number truly is. It's a number that transforms the abstract into something greatly substantial when applied to earthly measures.