Problem 33 - Chemistry: The Central Science in SI Units, Global Edition

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

The mass of an Avogadro's number of people, considering an average mass of 160 lb (72.57 kg), is approximately \(4.37 \times 10^{25} \mathrm{kg}\). This is about 7.31 times greater than the mass of Earth (\(5.98 \times 10^{24} \mathrm{kg}\)).

Step 1: Convert pounds to kilograms

To convert the average mass of a person from pounds to kilograms, we use the following conversion factor: 1 pound = 0.453592 kg So, the average mass of a person in kilograms is: \(160 \mathrm{lb} \times 0.453592 \frac{\mathrm{kg}}{\mathrm{lb}} \approx 72.57 \mathrm{kg}\)

Step 2: Calculate the mass of Avogadro's number of people

Now that we have established the average mass of a person in kilograms, we can multiply it by Avogadro's number (6.022 x 10^23) to find the mass of an Avogadro's number of people. To do this, simply use the formula: Mass = (average mass in kg) x (Avogadro's number) Mass ≈ \(72.57 \mathrm{kg} \times 6.022 \times 10^{23} = 4.37 \times 10^{25} \mathrm{kg}\) So, the mass of Avogadro's number of people is approximately \(4.37 \times 10^{25} \mathrm{kg}\).

Step 3: Compare with the mass of Earth

Now that we know the mass of Avogadro's number of people, we can compare it with the mass of Earth, which is given as \(5.98 \times 10^{24} \mathrm{kg}\). To find the ratio of the calculated mass to the mass of Earth, we can divide the mass of Avogadro's number of people by the mass of Earth: Ratio = \(\frac{4.37 \times 10^{25} \mathrm{kg}}{5.98 \times 10^{24} \mathrm{kg}} \approx 7.31\) This result means that Avogadro's number of people would have a mass approximately 7.31 times greater than the mass of Earth.

Key Concepts

Mass Conversion

When working with masses, converting between different units is often necessary. It's crucial to know the conversion factor between pounds (lb) and kilograms (kg) as they are both commonly used units of mass.
To convert a mass from pounds to kilograms, you use the conversion factor:

• 1 pound (lb) = 0.453592 kilograms (kg)

This means that each pound is equivalent to about 0.454 kg.
For example, if the average mass of a person is 160 lb, the mass in kilograms is calculated by multiplying 160 by the conversion factor, giving us approximately 72.57 kg. This conversion is often necessary to perform further calculations involving large numbers or comparing with values in kilograms.

Pounds to Kilograms

Understanding how to convert from pounds to kilograms is essential when dealing with scientific calculations or international contexts where metric units are preferred.
For our specific conversion:

• Start by noting the given weight in pounds, which is 160 lb for the average person.
• Use the conversion factor 0.453592 kg per lb.

Multiply 160 lb by 0.453592 kg/lb to get approximately 72.57 kg.
Converting weights to kilograms keeps calculations consistent, especially when multiplying by Avogadro's number or when comparing mass on a global scale with universal measurements.

Comparison with Earth's Mass

Once we have calculated the total mass, it becomes fascinating to compare it with Earth's mass. The Earth, a massive planetary body, has an enormous mass, approximately \(5.98 \times 10^{24}\) kg.
The idea of comparing the mass of Avogadro's number of people showcases how incredibly large Avogadro's number is.

• The total mass of Avogadro's number of people, calculated as \(4.37 \times 10^{25}\) kg, exceeds Earth's mass.
• When you divide this mass by Earth's mass, you obtain a ratio of about 7.31.

This means that collectively, these theoretical people weigh more than seven times that of Earth. This calculation puts into perspective the sheer volume involved when using Avogadro's number, making it a remarkable demonstration of scale in scientific terms.