Problem 31 - Chemistry The Central Science

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 Avogadro's number of people is \(4.367 \times 10^{25}\) kilograms, which is approximately 7.3 times the mass of Earth.

Step 1: Convert mass of one person from pounds to kilograms

To convert the mass of one person from pounds to kilograms, we will use the conversion factor that 1 pound is equivalent to 0.453592 kg. So, \(1 \mathrm{lb} = 0.453592 \mathrm{kg}\) Now, we will convert the given average mass of a person (160 lb) into kilograms by multiplying it by the conversion factor: \(160 \mathrm{lb} * 0.453592 \frac{\mathrm{kg}}{\mathrm{lb}}\)

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

To calculate the mass of Avogadro's number of people, we need to multiply the mass of one person (in kilograms) by Avogadro's number (6.022 x 10^23). The mass of one person (in kg) is: \(160 \mathrm{lb} * 0.453592 \frac{\mathrm{kg}}{\mathrm{lb}} = 72.57472 \mathrm{kg}\) Now, multiply this mass by Avogadro's number: \(72.57472 \mathrm{kg}\ * 6.022\times10^{23}\)

Step 3: Compare the mass of Avogadro's number of people to the mass of Earth

Now we will compare the mass of Avogadro's number of people obtained in Step 2 with the mass of Earth (5.98 x 10^24 kg). First, calculate the mass of Avogadro's number of people: \(72.57472 \mathrm{kg}\ * 6.022\times10^{23} = 4.367 \times 10^{25} \mathrm{kg}\) Now, compare this mass to the mass of Earth: \(\frac{4.367 \times 10^{25} \mathrm{kg}}{5.98 \times 10^{24} \mathrm{kg}}\) Finally, we have the mass of Avogadro's number of people and how it compares to the mass of Earth. To complete the solution, let's perform these calculations: Mass of one person in kg: \(160 \mathrm{lb} * 0.453592 \frac{\mathrm{kg}}{\mathrm{lb}} = 72.57472 \mathrm{kg}\) Mass of Avogadro's number of people: \(72.57472 \mathrm{kg}\ * 6.022\times10^{23} = 4.367 \times 10^{25} \mathrm{kg}\) Comparison of the masses: \(\frac{4.367 \times 10^{25} \mathrm{kg}}{5.98 \times 10^{24} \mathrm{kg}} = 7.3\) So, the mass of Avogadro's number of people is \(4.367 \times 10^{25}\) kilograms, which is approximately 7.3 times the mass of Earth.