Problem 33 - 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 an Avogadro's number of people is approximately \(4.37 \times 10^{25} \mathrm{~kg}\), which is about 7.31 times greater than the mass of Earth (\(5.98 \times 10^{24} \mathrm{~kg}\)).

Step 1: (Step 1: Convert mass from pounds to kilograms)

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

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

Now that we know the mass of one person, we can find the mass of Avogadro's number (\(N_A = 6.022 \times 10^{23}\)) of people by multiplying it by the mass of one person: Total mass \(= N_A \times\) mass per person \(= (6.022 \times 10^{23}) \times 72.57 \mathrm{~kg} \)

Step 3: (Step 3: Calculate the total mass in standard form)

To make it easier to compare the total mass with the mass of Earth, we can express the total mass in standard form: Total mass \(= (6.022 \times 10^{23}) \times 72.57 \mathrm{~kg} = 4.37 \times 10^{25} \mathrm{~kg} \)

Step 4: (Step 4: Compare the total mass with the mass of Earth)

Finally, let's compare the total mass of Avogadro's number of people with the mass of Earth (\(5.98 \times 10^{24} \mathrm{~kg}\)): The mass of an Avogadro's number of people is about \(4.37 \times 10^{25} \mathrm{~kg}\), and Earth's mass is \(5.98 \times 10^{24} \mathrm{~kg}\). To find how many times the mass of an Avogadro's number of people is greater than the mass of Earth, we can divide the total mass by the mass of Earth: \(\frac{4.37 \times 10^{25} \mathrm{~kg}}{5.98 \times 10^{24} \mathrm{~kg}} \approx 7.31\) So, the mass of an Avogadro's number of people is about 7.31 times greater than the mass of Earth.