Problem 68 - Chemistry The Molecular Science

Question

If you divide Avogadro's number of pennies among the nearly 300 million people in the United States, and if each person could count one penny each second every day of the year for eight hours per day, calculate how long it would take to count the pennies.

Step-by-Step Solution

Answer

It would take about 191 million years.

Step 1: Understanding the Problem

We need to find out the total number of pennies Avogadro's number represents and then determine how long it would take for each of the 300 million people in the U.S. to count their share when counting only 8 hours a day.

Step 2: Avogadro's Number Definition

Avogadro's number is defined as approximately \(6.022 \times 10^{23}\). This is the total number of pennies we have to distribute among the population.

Step 3: Division of Pennies

To divide \(6.022 \times 10^{23}\) pennies among 300 million people, calculate the number of pennies each person would receive: \(\frac{6.022 \times 10^{23}}{3 \times 10^8}\).

Step 4: Calculation of Pennies per Person

Perform the division: \(\frac{6.022 \times 10^{23}}{3 \times 10^8} = 2.0073 \times 10^{15}\). Each person would receive approximately \(2.0073 \times 10^{15}\) pennies.

Step 5: Rate of Counting Pennies

Each person can count 1 penny every second for 8 hours each day. Calculate the total seconds counted per day: \(8 \times 60 \times 60 = 28800\) seconds/day.

Step 6: Finding Total Days Needed

Each person needs to count \(2.0073 \times 10^{15}\) pennies. Calculate how many days it would take: \(\frac{2.0073 \times 10^{15}}{28800} \approx 6.97 \times 10^{10}\) days.

Step 7: Conversion to Years

Convert days to years knowing there are approximately 365.25 days per year: \(\frac{6.97 \times 10^{10}}{365.25} \approx 1.91 \times 10^{8}\) years.

Key Concepts

Counting Rate

When it comes to counting, understanding the rate is crucial. In the given problem, each person is tasked with counting one penny per second. This may seem slow, but over time, these pennies add up. It’s important to know how many seconds are in an hour and how many hours are spent counting each day.

• Each person counts for 8 hours a day.
• One hour equals 3,600 seconds.
• In total, each day provides 28,800 seconds for counting (8 hours × 3,600 seconds per hour).

What might surprise you is how quickly seconds multiply when extended over several hours. By calculating in seconds, we obtain a clearer understanding of the massive amount of time needed to accomplish this task.

Division of Large Numbers

Dividing large numbers can sometimes be intimidating, but with practice, it becomes a manageable task. The problem requires dividing Avogadro's number of pennies, which is an enormous figure, among the U.S. population. Consider Avogadro's number: approximately \[ 6.022 \times 10^{23} \]This represents the total pennies, and you need to share them among nearly 300 million people. Mathematically, this division is expressed as:\[ \frac{6.022 \times 10^{23}}{3 \times 10^8} = 2.0073 \times 10^{15} \]Here’s why breaking down such a division is helpful:- Simplifies large numbers into more manageable figures.- Helps estimate the scale of large quantities.Understanding these numbers is key to grasping the scale of distribution and further calculations needed, like the time it would take each person to count their share.

Time Conversion

Time conversion is a handy skill in solving real-world math problems. Once you’ve figured out the number of days each person would need to count their share of pennies, converting that time into years helps us better understand the duration. The formula converts days into years:\[ \text{Years} = \frac{\text{Total Days}}{365.25} \]Why use 365.25? It accounts for leap years, offering a slightly more accurate year estimate. From the problem:\[ \frac{6.97 \times 10^{10}}{365.25} \approx 1.91 \times 10^{8} \]By converting, you can express long periods in a more relatable unit. Using years instead of days offers perspective on how tasks of this magnitude truly span across centuries, making the enormity of such tasks more tangible to us.