Problem 56 - Introductory Chemistry Atoms First

Question

Jack is asked to determine the number of pennies in \(\$ 1.00\). Jill is asked to determine the number of liters in 1 gallon. Whose answer will be an exact number, and whose will be measured? Explain why.

Step-by-Step Solution

Answer

Jack's answer will be an exact number (100 pennies) because the conversion factor for dollars to pennies is exact. Jill's answer will be a measured value (approximately 3.78541 liters) because the conversion factor for gallons to liters is not an exact number.

Step 1: Identify the conversion factors for each task

For the first task, we know that there are 100 pennies in a dollar. Thus, the conversion factor is an exact value. In the second task, we have to convert gallons to liters. The conversion factor for this is approximately 3.78541 liters per gallon. Since this conversion factor is not an exact number, we can expect that the answer will be a measured value.

Step 2: Make the calculations for each task

Now that we have the conversion factors, we can proceed to calculate the answers for each task: - For Jack's task, we have: \( \$1.00 \times \frac{100 \text { pennies}}{\$1} = 100 \text { pennies} \) - For Jill's task, we have: \( 1 \text { gallon} \times \frac{3.78541 \text { liters}}{1 \text { gallon}} \approx 3.78541 \text { liters} \)

Step 3: Compare the exactness of the calculated answers

From our calculations above, we can see that: - Jack's answer (100 pennies) is an exact number, because the conversion factor for dollars to pennies is an exact value. - Jill's answer (approximately 3.78541 liters) is a measured value, because the conversion factor for gallons to liters is not an exact number.

Step 4: Conclusion

Jack's answer will be an exact number, while Jill's answer will be a measured value. This is because the conversion factor for dollars to pennies is exact, while the conversion factor for gallons to liters is not.

Key Concepts

Understanding Exact vs Measured Values

In the world of chemistry and in general scientific practice, distinguishing between exact and measured values is fundamental for precision and accuracy. Exact values are those that are known with complete certainty. They are often countable quantities or defined values, like the number of pennies in a dollar. These exact values don't change and are not subject to measurement error.

On the other hand, measured values are subject to the limits of the measuring equipment and the person performing the measurement. They are approximations and can vary slightly each time a measurement is taken. For example, while one gallon is technically the same volume anywhere in the world, the conversion to liters is not perfectly precise due to measurement limitations and conversion approximations. It's important to note that for precise scientific calculations, significant figures are used to reflect the accuracy of measured values.

Conversion Factors in Chemistry

Conversion factors are ratios that allow a quantity expressed in one unit to be converted into the equivalent quantity in another unit. They are absolutely essential in the field of chemistry where quantities need to be converted from one unit to another to perform calculations.

In the given exercises, conversion factors are used to convert monetary value to a countable number of pennies and volume from gallons to liters. When a conversion factor is an exact number, such as 100 pennies per dollar, it suggests that this conversion is absolute and does not vary. On the other hand, conversion factors like 3.78541 liters per gallon reflect a measured relationship that is determined by experiment or standard, which contains a level of uncertainty, hence the measured value.

Calculating Unit Conversions

The process of calculating unit conversions involves applying conversion factors to move from one unit of measurement to another. This is a critical skill for students in chemistry and other scientific fields. The method typically involves multiplication or division by the conversion factor, and it is essential to ensure that you are using the correct direction of the conversion factor to get the correct units in your answer.

Let's consider Jill's conversion from gallons to liters as an example of unit conversion in action. Here, Jill multiplies by the conversion factor of approximately 3.78541 liters per gallon. It's important for students to pay attention to the significant figures and realize that since the conversion factor is not exact, the final result should reflect the same level of precision as the least precise number used in the calculation.