Problem 86 - Chemistry The Molecular Nature of Matter

Question

An astronomy web site states that neutron stars have a density of \(1.00 \times 10^{8}\) tons per cubic centimeter. The site does not specify whether "tons" means metric tons (1 metric ton \(=1000 \mathrm{~kg}\) ) or English tons ( 1 English ton \(=2000\) pounds). How many grams would one teaspoon of a neutron star weigh if the density were in metric tons per \(\mathrm{cm}^{3}\) ? How many grams would the teaspoon weigh if the density were in English tons per \(\mathrm{cm}^{3}\) ? (One teaspoon is defined as \(5.00 \mathrm{~mL} .\) )

Step-by-Step Solution

Answer

One teaspoon of neutron star material would weigh 5.00 x 10^14 grams if the density were in metric tons per cubic centimeter, or 4.53592437 x 10^14 grams if the density were in English tons per cubic centimeter.

Step 1 - Calculate the volume of a teaspoon in cubic centimeters

The volume of one teaspoon corresponds to 5.00 mL and 1 mL is equivalent to 1 cm^3. Therefore, the volume in cubic centimeters is directly equal to the volume in milliliters: Volume of a teaspoon = 5.00 mL = 5.00 cm^3.

Step 2 - Convert the density from metric tons to grams per cubic centimeter

To convert the density from metric tons per cubic centimeter to grams per cubic centimeter, use the conversion 1 metric ton = 1,000,000 grams. Thus, the density in grams per cubic centimeter is: Density in metric tons = 1.00 x 10^8 tons/cm^3 = 1.00 x 10^8 tons/cm^3 x 1,000,000 g/ton = 1.00 x 10^14 g/cm^3.

Step 3 - Calculate the weight in grams using the metric ton density

Multiplying the density in grams per cubic centimeter by the volume in cubic centimeters gives the weight in grams: Weight (metric) = Density (metric) x Volume = 1.00 x 10^14 g/cm^3 x 5.00 cm^3 = 5.00 x 10^14 g.

Step 4 - Convert the density from English tons to grams per cubic centimeter

To convert English tons to grams, use the conversion 1 English ton = 2,000 pounds and 1 pound = 453.59237 grams. The density in grams per cubic centimeter is: Density in English tons = 1.00 x 10^8 tons/cm^3 = 1.00 x 10^8 tons/cm^3 x 2,000 lb/ton x 453.59237 g/lb = 9.07184874 x 10^13 g/cm^3.

Step 5 - Calculate the weight in grams using the English ton density

Now multiply the converted density in grams per cubic centimeter by the teaspoon's volume: Weight (English) = Density (English) x Volume = 9.07184874 x 10^13 g/cm^3 x 5.00 cm^3 = 4.53592437 x 10^14 g.

Key Concepts

Density Conversion

When dealing with concepts in physics or chemistry, understanding the concept of density conversion is essential. Density, which is the mass per unit volume of a substance, can be measured in various units depending on the context. In the case of the neutron star exercise, the density was initially given in tons per cubic centimeter. However, to make the value more comprehensible (considering that scientific measurements are often expressed in the metric system), a conversion to grams per cubic centimeter was required.

Since 1 metric ton equals 1,000,000 grams, the conversion involves multiplying the given density by this factor. Such conversions are necessary to align with the standard units of mass used in science, which are kilograms and grams. The conversion enables us to perform calculations and comparisons more conveniently and opens the door for better visualization of the extreme densities found in objects like neutron stars.

Mass-Volume Relationship

Understanding the mass-volume relationship is crucial when studying the properties of matter. This relationship allows us to calculate the mass of a substance given its volume and density. It is a direct application of the density formula: \( \text{Density} = \frac{\text{Mass}}{\text{Volume}} \).

To find the mass, simply rearrange the formula to solve for mass: \( \text{Mass} = \text{Density} \times \text{Volume} \). If you have the density of a substance (like the neutron star matter) and a certain volume (like the teaspoon in our exercise), multiplying these two gives you the mass of the substance in that volume. This relationship is essential in many scientific computations and engineering applications where you need to predict the weight or amount of a material and ensures accuracy in disciplines that require precise quantitative analysis. It's also a foundation for understanding how matter is structured in space, giving us a glimpse into the fascinating world of astronomical objects.

Metric and English Units Conversion

In scientific practice, the need to convert between metric and English units arises frequently, especially when collaborating across countries that use different measurement systems. As in our neutron star example, English tons had to be converted into grams to maintain consistency with the metric system. To perform this conversion, we use two steps: first converting English tons to pounds, as 1 English ton is equivalent to 2,000 pounds; and then converting pounds to grams, where 1 pound equals 453.59237 grams.

Performing these conversions requires an understanding of both the metric and English units, as well as the conversion factors that relate them. These factors are crucial for scientists, engineers, and students alike, enabling them to communicate and calculate in a universally understood manner. It's also an excellent example of how interconnected our world is, and the importance of having standards in scientific measurement to facilitate collaboration and knowledge sharing.