Problem 16 - Chemistry Matter and Change

Question

Challenge Calculate the areas and densities. Report the answers in the correct units. a. the area of a rectangle with sides measuring \(3 \times 10^{1} \mathrm{cm}\) and \(3 \times 10^{-2} \mathrm{cm}\) b. the area of a rectangle with sides measuring \(1 \times 10^{3} \mathrm{cm}\) and \(5 \times 10^{-1} \mathrm{cm}\) c. the density of a substance having a mass of \(9 \times 10^{5} \mathrm{g}\) and a volume of \(3 \times 10^{-1} \mathrm{cm}^{3}\) d. the density of a substance having a mass of \(4 \times 10^{-3} \mathrm{g}\) and a volume of \(2 \times 10^{-2} \mathrm{cm}^{3}\)

Step-by-Step Solution

Answer

a. \(Area = 9 \times 10^{-1} \mathrm{cm}^2\) b. \(Area = 5 \times 10^{2} \mathrm{cm}^2\) c. \(Density = 3 \times 10^{6} \mathrm{g/cm}^3\) d. \(Density = 2 \times 10^{-1} \mathrm{g/cm}^3\)

Step 1: Calculate the area of the first rectangle

To find the area of a rectangle, we multiply the length times the width. The area of the rectangle in part (a) is: \[ Area = (3 \times 10^{1} \mathrm{cm})(3 \times 10^{-2} \mathrm{cm}) \]

Step 2: Calculate the area of the second rectangle

For the rectangle in part (b), do the same calculation: \[ Area = (1 \times 10^{3} \mathrm{cm})(5 \times 10^{-1} \mathrm{cm}) \]

Step 3: Calculate the density of the first substance

The density of a substance can be found using the formula: \[ Density = \frac{Mass}{Volume} \] We are given the mass and volume of the substance in part (c). Applying the formula for density: \[ Density = \frac{9 \times 10^{5} \mathrm{g}}{3 \times 10^{-1} \mathrm{cm}^{3}} \]

Step 4: Calculate the density of the second substance

Finally, for the substance in part (d), we again use the density formula: \[ Density = \frac{4 \times 10^{-3} \mathrm{g}}{2 \times 10^{-2} \mathrm{cm}^{3}} \] Now, we just need to simplify the calculations and report the answers in the correct units for area (square centimeters) and density (grams per cubic centimeter).

Key Concepts

Area Calculation

Whenever you need to find the area of a rectangle, the formula to use is quite simple: multiply the length by the width. - In part (a), the rectangle has sides of sizes given in scientific notation: \(3 \times 10^{1} \mathrm{cm}\) and \(3 \times 10^{-2} \mathrm{cm}\). You multiply these values to get the area in square centimeters. - In part (b), the side lengths are \(1 \times 10^{3} \mathrm{cm}\) and \(5 \times 10^{-1} \mathrm{cm}\). Again, multiply these to find the area.Breaking down these multiplications with scientific notation can be easier if you handle the coefficients and powers of ten separately or use a calculator capable of scientific notation operations. As a result, the area will be in square centimeters \((\mathrm{cm}^2)\). Remember that because you are multiplying centimeters by centimeters, the result is always in square centimeters, which is crucial when reporting the measurement correctly.

Density Calculation

Density is a property of matter that measures how much mass is contained in a given volume. To calculate density, use the formula:\[Density = \frac{Mass}{Volume}\]For part (c), you have a mass of \(9 \times 10^{5} \mathrm{g}\) and a volume of \(3 \times 10^{-1} \mathrm{cm}^{3}\). Plug these values into the formula to find the density.For part (d), the mass is \(4 \times 10^{-3} \mathrm{g}\) with a volume of \(2 \times 10^{-2} \mathrm{cm}^{3}\). Again, you utilize the same formula to find the density.It's essential to remember that when you are dealing with density calculations, the resulting unit will be grams per cubic centimeter \((\mathrm{g/cm}^3)\) because you're dividing mass measured in grams by volume in cubic centimeters.

Scientific Notation

Scientific notation is a way of writing very large or small numbers in a manageable form. It's expressed as the product of a number between 1 and 10 and a power of ten. This notation makes calculations simpler, as you mostly need to handle the coefficients and then add or subtract the exponents of ten depending on the operation.For example, the side of length \(3 \times 10^{1} \mathrm{cm}\) represents 30 centimeters, and \(3 \times 10^{-2} \mathrm{cm}\) is 0.03 centimeters. In calculations, you handle the numbers (3, 30 or 0.03) and the exponents separately.When multiplying or dividing numbers in scientific notation:- Multiply/divide the base numbers.- Add/subtract the exponents of ten accordingly.This approach helps in maintaining precision and makes it easier to read lengthy calculations.

Units of Measurement

Units are the standard quantities used to specify measurements. Understanding and using correct units of measurement is crucial, especially in scientific fields like chemistry.In calculations involving area of rectangles:- The sides were measured in centimeters, \(\mathrm{cm}\).- The results for area are in square centimeters, \(\mathrm{cm}^2\), because area calculations involve multiplying two lengths.For density calculations:- Mass is generally given in grams, \(\mathrm{g}\).- Volume is typically in cubic centimeters, \(\mathrm{cm}^{3}\).- The resulting density is expressed in grams per cubic centimeter, \(\mathrm{g/cm}^{3}\).It's important to be consistent with your units to ensure accurate results. Always double-check your calculations, especially when converting between different unit types, to maintain precision and correctness.