Problem 74 - Chemistry Principles and Practice

Question

I Perform the conversions needed to fill in the blanks. Use scientific notation where appropriate. Do the operations first without a calculator or spreadsheet, to check your understanding of SI prefixes. (a) \(45 \mathrm{~s}=\underline{\mathrm{ms}}=\) minutes (b) \(550 \mathrm{nm}=\mathrm{cm}=\mathrm{m}\) (c) \(4^{\circ} \mathrm{C}=\underline{\mathrm{K}}=\underline{ }^{\circ} \mathrm{F}\) (d) \(2.00 \mathrm{~L}=\) \(-\mathrm{cm}^{3}=\underline{\mathrm{m}^{3}}=\underline{\mathrm{qt}}\)

Step-by-Step Solution

Answer

(a) 45,000 ms = 0.75 min; (b) 5.5×10^-5 cm = 5.5×10^-7 m; (c) 277.15 K = 39.2°F; (d) 2000 cm³ = 0.002 m³ = 2.113 qt.

Step 1: Convert Seconds to Milliseconds

To convert 45 seconds to milliseconds, use the conversion factor: 1 second = 1,000 milliseconds.\[ 45\, \text{s} = 45 \times 1,000 = 45,000 \, \text{ms} \]

Step 2: Convert Seconds to Minutes

To convert 45 seconds to minutes, recall that 1 minute = 60 seconds.\[ 45\, \text{s} = \frac{45}{60} = 0.75 \, \text{minutes} \]

Step 3: Convert Nanometers to Centimeters

To convert 550 nanometers to centimeters, use the conversion factors: 1 nm = \(10^{-9}\) meters and 1 cm = \(10^{-2}\) meters.First, convert nanometers to meters:\[ 550\, \text{nm} = 550 \times 10^{-9} \, \text{m} = 5.5 \times 10^{-7} \, \text{m} \]Then convert to centimeters:\[ 5.5 \times 10^{-7} \, \text{m} = 5.5 \times 10^{-7} \div 10^{-2} \, \text{cm} = 5.5 \times 10^{-5} \, \text{cm} \]

Step 4: Convert Nanometers to Meters

Using the previous conversion, 550 nanometers is already calculated as:\[ 550\, \text{nm} = 5.5 \times 10^{-7} \, \text{m} \]

Step 5: Convert Celsius to Kelvin

To convert 4°C to Kelvin, add 273.15 to the Celsius temperature:\[ 4 \, ^\circ \text{C} = 4 + 273.15 = 277.15 \, \text{K} \]

Step 6: Convert Celsius to Fahrenheit

To convert 4°C to Fahrenheit, use the formula:\[ ^\circ \text{F} = \left( \frac{9}{5} \times ^\circ \text{C} \right) + 32 \]\[ 4 \, ^\circ \text{C} = \left( \frac{9}{5} \times 4 \right) + 32 = 39.2 \, ^\circ \text{F} \]

Step 7: Convert Liters to Cubic Centimeters

To convert 2.00 liters to cubic centimeters, use the conversion: 1 L = 1,000 cm³.\[ 2.00 \, \text{L} = 2.00 \times 1,000 = 2,000 \, \text{cm}^3 \]

Step 8: Convert Liters to Cubic Meters

To convert liters to cubic meters, use the conversion: 1,000 liters = 1 cubic meter.\[ 2.00 \, \text{L} = \frac{2.00}{1,000} = 0.002 \, \text{m}^3 \]

Step 9: Convert Liters to Quarts

To convert liters to quarts, use the conversion: 1 liter = 1.05669 quarts.\[ 2.00 \, \text{L} = 2.00 \times 1.05669 = 2.11338 \, \text{qt} \]

Key Concepts

Scientific Notation

Scientific notation is a method used to express very large or very small numbers in a compact form. This is particularly helpful in science and engineering, where such numbers frequently occur. The format is simple: a number is expressed as the product of a number between 1 and 10, and a power of 10.
For instance, the number 550 nanometers can be written in scientific notation as \(5.5 \times 10^{-7} \) meters. Here, 5.5 is the coefficient and \(10^{-7}\) is the power of ten, indicating that the decimal is shifted 7 places to the left.
This style of notation allows ease of calculation and clarity, especially when dealing with multiple scales of measurement in a scientific context. It helps you manage large calculations without losing accuracy or precision.

Celsius to Fahrenheit Conversion

Converting temperature between Celsius and Fahrenheit is a common requirement in science and daily life. To convert from Celsius to Fahrenheit, use the formula: \[ ^\circ \text{F} = \left( \frac{9}{5} \times ^\circ \text{C} \right) + 32 \]Suppose we want to convert 4 degrees Celsius to Fahrenheit. First, multiply the Celsius temperature by 9, then divide by 5. Finally, add 32 to this result to get the temperature in Fahrenheit.
Applying the formula, we do:\[ ^\circ \text{F} = \left( \frac{9}{5} \times 4 \right) + 32 = 39.2 ^\circ \text{F} \]
The conversion reflects the different ways Celsius and Fahrenheit scales measure temperature, with important applications in scientific experiments and weather reporting.

Liters to Cubic Meters

Converting liters to cubic meters is straightforward when you understand the relationship between these units of volume. The metric system follows a base 10 relationship, so understanding conversions involves recalling small number changes.
The conversion factor here is: 1 cubic meter \(= 1,000\) liters, which means that 1 liter equals \(0.001\) cubic meters.
Consequently, to convert 2 liters to cubic meters, divide by 1,000.
More precisely:\[ 2.00 \, \text{L} = \frac{2.00}{1,000} = 0.002 \, \text{m}^3 \]
It simplifies understanding that cubic meters are a larger unit than liters, designed to measure vast volumes compared to everyday uses of liters. These conversions are crucial in fields like architecture, chemistry, and various engineering applications.

Seconds to Milliseconds

When converting seconds to milliseconds, remember that the metric system uses a base 10 relationship, where 1 second equals 1,000 milliseconds. This conversion is essential in many areas such as music production, computer science, and physics, where precise time measurements are necessary.
Given 45 seconds and wanting to convert them into milliseconds, multiply by 1,000.
This calculation is direct and results in:\[ 45 \, \text{s} = 45 \times 1,000 = 45,000 \, \text{ms} \]
Understanding this conversion helps in breaking down larger time units into smaller, more manageable parts which can be useful in various scientific and technical fields.