Problem 47 - Chemistry The Central Science

Question

Perform the following conversions: (a) 5.00 days to s, (b) 0.0550 mi to \(\mathrm{m}\), (c) $$\$ 1.89 /$$ gal to dollars per liter, (d) 0.510 in. \(/ \mathrm{ms}\) to (f) \(0.02500 \mathrm{ft}^{3}\) to \(\mathrm{cm}^{3}\) \(\mathrm{km} / \mathrm{hr},\) (e) \(22.50 \mathrm{gal} / \mathrm{min}\) to \(\mathrm{L} / \mathrm{s}\)

Step-by-Step Solution

Answer

In summary, the short version of the answers are: (a) 5.00 days = 432,000 seconds (b) 0.0550 mi = 88.5137 meters (c) $1.89/gal = $0.499 dollars per liter (approx.) (d) 0.510 in/ms = 55.384 km/hr (approx.) (e) 22.50 gal/min = 14.164 L/s (approx.) (f) 0.02500 ft³ = 707.218 cm³ (approx.)

Step 1: Identify conversion factors

1 day is equal to 24 hours, 1 hour is equal to 60 minutes, and 1 minute is equal to 60 seconds.

Step 2: Apply conversion factors

\(5.00 \,days \times \frac{24 \,hours}{1 \,day} \times \frac{60 \,minutes}{1 \,hour} \times \frac{60 \,seconds}{1\, minute}\)

Step 3: Calculate

After canceling out the units and completing the calculation, we find that 5.00 days is equal to 432,000 seconds. #b. Convert 0.0550 mi to meters#

Step 1: Identify conversion factors

1 mile is equal to 1609.34 meters.

Step 2: Apply the conversion factor

\(0.0550 \,mi \times \frac{1609.34 \,m}{1 \,mi}\)

Step 3: Calculate

After canceling out the units and completing the calculation, we find that 0.0550 mi is equal to 88.5137 meters. #c. Convert $$\$ 1.89 /$$ gal to dollars per liter#

Step 1: Identify conversion factors

1 gallon is equal to 3.78541 liters.

Step 2: Apply the conversion factor

\(\frac{\$ 1.89}{1 \,gal} \times \frac{1 \,gal}{3.78541 \,L}\)

Step 3: Calculate

After canceling out the units and completing the calculation, we find that \$ 1.89/gal is equal to \$ 0.499 dollars per liter (approximately). #d. Convert 0.510 in. \(/ \mathrm{ms}\) to \(\mathrm{km} / \mathrm{hr}\)#

Step 1: Identify conversion factors

1 inch is equal to 0.0254 meters, 1 kilometer is equal to 1000 meters, and 1 hour is equal to 3600 seconds.

Step 2: Apply conversion factors

\(0.510 \,in/ms \times \frac{0.0254 \,m}{1 \,in} \times \frac{1 \,km}{1000 \,m} \times \frac{3600 \,s}{1 \,hr}\)

Step 3: Calculate

After canceling out the units and completing the calculation, we find that 0.510 in/ms is equal to 55.384 km/hr (approximately). #e. Convert \(22.50 \mathrm{gal} / \mathrm{min}\) to \(\mathrm{L} / \mathrm{s}\)#

Step 1: Identify conversion factors

1 gallon is equal to 3.78541 liters, and 1 minute is equal to 60 seconds.

Step 2: Apply conversion factors

\(\frac{22.50 \,gal}{minutes} \times \frac{3.78541 \,L}{1 \,gal} \times \frac{1 \,minute}{60 \,seconds}\)

Step 3: Calculate

After canceling out the units and completing the calculation, we find that 22.50 gal/min is equal to 14.164 L/s (approximately). #f. Convert \(0.02500 \mathrm{ft}^{3}\) to \(\mathrm{cm}^{3}\)#

Step 1: Identify conversion factors

1 foot is equal to 30.48 centimeters.

Step 2: Apply conversion factors

\(0.02500 \,ft^{3} \times \frac{(30.48 \,cm)^{3}}{(1 \,ft)^{3}}\)

Step 3: Calculate

After canceling out the units and completing the calculation, we find that 0.02500 ft³ is equal to 707.218 cm³ (approximately).

Key Concepts

Conversion Factors

Conversion factors are essential in chemistry for transforming one unit of measurement into another. They allow you to keep track of the units and make sure you don’t miss conversions during calculations. Each conversion factor is a fraction that represents a relationship between two units. For example, the conversion factor to change miles to meters is \(\frac{1609.34 \, m}{1 \, mi}\), meaning one mile equals 1609.34 meters. Keeping track of these relationships helps ensure that calculations are accurate.

• Numerator and Denominator: The numerator and denominator should be directly proportional to the quantities you're converting.
• Multiplication: Always multiply the value you want to convert by the conversion factor.

During calculations, ensure the units you want to eliminate are set opposite in your conversion factor, allowing you to cancel them out. This way, only the desired units remain, simplifying problem-solving.

Dimensional Analysis

Dimensional analysis, also known as the factor-label method, is a powerful tool used to convert between different units. It involves multiplying by conversion factors, and tracking units throughout the calculation. Each step cancels out units you do not need, helping you reach the desired unit.

• Start with the given unit: Identify the starting measurement you need to convert.
• Set up the conversion: Arrange conversion factors so unwanted units cancel.
• Perform the operations: Multiply through to find the final answer.

For example, converting inches per millisecond (in/ms) to kilometers per hour (km/hr) involves multiple steps. You would convert inches to meters, then meters to kilometers, before adjusting time from milliseconds to hours. Breaking down conversions ensures accuracy and fluency in solving complex problems.

Metric System

The metric system is widely used in science because it is based on sets of ten, making calculations straightforward. It is preferred for scientific measurements due to its simplicity and universal acceptance. Common units in the metric system include meters for length, liters for volume, and grams for mass.

• Base Units: Recognize the base units associated with each measurement type (meter, liter, gram).
• Prefixes: Learn prefixes like "kilo," "centi," and "milli" to denote different magnitudes.

With familiarity in the metric system, converting between units like kilometers, meters, and centimeters becomes a consistent process. This system's logical structure makes it ideal for chemistry, where precision and clarity are vital.

SI Units

SI Units, short for the International System of Units, form the foundation for scientific measurement. This system standardizes measurements across disciplines, ensuring consistency. Each measurement type has a primary unit, such as the meter for length.

• Consistency: Ensures measurements can be understood internationally.
• Standardization: Helps in comparing and repeating experiments or calculations worldwide.

In chemistry, SI units simplify the communication and comparison of data. For example, converting from feet (a non-SI unit) to centimeters involves translating between systems using conversion factors. As a standardized system, it underscores the importance of learning and applying it effectively to facilitate accurate and meaningful scientific work.