Problem 59 - Chemistry The Central Science In Si Units

Question

(a) How many liters of wine can be held in a wine barrel whose capacity is 31 gal? \((\mathbf{b})\) The recommended adult dose of Elixophyllin \(^{\circ}\), a drug used to treat asthma, is \(6 \mathrm{mg} / \mathrm{kg}\) of body mass. Calculate the dose in milligrams for a \(185-\mathrm{lb}\) person. (c) If an automobile is able to travel \(400 \mathrm{~km}\) on \(47.3 \mathrm{~L}\) of gasoline, what is the gas mileage in miles per gallon? (d) When the coffee is brewed according to directions, a pound of coffee beans yields 50 cups of coffee \((4\) cups \(=1 \mathrm{qt}) .\) How many \(\mathrm{kg}\) of coffee are required to produce 200 cups of coffee?

Step-by-Step Solution

Answer

The barrel holds 117.35 liters. Elixophyllin dose is 503.49 mg. Mileage is 19.89 mpg. 0.45 kg coffee for 200 cups.

Step 1: Converting the Barrel's Capacity

To convert the wine barrel's capacity from gallons to liters, we use the conversion factor: \(1 \text{ gallon} = 3.78541 \text{ liters}\). Thus, for \(31 \text{ gallons}\), the volume in liters is obtained as follows: \[31 \times 3.78541 = 117.34871 \text{ liters}\]. Thus, the wine barrel can hold approximately 117.35 liters.

Step 2: Calculating the Elixophyllin Dose

First, convert the weight from pounds to kilograms using the conversion factor: \(1 \text{ pound} = 0.453592 \text{ kg}\). Therefore, \(185 \text{ pounds}\) corresponds to: \[185 \times 0.453592 = 83.91452 \text{ kg}\]. Then, using the drug dosage rate, we find the dose: \(6 \text{ mg/kg} \times 83.91452 \text{ kg} = 503.48712 \text{ mg}\). Thus, the dose for the person is approximately 503.49 mg.

Step 3: Converting Gas Mileage

First, convert the distance from kilometers to miles using the conversion factor: \(1 \text{ km} = 0.621371 \text{ miles}\), so \(400 \text{ km}\) becomes \[400 \times 0.621371 = 248.5484 \text{ miles}\]. Next, convert the fuel from liters to gallons: \(1 \text{ liter} = 0.264172 \text{ gallons}\), resulting in \[47.3 \times 0.264172 = 12.4939356 \text{ gallons}\]. Therefore, the mileage in miles per gallon is \[\frac{248.5484}{12.4939356} \approx 19.89 \text{ mpg}\].

Step 4: Converting Coffee Yield

First convert cups to quarts using the relation that \(4 \text{ cups} = 1 \text{ quart}\): \(200 \text{ cups} = \frac{200}{4} = 50 \text{ quarts}\) of coffee. As 1 pound of coffee yields 50 cups, or \(50 \text{ quarts}\), we need \(1\) pound for 200 cups. To find the mass in kilograms: \(1 \text{ pound} = 0.453592 \text{ kg}\). Thus, \(0.453592 \text{ kg}\) are required to make 200 cups.

Key Concepts

Volume Conversions

Volume conversions are essential in many real-world applications. Understanding how to convert between units such as gallons and liters is crucial, especially when dealing with liquids. For instance, if you're working with a wine barrel with a capacity of 31 gallons, converting it to liters can be done by using the conversion factor:

• 1 gallon = 3.78541 liters

By multiplying 31 gallons by this conversion factor:\[ 31 \times 3.78541 = 117.34871 \text{ liters} \]Thus, the wine barrel holds approximately 117.35 liters.
Being able to convert volume helps in accurately measuring and utilizing resources across different regions, as some countries use the metric system while others use the imperial system.

Mass Conversions

Mass conversions are a fundamental aspect of dosage calculations and other applications where body weight is a factor. Converting from pounds to kilograms, for example, is often necessary in medical settings.
To illustrate, if a person weighs 185 pounds, and we need to calculate a dosage that uses the metric system, the conversion factor is:

• 1 pound = 0.453592 kg

Calculating gives:\[ 185 \times 0.453592 = 83.91452 \text{ kg} \]With this conversion, if the correct dosage is 6 mg per kg of body weight, the total dose for this weight would be:\[ 6 \text{ mg/kg} \times 83.91452 \text{ kg} = 503.49 \text{ mg} \]Converting mass units ensures precision in medical dosing and helps in international communications where different systems are used.

Gas Mileage Calculation

Calculating gas mileage is integral to assessing a vehicle’s fuel efficiency. This calculation involves converting distance and fuel amounts into comparable units. For example, if a car travels 400 kilometers on 47.3 liters of fuel, we must first convert these to miles and gallons respectively.
The steps are as follows:

• Convert kilometers to miles: 1 km = 0.621371 miles, so
  \[ 400 \times 0.621371 = 248.5484 \text{ miles} \]
• Convert liters to gallons: 1 liter = 0.264172 gallons, hence
  \[ 47.3 \times 0.264172 = 12.49394 \text{ gallons} \]

Finally, dividing the distance by the fuel used gives us the mileage in miles per gallon:\[ \frac{248.5484}{12.49394} \approx 19.89 \text{ mpg} \]Understanding gas mileage helps in comparing vehicle efficiencies and budgeting for fuel costs.

Coffee Yield Conversion

Coffee yield conversion is practical when understanding how much coffee can be brewed from a given amount of beans. In this case, converting cups of coffee to a mass of coffee beans requires understanding of both volume and weight units.
Suppose 1 pound of coffee beans yields 50 cups. If you need to produce 200 cups, calculate how many pounds are needed. Since 4 cups = 1 quart, hence 200 cups is equivalent to 50 quarts:

• 1 pound is needed for 50 quarts, so 1 pound is sufficient for 200 cups.
• Convert pounds to kilograms: 1 pound = 0.453592 kg, so

\[ 1 \times 0.453592 = 0.453592 \text{ kg} \]Thus, 0.453592 kg of coffee is required to produce 200 cups. This helps in planning for large gatherings or in commercial settings where precise measurements are crucial.