Problem 53 - Chemistry: The Central Science

Question

(a) How many liters of wine can be held in a wine barrel whose capacity is 31 gal? (b) The recommended adult dose of Elixophyllin", a drug used to treat asthma, is \(6 \mathrm{mg} / \mathrm{kg}\) of body mass. Calculate the dose in milligrams for a 185 -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 kg of coffee are required to produce 200 cups of coffee?

Step-by-Step Solution

Answer

(a) The wine barrel can hold approximately 117.35 liters of wine. (b) The recommended Elixophyllin dosage for a 185-lb person is approximately 503.49 mg. (c) The gas mileage of the automobile is approximately 39.24 miles per gallon. (d) Approximately 1.81 kg of coffee is required to produce 200 cups of coffee.

Step 1: (a) Convert gallons to liters

First, we need to convert the wine barrel's capacity from gallons to liters. Recall that 1 gallon is equivalent to 3.78541 liters. To convert 31 gallons to liters, you can use the formula: Liters = Gallons × Conversion factor Liters = 31 gallons × 3.78541 L/gal = 117.34771 L The wine barrel can hold approximately 117.35 liters of wine.

Step 2: (b) Calculate Elixophyllin dosage

To calculate the Elixophyllin dosage for a 185-lb person, we first need to convert the weight from pounds to kilograms. Recall that 1 lb is equivalent to 0.453592 kg. Perform the conversion: Weight in kg = Weight in lb × Conversion factor Weight in kg = 185 lb × 0.453592 kg/lb ≈ 83.91452 kg Now, we can calculate the dosage: Dosage in mg = Weight in kg × Recommended dose (6 mg/kg) Dosage in mg = 83.91452 kg × 6 mg/kg ≈ 503.48712 mg The recommended Elixophyllin dosage for a 185-lb person is approximately 503.49 mg.

Step 3: (c) Convert gas mileage from km/L to mi/gal

To convert the gas mileage from kilometers per liter (km/L) to miles per gallon (mi/gal), we need two conversion factors: 1 km = 0.621371 mi and 1 gal = 3.78541 L. First, let's find the gas mileage in miles per liter: Miles per liter = 400 km × 0.621371 mi/km ≈ 248.5484 mi Now, let's find the gas mileage in miles per gallon by dividing the miles by the gallons: Miles per gallon = (248.5484 mi) / (47.3 L × 1 gal/3.78541 L) ≈ 39.23989 mi/gal The gas mileage of the automobile is approximately 39.24 miles per gallon.

Step 4: (d) Calculate required coffee in kg for 200 cups

Since 1 lb of coffee can yield 50 cups of coffee, we can determine the amount of coffee needed for 200 cups using a proportion: Amount of coffee (in lb) = (200 cups / 50 cups) × 1 lb = 4 lb Now, convert the weight of the coffee to kilograms using the conversion factor 1 lb = 0.453592 kg: Amount of coffee (in kg) = 4 lb × 0.453592 kg/lb ≈ 1.814368 kg Approximately 1.81 kg of coffee is required to produce 200 cups of coffee.

Key Concepts

Metric Conversions

When dealing with conversions between different measurement systems, it's essential to understand how to switch units accurately. This is often done using conversion factors. For instance, if you want to convert gallons to liters, you use the fact that 1 gallon = 3.78541 liters. By multiplying the number of gallons by this conversion factor, you obtain the equivalent in liters.

• Example: 31 gallons * 3.78541 L/gal = 117.35 liters. This is a direct application of dimensional analysis.
• Always check your units: After performing a conversion, ensure that you have correctly switched the units. This ensures the accuracy of your result.

Metric conversions are handy not only in scientific contexts but also in everyday scenarios, such as cooking with international recipes or understanding car fuel efficiency in different countries.

Gas Mileage Calculation

Calculating gas mileage allows you to understand how far a vehicle can travel on a particular amount of fuel, often helping in gauging fuel efficiency. This is typically measured in miles per gallon (mi/gal) in the United States, but it can be converted from kilometers per liter (km/L) through dimensional analysis.
The conversion requires the use of:

• 1 kilometer = 0.621371 miles.
• 1 gallon = 3.78541 liters.

To find the gas mileage, first convert the distance from kilometers to miles, then divide by the adjustment for liters to gallons:\[\text{Miles per gallon} = \frac{\text{Distance in miles}}{\text{Liters used / gallons}}\]This ensures you have a standard unit of measurement, vital for comparing different vehicles' fuel efficiency. For instance, a car traveling 400 kilometers using 47.3 liters of gas would equate to approximately 39.24 mi/gal when converted.

Dosage Calculation

Dosage calculations are crucial in medicine to ensure patients receive the correct amount of medication. These are typically based on body weight, and often, conversions are necessary to align units of weight with those provided for dosage.
For example, if a medication requires a dose of 6 mg/kg of body mass, first convert the person's weight into kilograms.

• Using 1 lb = 0.453592 kg, a 185-lb person is approximately 83.91 kg.

Once converted, multiply the weight by the dosage instruction:\[\text{Dosage in mg} = \text{Weight in kg} \times \text{Dose per kg}\]This will give you the precise dosage required. For this scenario, the correct dosage would be about 503.49 mg, ensuring the patient receives the accurate amount of medication.