Problem 49 - 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 \(^{\circledast}\), 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) A pound of coffee beans yields 50 cups of coffee \((4\) cups \(=1 \mathrm{qt}) .\) How many milliliters of coffee can be obtained from \(1 \mathrm{~g}\) of coffee beans?

Step-by-Step Solution

Answer

(a) The wine barrel can hold approximately \(117.33 L\) of wine. (b) The recommended dosage of Elixophyllin for a 185-lb person is approximately \(503.49 mg\). (c) The gas mileage of the automobile is approximately \(19.88 mi/gal\). (d) From 1 gram of coffee beans, approximately \(26.08 mL\) of coffee can be obtained.

Step 1: (a) Convert gallons to liters

In order to find out how many liters of wine the barrel can hold, we need to convert gallons to liters. 1 gallon is equal to approximately 3.78541 liters. So, the capacity in liters can be calculated by multiplying 31 gallons by the conversion factor: Capacity (in liters) = 31 gal * 3.78541 L/gal ≈ 117.32771 L

Step 2: (b) Calculate the drug dose in milligrams

First, we need to convert the person’s weight from pounds to kilograms. 1 pound is approximately equal to 0.453592 kg. Weight (in kg) = 185 lb * 0.453592 kg/lb ≈ 83.91452 kg Now, we can calculate the drug dose by multiplying the recommended dose of 6 mg/kg with their weight in kg: Drug dose = 6 mg/kg * 83.91452 kg = 503.48712 mg

Step 3: (c) Calculate the gas mileage in miles per gallon

First, we need to convert kilometers to miles and the volume of gasoline in liters to gallons. 1 kilometer is equal to 0.621371 miles, and 1 liter is equal to 0.264172 gallons. Distance (in miles) = 400 km * 0.621371 mi/km ≈ 248.5484 mi Gasoline volume (in gallons) = 47.3 L * 0.264172 gal/L ≈ 12.48918 gal Then we can calculate the gas mileage by dividing miles by gallons: Gas mileage (in miles per gallon) = 248.5484 mi / 12.48918 gal ≈ 19.878 mi/gal

Step 4: (d) Calculate coffee milliliters from grams of coffee beans

We are given that 1 pound of coffee beans yields 50 cups of coffee and 4 cups make 1 quart. Also, there are 453.592 grams in a pound, and 1 quart is equal to 946.353 milliliters. First, we'll determine how many milliliters of coffee can be produced from a pound of coffee beans: 1 pound = 50 cups of coffee * (1 quart / 4 cups) * 946.353 mL/quart ≈ 11829.4125 mL Now, we can calculate the milliliters of coffee produced from 1 gram of coffee beans by dividing the total milliliters of coffee from a pound by the number of grams in a pound: Milliliters of coffee per gram = 11829.4125 mL / 453.592 g ≈ 26.0804 mL/g So, 1 gram of coffee beans can yield approximately 26.0804 mL of coffee.

Key Concepts

Volume Conversion

Volume conversion is a pivotal concept in chemistry and everyday life when dealing with different units of measure. If an American recipe calls for a gallon of milk but you only have liter measurements, volume conversion comes to the rescue.

In the exercise, we converted the capacity of a wine barrel from gallons to liters. This conversion is frequently needed since the United States uses the imperial system, while most of the world uses the metric system. One gallon is approximately 3.78541 liters, and by multiplying the number of gallons by this factor, we can find out the equivalent volume in liters. It's important to note that this factor is a rounded-off value and can vary slightly based on the definition used (US gallon vs. UK gallon).

For tasks involving pouring, measuring, or assessing containers of liquids, a clear understanding of volume conversion is key. This allows for more accurate and consistent results in both laboratory and culinary settings.

Dosage Calculation

Dosage calculation ensures the safety and efficacy of medication administration. It is usually based on the individual's weight, making it a personalized form of treatment. In the exercise, the drug dosage for Elixophyllin, used to treat asthma, had to be calculated for a person weighing 185 pounds.

First, we converted the weight from pounds to kilograms since the dosage is given in milligrams per kilogram. It's crucial when performing a dosage calculation to verify that the units of weight match those in the dosage guidance. Incorrect conversion might result in a medical error, possibly leading to harmful consequences. After obtaining the weight in kilograms, we multiply it by the drug's dosage per kilogram to find the correct milligram dosage for the patient. Pharmacists and healthcare professionals routinely carry out these calculations to prescribe or administer the proper amount of medication.

Gas Mileage Calculation

Gas mileage calculation is a practical application of unit conversion that can significantly impact your budget and environmental footprint. Understanding how to calculate your car's gas mileage involves converting distances and volumes to a standard, in this case, miles per gallon (mpg).

In our exercise, we begin by converting the distance from kilometers to miles, using the conversion factor where 1 kilometer equals approximately 0.621371 miles. Next, we convert the gasoline volume from liters to gallons. Once both are in miles and gallons, we find the vehicle's gas mileage by dividing the distance traveled by the amount of gas used. This resulting value is an indicator of the vehicle's fuel efficiency, which is essential when comparing vehicles or tracking fuel costs.

Mass to Volume Conversion

Mass to volume conversion comes into play when you need to understand the yield of a certain mass of a substance. In culinary arts, this helps when a recipe specifies a certain weight of an ingredient and you need to know the corresponding volume.

In the context of our exercise, this conversion was crucial for determining how much coffee one could obtain from a given mass of coffee beans. Since different substances have different densities, there is no one-size-fits-all conversion factor. In the case of coffee beans, we are given the volume yield from a certain weight. After determining the milliliters of coffee made from a pound of beans, we use the mass of coffee in grams to find the volume of coffee produced per gram. Knowing how to convert mass to volume is indispensable for chemists, cooks, and anyone needing to measure ingredients accurately.