Problem 59 - 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 \(=1\) qt). How many kg of coffee are required to produce 200 cups of coffee?

Step-by-Step Solution

Answer

The wine barrel can hold approximately 117.45 liters of wine. The dose of Elixophyllin for a 185-lb person is approximately 503.49 mg. The gas mileage is approximately 19.88 miles per gallon. We need approximately 1.81 kg of coffee to produce 200 cups of coffee.

Step 1: (a) Convert gallons to liters

We know that 1 gallon is approximately equal to 3.78541 liters. The formula to convert gallons to liters is: \[ Liters = Gallons \times 3.78541 \] Given that the capacity of the wine barrel is 31 gallons, apply the formula to find the liters of wine it can hold. \[ Liters = 31 \times 3.78541 \] \[ Liters ≈ 117.44771 \] Thus, the wine barrel can hold approximately 117.45 liters of wine.

Step 2: (b) Calculate the dose in milligrams

First, we need to convert the person's weight from pounds to kilograms. We know that 1 pound is approximately equal to 0.453592 kg. The formula to convert pounds to kilograms is: \[ Kilograms = Pounds \times 0.453592 \] Given that the person's weight is 185 pounds, apply the formula to find the weight in kilograms. \[ Kilograms = 185 \times 0.453592 \] \[ Kilograms ≈ 83.91452 \] Next, we need to calculate the dose of Elixophyllin in milligrams. The recommended adult dose of Elixophyllin is 6 mg/kg of body mass. To find the dose for this person, use the formula: \[ Dose\,(mg) = Weight\,(kg) \times Dose\,(mg/kg) \] Plug in the calculated weight in kilograms and the given dose in mg/kg. \[ Dose\,(mg) = 83.91452 \times 6 \] \[ Dose\,(mg) ≈ 503.48712 \] Thus, the dose for a 185-lb person is approximately 503.49 mg.

Step 3: (c) Gas mileage in miles per gallon

First, we need to convert the distance from kilometers to miles. We know that 1 kilometer is approximately equal to 0.621371 miles. The formula to convert kilometers to miles is: \[ Miles = Kilometers \times 0.621371 \] Given that the automobile can travel 400 km, apply the formula to find the distance in miles. \[ Miles = 400 \times 0.621371 \] \[ Miles ≈ 248.548 \] Next, convert the fuel consumption from liters to gallons. We know that 1 liter is approximately equal to 0.264172 gallons. The formula to convert liters to gallons is: \[ Gallons = Liters \times 0.264172 \] Given that the automobile consumes 47.3 L of gasoline, apply the formula to find the fuel consumption in gallons. \[ Gallons = 47.3 \times 0.264172 \] \[ Gallons ≈ 12.4937196 \] Finally, to find the gas mileage in miles per gallon, divide the distance in miles by the fuel consumption in gallons. \[ Gas\,Mileage\,(mpg) = \frac{Miles}{Gallons} \] \[ Gas\,Mileage\,(mpg) = \frac{248.548}{12.4937196} \] \[ Gas\,Mileage\,(mpg) ≈ 19.884 \] Thus, the gas mileage is approximately 19.88 miles per gallon.

Step 4: (d) Convert pounds of coffee to kg required

We first find out how many pounds of coffee are required to produce 200 cups of coffee. Given that 1 lb of coffee yields 50 cups, \[ \frac{1\,lb}{50\,cups} = \frac{x\,lb}{200\,cups} \] Solving for x (pounds of coffee required), \[ x = \frac{1\,lb \times 200\,cups}{50\,cups} \] \[ x = 4\,lb \] Now, we need to convert the required coffee amount from pounds to kilograms. We know that 1 pound is approximately equal to 0.453592 kg. The formula to convert pounds to kilograms is: \[ Kilograms = Pounds \times 0.453592 \] Given that we need 4 pounds of coffee, apply the formula to find the required coffee amount in kilograms. \[ Kilograms = 4 \times 0.453592 \] \[ Kilograms ≈ 1.814368 \] Thus, we need approximately 1.81 kg of coffee to produce 200 cups of coffee.

Key Concepts

Conversion of Units

Converting units is an essential skill in chemistry and various scientific fields because measurements can be expressed in many different units. For instance, when converting gallons to liters, we use the fact that 1 gallon is approximately equivalent to 3.78541 liters. This knowledge allows us to convert volumes between these two units with a simple multiplication.

To perform the calculation, we multiply the number of gallons by the conversion factor: \[ Liters = Gallons \times 3.78541 \].

Correct unit conversion is crucial for accurate measurements and to ensure that formulas and equations are applied appropriately. Misconverting units can lead to errors in experiments, dosage calculations, and other practices that rely on precise measurements.

Dosage Calculation

Dosage calculation in pharmacology ensures that patients receive the correct amount of medication according to their body mass. This involves two steps: converting body weight from one unit to another, often from pounds to kilograms, and then applying the dosage formula. We usually convert using the factor where 1 pound is approximately equal to 0.453592 kilograms.

The calculation is performed as follows: \[ Kilograms = Pounds \times 0.453592 \],and then the dosage in milligrams (mg) can be determined by multiplying the person's weight in kilograms by the dosage prescribed per kilogram: \[ Dose\text{(mg)} = Weight(text{(kg)}) \times Dose\text{(mg/kg)} \].

This method ensures that a patient's dosage is personalized, taking into account their unique physiological characteristics.

Fuel Efficiency Calculation

Fuel efficiency is commonly measured by how many miles a vehicle can travel per gallon of gas (mpg). To calculate this for a car known to travel a certain distance on a specific volume of fuel, firstly, we convert the distance from kilometers to miles using the factor that 1 kilometer is approximately equal to 0.621371 miles. Secondly, we convert the fuel volume from liters to gallons, knowing that 1 liter is roughly equivalent to 0.264172 gallons.

The final step is to divide the number of miles by the number of gallons: \[ Gas\,Mileage\,(mpg) = \frac{Miles}{Gallons} \].

This calculation is critical for understanding a vehicle's fuel efficiency, which is not only important for personal cost-saving but also for environmental considerations related to fuel consumption.

Mass-Volume Conversion

Mass-volume conversions are frequently required in culinary arts, chemistry labs, and many other scenarios. One common example involves converting the mass of a substance to the required volume to produce a specific yield. The coffee brewing scenario from the exercise provides an excellent illustration of this.

First, determine the mass of the coffee needed to make a certain number of cups, followed by converting that mass from pounds to kilograms, knowing that 1 pound equals approximately 0.453592 kilograms. The conversion formula used is: \[ Kilograms = Pounds \times 0.453592 \].

Understanding mass-volume conversion allows for precise recipe formulations, chemical reactions, and manufacture specifications. Clear comprehension and accurate application of these conversions are vital to ensure successful outcomes in both everyday tasks and professional practices.