/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 6 You purchase six oranges that we... [FREE SOLUTION] | 91Ó°ÊÓ

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You purchase six oranges that weigh a total of \(2 \mathrm{Ib}_{\mathrm{f}}\) and 13 ounces. After cutting them open and squeezing all the juice your strength allows into a large measuring cup, you weigh the remaining pulp and orange peels. They weigh 1 Ib \(_{\mathrm{f}}\) and 12 ounces and the total volume of the juice is 1.75 cups. What is the specific gravity of orange juice? State any assumptions you make.

Short Answer

Expert verified
The specific gravity of the orange juice is approximately 1.21.

Step by step solution

01

Convert the weight of the oranges to ounces

This problem is working in ounces, not pounds, so to make the calculations easier, convert the weight of the oranges to ounces. 1 pound equals to 16 ounces. So, 2 pounds of oranges is \(2 \times 16 = 32\) ounces. Adding the 13 ounces, the total weight of the oranges is \(32 + 13 = 45\) ounces.
02

Calculate the weight of the orange juice

The weight of the orange juice equals the weight of the oranges minus the weight of the pulp and peels. To find the weight of the juice, first convert the weight of the pulp and peels to ounces. 1 pound of pulp and peels is 16 ounces, so adding the 12 ounces, the total weight of the pulp and peels is \(16 + 12 = 28\) ounces. Now subtract this from the total weight of the oranges, so the weight of the juice is \(45 - 28 = 17\) ounces.
03

Find the specific gravity

Specific gravity is the weight of a substance divided by the weight of an equal volume of water. In this case, we know that 1 cup of water weighs 8 ounces, so 1.75 cups of water weighs \(1.75 \times 8 = 14\) ounces. Now to compute the specific gravity, divide the weight of the orange juice by the weight of the equal volume of water, so the specific gravity is \(17 / 14 \approx 1.21\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Unit Conversion
Understanding unit conversion is crucial when dealing with a variety of measurements, as it enables clear communication and accurate calculations across different units of measure.

In the context of specific gravity calculations, unit conversion often plays a vital role. For example, when working with weight and volume measurements that are not aligned (like pounds and ounces versus cups), converting them into a common unit allows for a straightforward comparison. In the provided problem, we had to convert pounds into ounces because the weight of the orange peels and the orange juice were initially given in different units.

Unit conversion typically involves a conversion factor, which is a numerical factor used to multiply or divide a quantity when converting from one unit to another. This process might look simple, but being meticulous with conversion factors is necessary to avoid errors. For example, knowing that 1 pound equals to 16 ounces makes it possible to convert 2 pounds of oranges to ounces by multiplying by the conversion factor (2 pounds * 16 ounces/pound = 32 ounces).

These calculations may also involve more complex units such as converting from metric to imperial systems, or from volume to mass, depending on the substance's density. The key to mastering unit conversion is understanding the relationship between various units and practicing the conversions regularly.
Weight Measurement
Weight measurement is a fundamental aspect of science and everyday life, serving as a basis for numerous calculations and assessments. We measure weight to determine the heaviness of an object, and in scientific terms, it’s the gravitational force exerted on an object's mass.

In our specific gravity calculation exercise, weight measurement was a critical step. We measured the total weight of the oranges before and after juicing, as well as the weight of the juice itself. We saw that these weights were expressed in both pounds and ounces. This is where understanding unit conversions becomes useful – by converting these weights into a single unit (ounces in our case), we were able to perform consistent calculations.

To measure weight accurately, a balance or scale is commonly used. These devices can range from simple spring scales to more complex digital scales that provide readings in various units. When measuring for scientific purposes, it's important to use a properly calibrated scale and to record measurements in a standard unit that aligns with the context of the problem we are solving. Also, always take into account the precision of the measuring device, as this can affect the accuracy of your results.
Volume Measurement
Volume measurement represents the amount of space taken up by a substance, be it solid, liquid, or gas. For liquids and gases, volume is often measured in liters, milliliters, gallons, or cups, depending on the context and region.

In the context of the specific gravity calculation for orange juice, we dealt with measuring the liquid's volume after juicing the oranges. The result was conveniently provided in cups, which is a standard unit for volume in the kitchen but not always used in scientific calculations. This illustrates the ubiquitous nature of volume measurement - it appears in both domestic and scientific settings.

When it comes to measuring the volume of liquids, we use measuring cups, graduated cylinders, or pipettes, each suited for different levels of precision. Graduated cylinders and pipettes are particularly common in laboratory settings for their accuracy. In our exercise, 1.75 cups of orange juice were measured, which was crucial for determining its specific gravity. Knowing the weight of the water volume equivalent to the juice volume enabled us to calculate the specific gravity by providing a comparison. To further understand the relationship between weight and volume, it’s key to grasp the concept of density, which directly ties into the principle of specific gravity.

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Most popular questions from this chapter

The specific gravity of gasoline is approximately 0.70. (a) Estimate the mass (kg) of 50.0 liters of gasoline. (b) The mass flow rate of gasoline exiting a refinery tank is \(1150 \mathrm{kg} / \mathrm{min}\). Estimate the volumetric flow rate in liters/s. (c) Estimate the average mass flow rate ( \(\left(\mathrm{lb}_{\mathrm{m}} / \mathrm{min}\right)\) delivered by a gasoline pump. (d) Gasoline and kerosene (specific gravity \(=0.82\) ) are blended to obtain a mixture with a specific gravity of 0.78. Calculate the volumetric ratio (volume of gasoline/volume of kerosene) of the two compounds in the mixture, assuming \(V_{\text {blend }}=V_{\text {gasoline }}+V_{\text {kerosene. }}\)

The liquid level in a tank is determined by measuring the pressure at the bottom of the tank. A calibration curve is prepared by filling the tank to several known levels, reading the bottom pressure from a Bourdon gauge, and drawing a plot of level (m) vs. pressure (Pa). (a) Would you expect the calibration curve to be a straight line? Explain your answer. (b) The calibration experiment was done using a liquid with a specific gravity of \(0.900,\) but the tank is used to store a liquid with specific gravity of \(0.800 .\) Will the liquid level determined from the calibration curve be too high, too low, or correct? Explain. (c) If the actual liquid level is 8.0 meters, what value will be read from the calibration curve? If the tank has a height of \(10.0 \mathrm{m},\) what value will be read from the curve when the tank overflows?

In the manufacture of pharmaceuticals, most active pharmaceutical ingredients (APIs) are made in solution and then recovered by separation. Acetaminophen, a pain-killing drug commercially marketed as Tylenol", is synthesized in an aqueous solution and subsequently crystallized. The slurry of crystals is sent to a centrifuge from which two effluent streams emerge: ( 1 ) a wet cake containing 90.0 wt\% solid acetaminophen \((\mathrm{MW}=\) 151 g/mol) and 10.0 wt\% water (plus some acetaminophen and other dissolved substances, which we will neglect), and (2) a highly dilute aqueous solution of acetaminophen that is discharged from the process. The wet cake is fed to a dryer where the water is completely evaporated, leaving the residual acetaminophen solids bone dry. If the evaporated water were condensed, its volumetric flow rate would be \(50.0 \mathrm{Lh}\). Following is a flowchart of the process, which runs 24 h/day, 320 days/yr. A denotes acetaminophen. (a) Calculate the yearly production rate of solid acetaminophen (tonne/yr), using as few dimensional equations as possible. (b) A proposal has been made to subject the liquid solution leaving the centrifuge to further processing to recover more of the dissolved acetaminophen instead of disposing of the solution. On what would the decision depend?

The feed to an ammonia synthesis reactor contains 25 mole \(\%\) nitrogen and the balance hydrogen. The flow rate of the stream is \(3000 \mathrm{kg} / \mathrm{h}\). Calculate the rate of flow of nitrogen into the reactor in \(\mathrm{kg} / \mathrm{h}\). (Suggestion: First calculate the average molecular weight of the mixture.)

A mixture is 10.0 mole \(\%\) methyl alcohol, 75.0 mole \(\%\) methyl acetate \(\left(\mathrm{C}_{3} \mathrm{H}_{6} \mathrm{O}_{2}\right),\) and 15.0 mole \(\%\) acetic acid. Calculate the mass fractions of each compound. What is the average molecular weight of the mixture? What would be the mass (kg) of a sample containing 25.0 kmol of methyl acetate?

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