/*! 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 73 Diamonds are measured in carats,... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 73

Diamonds are measured in carats, and 1 carat \(=0.200 \mathrm{g} .\) The density of diamond is 3.51 \(\mathrm{g} / \mathrm{cm}^{3}\) . a. What is the volume of a 5.0 -carat diamond? b. What is the mass in carats of a diamond measuring 2.8 \(\mathrm{mL} ?\)

Short Answer

Expert verified
a. The volume of a 5.0-carat diamond is approximately \(0.285 \ \text{cm}^3\). b. The mass of a diamond with a volume of \(2.8 \ \text{mL}\) is approximately \(49.14 \ \text{carats}\).

Step by step solution

01

Convert the mass of the diamond to grams.

To find the volume, we should first convert the given mass in carats to grams and use the density formula. To do that, multiply the mass in carats by the conversion factor (1 carat = 0.200 g). Mass in grams = (5.0 carats) 脳 (0.200 g / 1 carat) = 1.0 g
02

Calculate the volume of the diamond.

Now that we have the mass of the diamond in grams, it's time to calculate the volume. We can rearrange the density formula to find the volume: V = m / 蟻. Volume = Mass / Density V = 1.0 g / (3.51 g/cm鲁) V 鈮� 0.285 cm鲁 The volume of the 5.0-carat diamond is approximately 0.285 cm鲁. b. We want to find the carats of a diamond measuring 2.8 mL. Given information: - The diamond's volume is 2.8 mL. - 1 mL = 1 cm鲁 (since we are working with cm鲁 in the rest of the problem, we will use this conversion). - The diamond's density is 3.51 g/cm鲁. - 1 carat = 0.200 g
03

Convert the volume of the diamond to cm鲁.

We know that the diamond has a volume of 2.8 mL, which is equal to 2.8 cm鲁 since 1 mL is equal to 1 cm鲁. So the volume of the diamond is 2.8 cm鲁.
04

Calculate the mass of the diamond.

Using the density formula, we can find the mass of the diamond. To do so, rearrange the density formula to m = 蟻 * V. Mass = Density * Volume m = (3.51 g/cm鲁) * (2.8 cm鲁) m 鈮� 9.828 g The mass of the diamond is approximately 9.828 g.
05

Convert the mass to carats.

Now, we want to find the mass of the diamond in carats. To do that, divide the mass in grams by the conversion factor (0.200 g = 1 carat). Mass in carats = 9.828 g 梅 (0.200 g/carats) 鈮� 49.14 carats The mass of the diamond with a volume of 2.8 mL is approximately 49.14 carats.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Volume
When we talk about volume, we are referring to the amount of space an object occupies. In the exercise, we're asked to find the volume of a diamond. This involves using the density of the diamond and its mass.
To find volume, we use the formula:
  • Volume (\(V\)) = Mass (\(m\)) / Density (\(\rho\))
For our diamond, with a known density, you first convert the carats to grams. Then, you divide by the density to find the volume in cubic centimeters (\(\text{cm}^3\)). This gives us an insight into the diamond's size based on its weight.
Mass conversion
Mass conversion is a crucial step in dealing with diamonds or any scientific calculations involving mass. A diamond's mass is typically given in carats, but for calculations, especially involving density, converting it to grams is necessary.
This conversion involves a simple multiplication:
  • 1 carat = 0.200 grams
In the original exercise, we start with carats and use this factor to obtain the mass in grams. It allows us to plug these values into further formulas, especially when other metrics like volume are involved. Understanding this basic conversion is key to solving related physics and chemistry problems.
Carats
Carats are a unit of weight specifically used for gemstones and pearls. A single carat corresponds to 0.200 grams. This specific measurement is significant in industries like jewelry, where diamond weight can greatly affect value.
To work with diamonds in scientific contexts, carats are often converted to grams using a simple calculation:
  • Carats to grams: Multiply carats by 0.200
  • Grams to carats: Divide grams by 0.200
This kind of conversion ensures that your calculations are accurate, whether you're figuring out volume from weight or the weight based on volume and density.
Understanding carats also helps in comparing gemstones of different sizes and determining the suitability of gemstones for different purposes.
Density formula
The density formula is a fundamental concept in both physics and chemistry. Density (\(\rho\)) is defined as mass (\(m\)) divided by volume (\(V\)):
  • \[ \rho = \frac{m}{V} \]
Density indicates how much mass is contained in a given volume. For example, diamonds have a high density, meaning they're fairly heavy for their size.
This formula proves very useful for calculating either the mass, volume, or density of an object if you know the other two quantities. In the exercise, once we know the diamond's density and either its mass or its volume, we can find the missing value. Using the density formula helps in discerning relationships between these physical properties.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

A rule of thumb in designing experiments is to avoid using a result that is the small difference between two large measured quantities. In terms of uncertainties in measurement, why is this good advice?

To determine the volume of a cube, a student measured one of the dimensions of the cube several times. If the true dimension of the cube is 10.62 cm, give an example of four sets of measurements that would illustrate the following. a. imprecise and inaccurate data b. precise but inaccurate data c. precise and accurate data Give a possible explanation as to why data can be imprecise or inaccurate. What is wrong with saying a set of measurements is imprecise but accurate?

Consider the addition of 15.4 to 28. What would a mathematician say the answer is? What would a scientist say? Justify the scientist鈥檚 answer, not merely citing the rule, but explaining it.

You have two beakers, one filled to the 100-mL mark with sugar (the sugar has a mass of 180.0 g) and the other filled to the 100-mL mark with water (the water has a mass of 100.0 g). You pour all the sugar and all the water together in a bigger beaker and stir until the sugar is completely dissolved. a. Which of the following is true about the mass of the solution? Explain. i. It is much greater than 280.0 g. ii. It is somewhat greater than 280.0 g. iii. It is exactly 280.0 g. iv. It is somewhat less than 280.0 g. v. It is much less than 280.0 g. b. Which of the following is true about the volume of the solution? Explain. i. It is much greater than 200.0 mL. ii. It is somewhat greater than 200.0 mL. iii. It is exactly 200.0 mL. iv. It is somewhat less than 200.0 mL. v. It is much less than 200.0 mL.

You go to a convenience store to buy candy and find the owner to be rather odd. He allows you to buy pieces in multiples of four, and to buy four, you need $0.23. He only allows you to do this by using 3 pennies and 2 dimes. You have a bunch of pennies and dimes, and instead of counting them, you decide to weigh them. You have 636.3 g of pennies, and each penny weighs 3.03 g. Each dime weighs 2.29 g. Each piece of candy weighs 10.23 g. a. How many pennies do you have? b. How many dimes do you need to buy as much candy as possible? c. How much should all these dimes weigh? d. How many pieces of candy could you buy? (number of dimes from part b) e. How much would this candy weigh? f. How many pieces of candy could you buy with twice as many dimes?
See all solutions

Recommended explanations on Chemistry Textbooks

Organic Chemistry

Read Explanation

Inorganic Chemistry

Read Explanation

Ionic and Molecular Compounds

Read Explanation

Chemical Reactions

Read Explanation

Kinetics

Read Explanation

Physical Chemistry

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.