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Chapter 1: Problem 25

A solid piece of lead has a mass of \(23.94 \mathrm{g}\) and a volume of \(2.10 \mathrm{cm}^{3} .\) From these data, calculate the density of lead in SI units \(\left(\mathrm{kg} / \mathrm{m}^{3}\right)\).

Short Answer

Expert verified
The density of the lead is \(11352 kg/m^{3}\).

Step by step solution

01

Convert mass from grams to kilograms

Since 1 kilogram contains 1000 grams, use this conversion ratio:\n \(23.94 g * (1 kg / 1000 g)= 0.02394 kg\).
02

Convert Volume from cubic centimeters to cubic meters

Since 1 cubic meter contains \(1 * 10^{6}\) cubic centimeters, use this conversion ratio: \n\(2.10 cm^{3} * (1 m^{3} / 10^{6} cm^{3}) = 2.10 * 10^{-6} m^{3}\).
03

Formula for density and calculation

The density, D, is calculated by using formula D = m/V. Substituting the values into the formula gives: \n\(D = 0.02394 kg / 2.10 * 10^{-6} m^{3} = 11352 kg/m^{3}\).

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Key Concepts

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

SI units
When discussing scientific measurements, it's essential to use the International System of Units, or SI units. These units simplify information sharing and make scientific computations consistent across different regions and fields.
For this exercise, we are specifically interested in the SI units for mass and volume. The standard SI unit for mass is the kilogram (kg), while for volume, it's the cubic meter (m^3). Using SI units allows measurements to be universally understood and accurately compared in scientific work.
  • Mass: Always measured in kilograms in the SI system.
  • Volume: Measured in cubic meters for three-dimensional space.
It's important to start any calculation process by converting given measurements into these SI units to maintain consistency and accuracy in your calculations.
Unit conversion
Unit conversion is a crucial skill in science and engineering that involves changing measurements from one unit to another. This is particularly important when converting between metric units and SI units.
In our exercise, the mass was initially given in grams, and the goal was to convert it to kilograms.
  • To convert grams to kilograms, divide the mass by 1000, since 1000 grams equals 1 kilogram.
Similarly, to convert the volume from cubic centimeters to cubic meters:
  • Use the fact that a cubic meter is equivalent to a million cubic centimeters. So, divide the volume by 1,000,000.
Perfecting these conversions ensures you operate with the correct measurements and achieve accurate scientific results.
Mass and volume
Mass and volume are fundamental concepts in physics and chemistry that are often used to calculate density. They represent different aspects of matter:
  • Mass: It indicates the amount of matter in an object and is typically measured in kilograms or grams.
  • Volume: It shows the space that an object occupies, generally measured in cubic meters or cubic centimeters.
To determine density, you divide mass by volume. Therefore, understanding and accurately measuring both mass and volume is essential. These two quantities interrelate via the formula for density, allowing us to understand the object's compactness or how much matter is packed within a certain volume.
By obtaining accurate measurements of mass and volume and converting them to SI units, you'll be able to calculate density effectively without errors.

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Most popular questions from this chapter

A child loves to watch as you fill a transparent plastic bottle with shampoo. Every horizontal cross-section is a circle, but the diameters of the circles have different values, so that the bottle is much wider in some places than others. You pour in bright green shampoo with constant volume flow rate \(16.5 \mathrm{cm}^{3} / \mathrm{s} .\) At what rate is its level in the bottle rising (a) at a point where the diameter of the bottle is \(6.30 \mathrm{cm}\) and \((\mathrm{b})\) at a point where the diameter is \(1.35 \mathrm{cm} ?\)

At the time of this book's printing, the U.S. national debt is about \(\$ 6\) trillion. (a) If payments were made at the rate of \(\$ 1000\) per second, how many years would it take to pay off the debt, assuming no interest were charged? (b) A dollar bill is about \(15.5 \mathrm{cm}\) long. If six trillion dollar bills were laid end to end around the Earth's equator, how many times would they encircle the planet? Take the radius of the Earth at the equator to be \(6378 \mathrm{km} .\) (Nole: Before doing any of these calculations, try to guess at the answers. You may be very surprised.)

A small cube of iron is observed under a microscope. The edge of the cube is \(5.00 \times 10^{-6} \mathrm{cm}\) long. Find (a) the mass of the cube and (b) the number of iron atoms in the cube. The atomic mass of iron is \(55.9 \mathrm{u},\) and its density is \(7.86 \mathrm{g} / \mathrm{cm}^{3}\).

Suppose your hair grows at the rate \(1 / 32\) in. per day. Find the rate at which it grows in nanometers per second. Because the distance between atoms in a molecule is on the order of \(0.1 \mathrm{nm},\) your answer suggests how rapidly layers of atoms are assembled in this protein synthesis.

The mean radius of the Earth is \(6.37 \times 10^{6} \mathrm{m},\) and that of the Moon is \(1.74 \times 10^{8} \mathrm{cm} .\) From these data calculate (a) the ratio of the Earth's surface area to that of the Moon and (b) the ratio of the Earth's volume to that of the Moon. Recall that the surface area of a sphere is \(4 \pi r^{2}\) and the volume of a sphere is \(\frac{4}{3} \pi r^{3}\).
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