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Chapter 1: Problem 102

Bronze is an alloy made of copper (Cu) and tin (Sn) used in applications that require low metalon-metal friction. Calculate the mass of a bronze cylinder of radius \(6.44 \mathrm{~cm}\) and length \(44.37 \mathrm{~cm} .\) The composition of the bronze is 79.42 percent Cu and 20.58 percent Sn and the densities of Cu and \(\mathrm{Sn}\) are \(8.94 \mathrm{~g} / \mathrm{cm}^{3}\) and \(7.31 \mathrm{~g} / \mathrm{cm}^{3},\) respec- tively. What assumption should you make in this calculation?

Short Answer

Expert verified
The mass of the bronze cylinder is determined by multiplying its volume by its density, derived using volume percentage method with Cu and Sn as its components. One key assumption is that the alloy's density can be calculated using the densities and volume percentages of its component elements.

Step by step solution

01

Volume Calculation

Using the formula for the volume of a cylinder \(V = \pi r^{2} h\), where \(r\) is the cylinder's radius and \(h\) its length, the volume of the bronze cylinder is calculated as follows: \(V = \pi \times (6.44 cm)^{2} x 44.37 cm\).
02

Calculation of Bronze's density

The density of bronze is calculated using the proportions and densities of its components. Using the formula \( \rho _{bronze} =_PCu\rho _{Cu} + _PSn \rho _{Sn}\), where \(_PCu = 0.7942\) and \(_PSn = 0.2058\) are the volume percentages of Cu and Sn, and \(\rho _{Cu} = 8.94 g/cm^{3}\) and \(\rho _{Sn} = 7.31 g/cm^{3}\) are their respective densities.
03

Mass Calculation

The mass of the bronze cylinder is then calculated using the formula: \(Mass = Density \times Volume\) using the derived density of bronze and the calculated volume of the cylinder from Step 1.

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Most popular questions from this chapter

Calculate the mass of each of the following: (a) a sphere of gold with a radius of \(10.0 \mathrm{~cm}\) [ the volume of a sphere with a radius \(r\) is \(V=(4 / 3) \pi r^{3} ;\) the density of gold \(\left.=19.3 \mathrm{~g} / \mathrm{cm}^{3}\right],\) (b) a cube of platinum of edge length \(0.040 \mathrm{~mm}\) (the density of platinum \(=\) \(\left.21.4 \mathrm{~g} / \mathrm{cm}^{3}\right),\) (c) \(50.0 \mathrm{~mL}\) of ethanol (the density of cthanol \(=0.798 \mathrm{~g} / \mathrm{mL}\) ).

Lithium is the least dense metal known (density: \(0.53 \mathrm{~g} / \mathrm{cm}^{3}\) ). What is the volume occupied by \(1.20 \times 10^{3} \mathrm{~g}\) of lithium?

Carry out the following conversions: (a) \(70 \mathrm{~kg}\), the average weight of a male adult, to pounds. (b) 14 billion years (roughly the age of the universe) to seconds. (Assume there are 365 days in a year.) (c) \(7 \mathrm{ft} 6 \mathrm{in},\) the height of the basketball player \(\mathrm{Yao}\) Ming, to meters. (d) \(88.6 \mathrm{~m}^{3}\) to liters.

What units do chemists normally use for density of liquids and solids? For gas density? Explain the differences.

A chemist mixes two liquids \(A\) and \(B\) to form a homogeneous mixture. The densities of the liquids are \(2.0514 \mathrm{~g} / \mathrm{mL}\) for \(\mathrm{A}\) and \(2.6678 \mathrm{~g} / \mathrm{mL}\) for \(\mathrm{B}\). When she drops a small object into the mixture, she finds that the object becomes suspended in the liquid; that is, it neither sinks nor floats. If the mixture is made of 41.37 percent \(A\) and 58.63 percent \(B\) by volume, what is the density of the metal? Can this procedure be used in general to determine the densities of solids? What assumptions must be made in applying this method?
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