Brass is a copper-zinc alloy. What is the mass in grams of a...

Brass is a copper-zinc alloy. What is the mass in grams of a brass cylinder having a length of $1.62 \mathrm{in}$. and a diameter of $0.514 \mathrm{in}$. if the composition of the brass is $67.0 \%$ copper and $33.0 \%$ zinc by mass? The density of copper is $8.92 \mathrm{~g} / \mathrm{cm}^{3}$, and the density of zinc is $7.14 \mathrm{~g} / \mathrm{cm}^{3}$. Assume that the density of the brass varies linearly with composition.

Key Concepts

Linear Interpolation

Linear interpolation is a mathematical method used to estimate an unknown value that falls between two known values. In the case of brass, where the density changes linearly with copper and zinc composition, the density of brass can be determined by interpolating between the density of copper and the density of zinc according to the given mass percentages.

Density of Alloys

The density of an alloy, such as brass, is a property that depends on the densities of the constituent metals and their relative proportions. When the density is said to vary linearly with composition, it implies that the overall density of the alloy can be obtained as a weighted average of the densities of its individual components based on their mass fractions.

Volume Calculation

Volume calculation involves determining the space occupied by a three-dimensional object. For a cylinder, this is done using the formula V = ?r²h, where r is the radius and h is the height (or length). This concept is important as it provides the geometric basis for calculating the mass once the density is known.

Unit Conversion

Unit conversion is the process of converting measurements from one set of units to another. In this context, converting the dimensions from inches to centimeters is essential since the density values are given in grams per cubic centimeter. Accurate conversion is crucial to ensure that the calculated volume is in the same unit system as that used for the density of the materials.

Transcript

00:01 We're going to be considering brass which is a copper zinc alloy.

00:11 Okay, so what we're going to find what is mass in grams of a brass cylinder that it has a length of 1 .62 inches a diameter of 0 .514 inches it is 67 if it's 67 percent 67 .0 let's get the sig figs in there which one copper and 33 .0 percent zinc by mass the density of copper is 8 .92 grams per cubic centimeter and the density of zinc is 7 .14 grams per cubic centimeter and we're going to assume that it varies linearly the density varies linearly okay this one shouldn't be too hard to do um let me go ahead and highlight we've got 1 .62 inches length and we've got this is our diameter this this this and this okay so let's call these l let's call this d and these are percents those are pretty easy and that okay so first we're going to find the volume of the cylinder which will be equals pi d squared l over four so this will equal pi times 0 .514 inches squared times 1 .62 inches divided by four i'm going to see if my other calculator i unplugged it shoot i don't know what happened here okay but it's dark in where i'm working now it's raining outside so i'm doing my math and i get 0 .33615 inches cubed okay so we're going to have to get this to centimeters cubed because of our densities so we're going to take this times 2 .54 we'll have to cube this whole thing take this number times 2 .54 math down down enter enter and i get 5 .5085 cubic centimeters okay so now the volume of our copper plus the volume of our zinc is going to equal 5 .5085 cubic centimeters so let's figure out the mass of copper and that's going to be 0 .670 times x and the mass of our zinc will be expressed as 0 .330 times x okay and we know that the density of copper will equal its mass divided by its volume and it's 8 .92 grams per cubic centimeter and we know that v1 our volume will be equal to our mass divided by 8 .92 and that will equal 0 .670 x over 8 .92...

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