In 1983, the United States began coining the cent piece out...

In $1983,$ the United States began coining the cent piece out of copper-clad rinc rather than pure copper. The mass of the old copper penny is $3.083 \mathrm{g},$ while that of the new cent is 2.517 g. Calculate the percentage of sinc (by volume) in the new cent. The density of copper is $8.960 \mathrm{g} / \mathrm{cm}^{3}$ and that of zinc is $7.133 \mathrm{g} / \mathrm{cm}^{3} .$ The new and old coins have the same volume.

Key Concepts

Volume Percent Composition

Volume percent composition measures the proportion of each component in an alloy by volume, rather than by mass. This concept is important in scenarios where different materials may have significantly different densities, such that a simple mass percentage does not reflect their contribution to the overall volume. It provides a means of understanding how much physical space each component occupies within the alloy.

Alloy Composition Calculations

Alloy composition calculations involve determining the proportion of each component in an alloy based on their individual masses, densities, or volumes. When the total volume of the alloy is known or fixed, the calculation typically requires finding the volume contributed by each constituent material using their respective masses and densities, ensuring that the sum of these individual volumes equals the total volume.

Mass-Volume Relationship

The mass-volume relationship is described by the equation mass = density × volume. When calculating the composition of an alloy by volume, one can use the known mass and density of each component to compute the volume it occupies. This relationship is essential in connecting the physical properties of the materials involved in an alloy.

Density

Density is a fundamental property that relates the mass of a substance to its volume. It serves as a conversion factor between the two and is pivotal when determining how much space a given mass of a material will occupy. In alloy calculations, the densities of the constituent metals help determine the individual volumes that combine to form the final alloy.

Transcript

00:01 Hello, so in this problem, we're trying to figure out the percentage of zinc that is in the new penny compared to what was in the old penny, which was pure made by pure copper.

00:19 And we are given the mass of the old and the new penny and the densities of zinc and copper.

00:25 So how do we go about figuring this out? well, first of all, we know that our mass is going to equal in general the volume times the density.

00:37 So in our old penny, it's going to equal the volume times the density of copper because in our old penny was pure made up of pure copper.

00:48 And then our new mass is going to be the sum percentage, we're going to say a, some percentage of the volume.

00:59 Is going to still be made up by copper plus some percentage of volume.

01:06 I'm going to denote that as 1 minus a so that the total volume is going to be the same for both the old and the new pennies.

01:16 And some percentage of volume is going to equal zinc.

01:22 Okay.

01:23 So now we have two equations with two unknown variables.

01:29 The volume and a, our percentage.

01:33 So we should be able to just solve the systems of equations and find out what our percentage is going to be.

01:43 So first of all, let's figure v in terms of the first expression.

01:47 So we get v equals the mass of our old penny divided by the density of copper.

01:54 Cool.

01:54 So let's keep that in mind.

01:56 And in our second equation, let's try solving this one for a.

02:00 You can plug in the v later, just so that we don't have a bunch of variables flying around.

02:06 So let's try doing that.

02:09 We can first expand this parentheses on the right.

02:13 So this is going to equal v.

02:17 Ro zinc minus a.

02:21 V.

02:22 Rosink.

02:24 Okay.

02:25 And now we can put together, we can combine terms.

02:29 So we have m -new minus v -ros -ink...

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