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Density Problem involving solid metal added to mercury in a cylinder

Submitted by futbolher on Thu, 01/29/2009 - 09:57

Hi! I've been staring at this problem forever now. I feel like I don't have enough information to solve it. Help would be greatly appreciated. Thanks!

Q: Solid metal with a mass of 225 grams is placed into a graduated cylinder and the rest is filled with mercury. The metal is then removed and mercury is added to fill the rest of the cylinder. Now the cylinder weighs 101 grams less than before. The density of mercury is 13.6 grams per cm cubed. What is the density of the solid metal?

I answered a question almost identical to this one a while back. Let me try to find it for you...

They try to trick you by not giving you a specific volume. I think this is throwing you off. Since density is an intensive property ( it doesn't matter how much we have) assume that the graduated cylinder has a capacity of 100mL. Since they don't give you a specific volume, we can make any volume up we want (I chose 100 since it's a relatively easy number to work with). We can do this because density is constant.(The density of 1 ml of pure water has the same density as 100ml of pure water).
If you want to see this for yourself, work the problem again by choosing a totally random volume instead of 100 ml.

Here's how you do it

I'm going to abbreviate mercury as Hg and the unknown metal as M

Mass of Hg  +  Mass of M = x grams     

It says she fills the cylinder to capacity with Hg, so the total volume is 100ml
(100mL= volume Hg + volume of M)

Mass of Hg = x grams -101 grams         
Again, it is filled to capacity, so the volume is 100mL (volume= volume Hg)

Determine the mass of Hg without the metal, M. In the second part, the volume is again filled to capacity, but only with mercury. So the mass of the mercury= Density X volume  =  13.6g/cm^3  X 100mL = 1360g Hg

Now we can solve for x.

Mass of Hg = x grams -101 grams

1360 g  = x -101

x = 1461g

Recall that x is the total mass of the metal and mercury.

We already know the mass of metal M is 255 g. So the mass of mercury in the first part is:

Mass of Hg  +  Mass of M = x grams

Mass of Hg  +  255g = 1461g

Mass of Hg= 1206g Hg

Now determine the volume of mercury:

1206g Hg / 13.6g/cm^3  = 88.676 mL Hg

But the graduated cylinder was filled to capacity (100 mL) right? The missing volume comes from metal M:

volume of metal + volume of Hg = 100mL

volume of metal + 88.676mL =100 mL

volume of metal= 11.32 mL

from there, calculate the density of the metal

density = mass/volume

density = 255g/ 11.32mL

            = 22.52g/cm^3