Density calculation sometimes can be confusing

[Rev. Cheeseman]

Ok, let's compare two cubes of lead. First lead cube weigh 6078 grams and its area is 3376 cm. The second lead cube is smaller and lighter at 5216 grams and 2713 cm area. The density of the first lead cube which is the bigger and heavier lead (by dividing its weight with the area) is 1.800 g/cm2 while the second lead cube which is lighter and smaller is 1.922 g/cm2.

This is confusing because in real life, the bigger and heavier lead cube will be much denser than the smaller and lighter cube. Is there anything wrong with the calculation? Thank you.

 

[Orodruin]

First of all you should get your units and nomenclature correct.

area is 3376 cm

3376 cm is a length, not an area

2713 cm area

Same here, length not area

by dividing its weight with the area

Density is mass/volume, not mass/area.

This is confusing because in real life, the bigger and heavier lead cube will be much denser than the smaller and lighter cube.

Finally, this is false. The density of a material does not depend on the object made out of the material. The bigger object will be heavier, but not denser. Density is the mass/volume.

 

[Rev. Cheeseman]

Finally, this is false. The density of a material does not depend on the object made out of the material. The bigger object will be heavier, but not denser. Density is the mass/volume.

Ok, so it should be volume instead of area. Thus, even in reality the smaller and lighter lead cube will be denser than that bigger and heavier lead cube in the question? Does denser means stronger? Sorry, I have a very limited background in chemistry.

 

[Orodruin]

Ok, so it should be volume instead of area. Thus, even in reality the smaller and lighter lead cube will be denser than that bigger and heavier lead cube in the question? Does denser means stronger? Sorry, I have a very limited background in chemistry.

No, they will have the same density if they are made of the same material. Density tells you how much mass a material has per volume. You take a cube that has twice the volume of another cube of the same material - it will have twice the mass. Density is a property of the material itself independent of the object made from it.

Where are you getting the numbers from?

 

[Rev. Cheeseman]

No, they will have the same density if they are made of the same material. Density tells you how much mass a material has per volume. You take a cube that has twice the volume of another cube of the same material - it will have twice the mass. Density is a property of the material itself independent of the object made from it.

Where are you getting the numbers from?

Ok, so the density of the big lead cube and smaller lead cube is actually just the same although the bigger lead cube is heavier and obviously bigger and the smaller lead cube is lighter. I just made up those numbers. Is this also applies to other metals such as calcium, potassium, etc.?

 

[Orodruin]

Ok, so the density of the big lead cube and smaller lead cube is actually just the same although the bigger lead cube is heavier and obviously bigger and the smaller lead cube is lighter. I just made up those numbers. Is this also applies to other metals such as calcium, potassium, etc.?

How do you just make up numbers and pretend to learn from that?

Yes, it goes for other materials too (with the cavet that they need to be under the same conditions - pressure, temperature, etc - eg, materials typically expand when heated, leading to lower density).

 

[Rev. Cheeseman]

How do you just make up numbers and pretend to learn from that?

Yes, it goes for other materials too (with the cavet that they need to be under the same conditions - pressure, temperature, etc - eg, materials typically expand when heated, leading to lower density).

I copy these numbers from somewhere else. It is a question from a learning group on Facebook but I can no longer find the question but I managed to copy and paste the numbers into a Word file. So, a bigger and heavier bone with no organic composition remaining will have the same density as a smaller and lighter bone. How can the density remain the same?

 

[Orodruin]

How can the density remain the same?

Because if you take twice the amount of some mateterial it will have twice the mass. One liter of water has half the mass of two liters of water. Density tells you the mass of something per how much you have of it. Larger objects weigh proportionally more. As simple as that.

I copy these numbers from somewhere else.

Then you should not say you made them up. You should say where they came from. Based on your description it is unclear if they are experimental values or just made up ones - nor if there are any uncertainties attached to them. Without that information, the resulting densities are close enough to suspect that something like experimental uncertainties could account for the difference.

 

[Rev. Cheeseman]

Because if you take twice the amount of some mateterial it will have twice the mass. One liter of water has half the mass of two liters of water. Density tells you the mass of something per how much you have of it. Larger objects weigh proportionally more. As simple as that.

This part is confusing to me, because the bigger and heavier lead cube or bone that is stripped off of organic components will have the same density as the smaller and lighter lead cube or bone with no organic components as we have discussed earlier.

Then you should not say you made them up. You should say where they came from. Based on your description it is unclear if they are experimental values or just made up ones - nor if there are any uncertainties attached to them. Without that information, the resulting densities are close enough to suspect that something like experimental uncertainties could account for the difference.

Made them up by copying these numbers from somewhere, that's what I mean. Sorry, what do you mean by experimental uncertainties?

 

[Orodruin]

This part is confusing to me, because the bigger and heavier lead cube or bone that is stripped off of organic components will have the same density as the smaller and lighter lead cube or bone with no organic components as we have discussed earlier.

Why is that confuaing to you? Are you fine with the statement that two liters of water weigh twice as much as one liter?

Sorry, what do you mean by experimental uncertainties?

Any measurement is connected to a measurement error telling you how precise that measurement is. For example, if you measure the side of a cube with a precision of 1 cm, that would translate to an error in the side length of your smallest cube of about 7%. Taking the cube of this length to get volume would then lead to an error larger than 20% in the volume. Such an error would be more than enough to account for the difference in the resulting densities.

 

[Rev. Cheeseman]

Why is that confuaing to you? Are you fine with the statement that two liters of water weight twice as much as one liter?

I'm just fine with the bigger bone that is stripped off of organic components or just say hydroxyapatite being heavier than the smaller and lighter bone but the density remains the same. Looks like right now density is the same as weight or mass.

Any measurement is connected to a measurement error telling you how precise that measurement is. For example, if you measure the side of a cube with a precision of 1 cm, that would translate to an error in the side length of your smallest cube of about 7%. Taking the cube of this length to get volume would then lead to an error larger than 20% in the volume. Such an error would be more than enough to account for the difference in the resulting densities.

So that is the mistake of my question because I was using area instead of volume. I don't know the volume of the lead cubes in my example, only the area.

 

[Borek]

Looks like right now density is the same as weight or mass.

No, it is not. Mass of one liter of water is 1 kg, mass of two liters of water is 2 kg, but the density is in each case the same - 1 kg/liter.

There are basically two types of properties matter has - we call them extensive properties and intensive properties. Extensive properties depend on the amount of matter - so the higher the volume of water, the larger its mass is. Intensive properties don't depend on the amount of matter - density of each sample of water*, no matter what its size is, is always 1 kg/1L (or 1g/mL).

*There are some slight differences related to the water purity and temperature, but we can safely ignore them for now.

 

[Orodruin]

There are some slight differences related to the water purity and temperature, but we can safely ignore them for now.

Just to add to this: ”Bone” (as used as an example by the OP earlier in the thread) is not of a fixed density because it is much less specific than ”water”. Bone density is not only minorly affected by temperature and pressure but also by composition, which can be affected by things like medical conditions (eg osteoporosis).

 

[symbolipoint]

Ok, so it should be volume instead of area. Thus, even in reality the smaller and lighter lead cube will be denser than that bigger and heavier lead cube in the question? Does denser means stronger? Sorry, I have a very limited background in chemistry.

No. That makes no sense. One given material? Is it the same form for both samples? Same density!

 

[symbolipoint]

Because if you take twice the amount of some mateterial it will have twice the mass. One liter of water has half the mass of two liters of water. Density tells you the mass of something per how much you have of it. Larger objects weigh proportionally more. As simple as that.

You would benefit to understand that density is a ratio.

 

[symbolipoint]

Looks like right now density is the same as weight or mass.

No. Three different concepts. One of them is a ratio.

 

[Rev. Cheeseman]

Just to add to this: ”Bone” (as used as an example by the OP earlier in the thread) is not of a fixed density because it is much less specific than ”water”. Bone density is not only minorly affected by temperature and pressure but also by composition, which can be affected by things like medical conditions (eg osteoporosis).

Yes, what about bone especially bones that had been stripped off of their organic components leaving just the minerals? For example, person A has a total body bone mineral content of 6078 grams and its area is 3376 cm. Person B has a total body bone mineral content of 5216 grams and 2713 cm area. Here, we are talking about areal density which is weight divided by 2D width of the bone which is how DEXA scan machines calculated bone mineral density.

Noted that I simply replaced the lead cubes with bones while keeping the original numbers from my original question.

The areal bone mineral density for person A (6078 gram/3376 cm) is 1.800 g/cm2 and person B (5216 gram/2713 cm) is 1.922 g/cm2.

Using areal density calculation, the smaller and lighter bones of person B is denser than the bigger and heavier bones of person A which is strange and confusing despite both are composed of same material which is hydroxyapatite which is a naturally occurring mineral form of calcium apatite with the formula Ca₅(PO₄)₃, often written Ca₁₀(PO₄)₆(OH)₂ to denote that the crystal unit cell comprises two entities. As we have discussed earlier that the correct measurement for the density of both of these bones is by using volumetric density, is the real density between these bones still the same?

 

[symbolipoint]

#17, now very difficult to understand. Too technical for just the two of us is my best grasp. You talk about aereal density but not you understand Density as the ratio of mass to volume?

 

[Borek]

Bone density as measured by clinical densitometry is different from the density understand as a physical property of matter. Names are similar but they mean different things.

There is some lousy relationship between both and optical density (used when scanning bones in living humans) expressed as mass/surface is used as a proxy for a density expressed as mass/volume, but there is plenty of assumptions behind, so don't mix these things before you understand what the density really is.

 

[Rev. Cheeseman]

#17, now very difficult to understand. Too technical for just the two of us is my best grasp. You talk about aereal density but not you understand Density as the ratio of mass to volume?

Bone density as measured by clinical densitometry is different from the density understand as a physical property of matter. Names are similar but they mean different things.

There is some lousy relationship between both and optical density (used when scanning bones in living humans) expressed as mass/surface is used as a proxy for a density expressed as mass/volume, but there is plenty of assumptions behind, so don't mix these things before you understand what the density really is.

As I have discussed earlier with Orodruin, I eventually got it that density should be mass or weight divided by volume. Not mass or weight/surface or area.

So, density changes for something that is not uniform or fixed like the mineral content of bones. Thus, the bigger and heavier bones that I talked earlier were denser than the smaller and lighter bones then? Or, the previous surface or areal bone mineral density that we got from the smaller and lighter bones being higher than the bigger and heavier bones do not actually mean that the real density of the smaller and lighter bones is actually higher in reality?

Noted that we're talking about things that are dynamic and flexible not uniform and fixed like the lead cubes.

 

[Borek]

Thus, the bigger and heavier bones that I talked earlier were denser than the smaller and lighter bones then?

Not necessarily. They can be, but they don't have to be.

Larger bones do have higher optical density, but it doesn't mean they are denser. This is one of the reasons why optical density has to be treated with caution, as a proxy only.

 

[Rev. Cheeseman]

Not necessarily. They can be, but they don't have to be.

Larger bones do have higher optical density, but it doesn't mean they are denser. This is one of the reasons why optical density has to be treated with caution, as a proxy only.

What we understand is that bigger and heavier bone with more mineral contents (that resulted to bones with heavier weight) will be stronger and therefore denser (as in the context of bone which are not uniform and fixed, denser means the bone is not easily broken compared to another).

In what way the smaller and lighter bone is denser than the bigger and heavier bone in reality? Less porous? Less porous will result to heavier bone but it is not because the bigger bone is heavier.

What is more confusing, even without taking into account the porosity of the bone, the formula of areal density itself will still give you a higher density figure of the smaller and lighter bone.

I agree that areal density, such as what is produced by clinical densitometry, gives ambiguous and vague results that are open to multiple interpretations. Especially the higher areal density figure of the smaller and lighter bone than the heavier and bigger bone that we got earlier.

 

[Tom.G]

It sounds like there is a mix-up between Mass-density and Optical-density.

The OPs question seems to be about Optical Density.

Even though they both use the same word, density, in their name, they are two different things.

As @Borek pointed out above:

Mass of one liter of water is 1 kg, mass of two liters of water is 2 kg, but the density is in each case the same - 1 kg/liter.

This is the definition generally used in physics, mechanics, and many other fields.

The optical density (or X-Ray density) of a substance is how much radiation, light or X-Ray, is blocked by the substance.

For instance if you take that 1kg of water and put a few drops of ink in it, the optical density will increase substantially (less light will get thru it), but the mass density will be (almost) unchanged at 1kg per liter.

Hope this helps!

Cheers,
Tom

 

[Rev. Cheeseman]

It sounds like there is a mix-up between Mass-density and Optical-density.

The OPs question seems to be about Optical Density.

Even though they both use the same word, density, in their name, they are two different things.

As @Borek pointed out above:

This is the definition generally used in physics, mechanics, and many other fields.

The optical density (or X-Ray density) of a substance is how much radiation, light or X-Ray, is blocked by the substance.

For instance if you take that 1kg of water and put a few drops of ink in it, the optical density will increase substantially (less light will get thru it), but the mass density will be (almost) unchanged at 1kg per liter.

Hope this helps!

Cheers,
Tom

Looks like if the organic components of bones are decreasing or increasing, the inorganic components such as minerals also follow. But to repeat my previous statement, even excluding these mobile organic and inorganic components, using the calculation of surface/areal density we still get higher density figure for the lighter mass/smaller surface. Which makes me think to not depends too much on the surface density calculation when it comes to assessing whether this bone is dense or not.

 

[Rev. Cheeseman]

My point here is that, instead of relying on surface density, why don't we rely on the size and weight/mass of the bone alone to assess whether the bone is less likely to break or fracture or whether this or that bone has osteoporosis?

 

[Tom.G]

...why don't we rely on the size and weight/mass of the bone alone to assess whether the bone is less likely to break...

Uhmm, would you like your thigh bone removed so it can be weighed and measured?

 

[Rev. Cheeseman]

Uhmm, would you like your thigh bone removed so it can be weighed and measured?

Of course not, we have no choice but to assume with using indirect methods. Even before getting the answer with surface/areal density, we already get the weight/mass and the size before we divide the weight/mass and the size.

 

[Borek]

Even before getting the answer with surface/areal density, we already get the weight/mass and the size before we divide the weight/mass and the size.

"We already get" - do we?

Please elaborate on where we have this information from.

 

[Rev. Cheeseman]

"We already get" - do we?

Please elaborate on where we have this information from.

Sorry, which information?

 

[Borek]

Sorry, which information?

"weight/mass and the size", just like you stated in what I quoted.

 

[Rev. Cheeseman]

"weight/mass and the size", just like you stated in what I quoted.

That's how those DEXA scan machines calculated the bone mineral density. You stated previously "Bone density as measured by clinical densitometry is different from the density understand as a physical property of matter. Names are similar but they mean different things." I agree with it.

 

[Rev. Cheeseman]

"weight/mass and the size", just like you stated in what I quoted.

I mean, the mass or weight and the two-dimensional surface or area which is basically two-dimensional representation of the bone size are already there before we divide them to get the surface/areal density.

 

[Borek]

That's how those DEXA scan machines calculated the bone mineral density.

Nope. They _measure_ optical density and bone cross section (these things are doable using xray image) and _estimate_ bone mass and bone volume from these numbers. You got it reversed.

 

[Rev. Cheeseman]

Nope. They _measure_ optical density and bone cross section (these things are doable using xray image) and _estimate_ bone mass and bone volume from these numbers. You got it reversed.

By dividing mass/weight with the surface/area. Correct?

 

[Orodruin]

By dividing mass/weight with the surface/area. Correct?

No. You do not have the mass or weight. That is why you need to estimate it from what you can infer.