[ethyl ethanoate]

Hello,

If the surface to volume ratio of a sphere with radius r is (4πr²) / (4/3 * πr³) = 3/r,

why is it that there are more atoms on the surface when the radius decreases? And why are there more atoms on surface when the number of atoms increase?

[Borek]

So basically you are asking why 3/r grows when r gets smaller?

What is larger: 1/2 or 1/3? 3/2 or 3/3?

[Corribus]

Also, there aren't more atoms on the surface when the particle gets smaller. Same as the fact that the surface area gets smaller as the radius gets smaller. Rather, the fraction of total atoms in the particle that are on the surface increases as the radius decreases. This is related to the concept called the specific surface area, the surface area of a solid divided by its mass. Since the surface area of a sphere scales as r² and the mass scales as r³ (mass is proportional to volume), then specific surface area scales as 1/r, has (SI) units of m^-1, and gets larger as r gets smaller. Since the nanoparticle mass can be thought of as in terms of "number of atoms", then the number of atoms on the surface divided by the total number of atoms in the particle also gets larger as r gets smaller. You can see it is the case by imagining a "particle" of exactly 1 atom. In this case, 100% of the atoms are on the surface. And in the case of a huge particle a mile wide, the fraction of particles on the surface is effectively zero.

[jeffmoonchop]

Check the graph. The Y axis is not number of atoms on surface. Its the percentage of atoms on the surface.

[ethyl ethanoate]

Sorry, I was just so confused because in the script it was basically talking about the ratio of surface area to volume. It showed that it was 3/r and then concluded from the equation that "with decreasing radius the ratio of surface area to volume increases and more and more atoms are on the surface".

Also, there aren't more atoms on the surface when the particle gets smaller. Same as the fact that the surface area gets smaller as the radius gets smaller. Rather, the fraction of total atoms in the particle that are on the surface increases as the radius decreases. This is related to the concept called the specific surface area, the surface area of a solid divided by its mass. Since the surface area of a sphere scales as r² and the mass scales as r³ (mass is proportional to volume), then specific surface area scales as 1/r, has (SI) units of m^-1, and gets larger as r gets smaller. Since the nanoparticle mass can be thought of as in terms of "number of atoms", then the number of atoms on the surface divided by the total number of atoms in the particle also gets larger as r gets smaller. You can see it is the case by imagining a "particle" of exactly 1 atom. In this case, 100% of the atoms are on the surface. And in the case of a huge particle a mile wide, the fraction of particles on the surface is effectively zero.

Ah okay I get it now, so basically if we decrease r that basically means increasing number of atoms?
For example if we imagine one particle as exactly one atom, then if we decrease its radius, it splits up into many tiny atoms which still occupies the same volume (like the picture above) which leads to the fact that the ratio of the atoms on the surface to the total number of atoms is very very little?

Is my understanding correct?

[Borek]

Ah okay I get it now, so basically if we decrease r that basically means increasing number of atoms?

No, smaller radius means a smaller sample, that always means less atoms.

For example if we imagine one particle as exactly one atom, then if we decrease its radius, it splits up into many tiny atoms which still occupies the same volume (like the picture above) which leads to the fact that the ratio of the atoms on the surface to the total number of atoms is very very little?

Nope. Atoms are indivisible (that's actually what their name means).

Imagine atoms being small cubes.

Combine them into a larger cube 10x10x10 - how many atoms there?

How many of them are on the surface? How many of them are inside?

What is the ratio of these numbers?

Now combine identical atoms into a smaller cube 5x5x5 - how many atoms there?

How many of them are on the surface? How many of them are inside?

What is the ratio of these numbers?

For which cube - the large 10x10x10, or the small 5x5x5 - is the ratio of those on the surface to those inside larger?

[ethyl ethanoate]

Imagine atoms being small cubes.

Combine them into a larger cube 10x10x10 - how many atoms there?

How many of them are on the surface? How many of them are inside?

What is the ratio of these numbers?

Now combine identical atoms into a smaller cube 5x5x5 - how many atoms there?

How many of them are on the surface? How many of them are inside?

What is the ratio of these numbers?

For which cube - the large 10x10x10, or the small 5x5x5 - is the ratio of those on the surface to those inside larger?

So the ratio of atoms on the surface area to volume for the 10x10x10 cube is < the 5x5x5 cube.

Ah okay I get it now, so basically if we decrease r that basically means increasing number of atoms?

No, smaller radius means a smaller sample, that always means less atoms.

Thank you! I understand now.  I don't know why I initially thought that the radius here refers to the radius of an atom.

So basically higher radius of an object --> the number of atoms the object is made of is higher--> ratio of atoms on surface area to volume of object decreases.

The radius of the 10x10x10 cube is larger than the radius of the 5x5x5 cube therefore the ratio of atoms on the surface area to volume for the 10x10x10 cube is < the 5x5x5 cube.