Pycnometer Equation: Understanding Specific Gravity

[nobahar]

hello!

On wiki, regarding specific gravity, it gives the derivation for the pycnometer equation for determining specific gravity. However, the first equation confuses me. It goes as follows:
The pycnometer, placed on a balance, will exert a force:
$$F = g(m_b - \frac{ \rho_a m_b}{ \rho_b})$$
The subscript b is for the bottle, and a is for air. rho is density and m is mass; g is the acceleration due to gravity
The force on the balance would be F = gm. The above equation suggests that the mass of the air displaced by the bottle needs to be subtracted from the mass of the bottle; so the mass used in the equation is the difference in mass between the bottle and the air that would occupy the space of the material used for the bottle. I don't understand why this is the case. Why does the mass of air that would occupy that space if the bottle wasn't there even matter? It seems irrelevant. Surely we need to consider the mass of the bottle, all of its mass.

Any help appreciated.

 

[Borek]

You have to go back to Archimedes.

 

[tadchem]

hello!

On wiki, regarding specific gravity, it gives the derivation for the pycnometer equation for determining specific gravity. However, the first equation confuses me. It goes as follows:
The pycnometer, placed on a balance, will exert a force:
$$F = g(m_b - \frac{ \rho_a m_b}{ \rho_b})$$
The subscript b is for the bottle, and a is for air. rho is density and m is mass; g is the acceleration due to gravity
The force on the balance would be F = gm. The above equation suggests that the mass of the air displaced by the bottle needs to be subtracted from the mass of the bottle; so the mass used in the equation is the difference in mass between the bottle and the air that would occupy the space of the material used for the bottle. I don't understand why this is the case. Why does the mass of air that would occupy that space if the bottle wasn't there even matter? It seems irrelevant. Surely we need to consider the mass of the bottle, all of its mass.

Any help appreciated.

The second term:

$$- \frac{ \rho_a m_b}{ \rho_b}$$

is the correction to the weight of the bottle for the buoyancy of air. In the most precise work, this must be determined using air temperature, pressure, and humidity. This can amount to a correction of a few parts per million.

 

[nobahar]

Thanks for the responses. I have a really dumb question: if something is flat on the surface, how can the air "push up" underneath the bottle? Dumb question, I know, but I am trying to visualise where the pressure comes from! Thanks for the responses so far!

 

[LudusRex]

Hello! Thank you for bringing up this question about the pycnometer equation for determining specific gravity. This equation may seem confusing at first, but it is important to understand the reasoning behind it.

The pycnometer is a device used to measure the density of a substance by comparing it to the density of water. The first equation you mentioned is used to calculate the force exerted by the pycnometer on the balance, which is equal to the weight of the pycnometer. This force is equal to the difference between the weight of the pycnometer and the weight of the air that would occupy the space of the pycnometer if it were not there.

This may seem irrelevant, but it is actually a crucial component of the equation. The reason we need to consider the mass of the air is because the pycnometer is measuring the density of a substance in comparison to water. When the pycnometer is filled with the substance, it displaces a certain volume of water. This displaced water creates a buoyant force on the pycnometer, which must be taken into account in order to accurately measure the density of the substance.

So, in summary, the mass of air that would occupy the space of the pycnometer is included in the equation because it helps us account for the buoyant force that is present when measuring the density of a substance in comparison to water. I hope this explanation helps clarify the reasoning behind the pycnometer equation. Let me know if you have any further questions!