How Can We Calculate the Pressure Inside a Sphere With Varying Density Gas?

[78141471]

Homework Statement

I have a sphere filled with unknown gas. I have to estimate the pressure inside of it.

• The tricky part is that density of the sphere changes as we go deeper inside of it with the given formula:
  ρ=ρ_o*(1-r/R), where R is radius of the sphere and is known: R=5*10¹⁰m, and r is a variable.
  
• Gas has mol mass m=2.7 g/mol.
  
• Mass of the whole sphere is M=3.3*10⁵⁰ kg.
  
• Gas is ideal.

Homework Equations

• pV=nRT
  
• p=F/A

The Attempt at a Solution

The force of gravity of an infinitesimal layer of thickness dr inside the sphere at some radius r caused by the inner sphere is:

dF=GmM/r² where m is the mass of the infinitesimal layer and M is the mass of inner part of the sphere.

Mass m=V*ρ=4*PI*r²*ρ_o*(1-r/R)*dr
Mass M=4/3*PI*r³*ρ_o*(1-r/R)

so dF=G*16/3*PI²*r³*ρ_o²*(1-r/R)²*dr

dp=dF/A
dp=dF/4*PI*r²
dp=4/3*PI*G*ρ_o²*(1-r/R)²*r*dr

so
p(r)=4/3*PI*G*ρ_o²*$$\int$$(1-r/R)²*r*dr from r to R

After making the calculations I will put p(r=0) and get the pressure inside the sphere. Is my method good?

 

[78141471]

too hard or did I do something wrong?

 

[nlightner]

I would say that your method is a good start, but it may not give an accurate estimation of the pressure inside the sphere. There are a few assumptions and simplifications that may affect the accuracy of your calculation.

Firstly, assuming that the gas is ideal may not accurately represent the behavior of the unknown gas inside the sphere. Real gases can deviate from ideal behavior, especially at high pressures.

Secondly, using the formula for density changing with depth may not be accurate for a gas, as gases tend to expand to fill their container. This formula is more commonly used for liquids or solids.

Additionally, the force of gravity may not be the only force acting on the gas inside the sphere. There may be other external forces, such as electromagnetic forces, that could affect the pressure.

Overall, your method provides a good starting point, but to get a more accurate estimation of the pressure inside the sphere, you may need to consider these factors and possibly use more advanced equations and techniques. It may also be helpful to conduct experiments or gather more information about the gas inside the sphere to improve the accuracy of your estimation.