How Does an Ideal Gas Sphere Expand in a Vacuum?

[afallingbomb]

I thought of what seems to be a very classical problem but I can't find a solution for it.

At t=0 you have a sphere of radius, R, made up of an ideal gas at a temperature, T, and pressure, P. The sphere is sitting in an infinite vacuum. At t>0 you allow the sphere to expand freely outward. Are there analytical solutions to the velocity (of the surface or perhaps along any point on the radius), density, and other state variables (P, T, etc) as they evolve through time? Thank you!

 

[PhaseShifter]

Essentially you have an area of low pressure (vacuum, or "gas with zero pressure") containing a smaller region of higher pressure (the sphere). Wouldn't this propagate at the same speed sound propagates within the gas, like other pressure disturbances?

I suppose you would get a better picture of what would really happen by using statistical mechanics, but then you have to decide what the surface of the sphere is, since there is no upper limit on the speed of gas molecules. (They do become exponentially more uncommon at higher speeds, though.)

 

[charng]

Dear colleague,

Thank you for bringing up this interesting problem. The expansion of a sphere of ideal gas in a vacuum is indeed a classical problem that has been studied extensively in thermodynamics and fluid mechanics. The solution to this problem involves several important physical concepts, including the ideal gas law, adiabatic expansion, and the conservation of mass and energy.

To answer your question, yes, there are analytical solutions to the velocity, density, pressure, and temperature as they evolve through time during the expansion of the sphere. However, the exact solutions depend on the specific assumptions and conditions of the problem, such as the initial temperature and pressure of the gas, the composition of the gas, and the type of expansion (isobaric, isothermal, or adiabatic).

In general, the velocity of the surface of the sphere can be calculated using the ideal gas law and the conservation of mass, which states that the volume of the gas remains constant during the expansion. The density and other state variables can then be determined from the velocity using the equations of state for an ideal gas.

For adiabatic expansion, which is the most common assumption for this problem, the temperature and pressure of the gas decrease as the volume increases. This can be described by the adiabatic process equation, which relates the temperature and pressure at different volumes. The density can also be calculated using the ideal gas law and the conservation of mass.

In conclusion, while there are analytical solutions to the velocity, density, and other state variables for the expansion of a sphere of ideal gas in a vacuum, the exact solutions depend on the specific conditions and assumptions of the problem. I hope this helps in your research and I am happy to discuss this topic further.

Best regards,