If a packet of gas rising through the interior of the Sun...

If a packet of gas rising through the interior of the Sun reduces its temperature from 1.5 million $\mathrm{K}$ to $10,000 \mathrm{K}$ and has a density decrease from $120 \mathrm{kg} / \mathrm{L}$ to $5.5 \mathrm{kg} / \mathrm{L}$, by what factor has the pressure on it changed?

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Key Concepts

Proportional Relationships in Thermal Physics

Understanding proportional relationships is crucial when calculating changes in a system's properties. In problems where both density and temperature vary, the pressure change can be determined by multiplying the individual change factors, assuming direct proportionality. This method is widely used in solving problems involving gas dynamics and energy transport in astrophysical contexts.

Ideal Gas Law

The ideal gas law relates pressure, density, and temperature of a gas through the equation P ? ?T when the composition (and thus the gas constant) remains constant. This principle is fundamental in understanding how variations in temperature and density affect the pressure of a gas, which is essential in many areas of physics including astrophysics and thermodynamics.

Transcript

00:01 In this problem, we're going to talk about adiabatic processes in ideal gases.

00:06 So for ideal gases, when an adiabatic process happens, the quantity p times v to the gamma is a constant quantity, where gamma is the adiabatic index and is equal to the heat capacity at constant pressure divided by the heat capacity.

00:33 Capacity at constant volume.

00:36 It's important for us to notice that for monoatomic gases, the value of the adabatic index is 5 thirds.

00:50 Also for ideal gases, pv is equal to nrt, where n is the number of moles, r is the ideal gas constant, and t is the temperature.

01:02 And what we have in our problem is helium.

01:06 Remember that the helium is an oboe gas, so the molecule is monoatomic.

01:15 And also this helium gas undergoes a process where its temperature bellows.

01:24 So the final temperature is equal to two times the initial temperature in kelvin.

01:31 And our goal is to find by what factor the pressure changes.

01:36 So basically what we want to find is what is the ratio between the final pressure and the initial pressure.

01:44 So first i'm going to use the ideal gas law in order to isolate v.

01:52 So we have that pv is equal to nrt.

01:55 So v, the volume, is nrt divided by v pressure.

02:00 Now p v to the gamma is equal to p0 v0 to the gamma and i'm gonna substitute v here...

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