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Show that, for an ideal gas,$$P=\frac{1}{3} \rho v_{\mathrm{rms}}^{2}$$where $P$ is the pressure, $\rho$ is the mass density, and $v_{\mathrm{rms}}$ is the rms speed of the gas molecules.

$$P=\frac{1}{3} \rho v_{r . m . s}^{2}$$

Physics 101 Mechanics

Chapter 13

Temperature and the Ideal Gas

Fluid Mechanics

Temperature and Heat

Thermal Properties of Matter

The First Law of Thermodynamics

The Second Law of Thermodynamics

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Rutgers, The State University of New Jersey

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in this question, we have to show that for India last, the pressure is given by 1/3 off the mass density times. They are a mass velocity squared. So how can be sure that I will begin by using this equation? So the idea of God's law, the Indio God's Law tells us that the pressure times the volume is equal to the number off more your kings or atoms times boatmen's constant times the temperature. Then we can solve this equation for the pressure to get that the pressure is given by an times k times T divided by the volume. Now take a look at this factor off K Times T right here. So this factor off King Times t also appear in the expression for the average kinetic energy off a single Moya Q or atom off ideal gas. So we can actually serve this equation for K time steep by doing that to get the following. So the average kinetic energy is three divided by two times K time. Steve. It means that Kate Times t is across the truth, divided by three times the average kinetic energy. Then plug in these results into these equation results in the following The pressure is given by AM, divided by V. Times two divided by three times the average kinetic energy. Now we must remember from classical mechanics that the kinetic energy is the mass times the velocity squared, divided by True as we're dealing with a gas, I write that the average kinetic energy off a single molecule off Mass M is it close to the mass times the average square velocity divided by True. So we can plugging these results into these equations, and by doing that, we get the following. Pressure is an divided by V times two divided by three times en masse times average off the squared, divided by two. Then we can simplify this factor and this factor to get the following the pressure is n times m divided by we kinds. 1/3 finds the average squared velocity. No, take a look at this factor. What we have here, well, we have the number of Michael's times, the mass off each one, a coup. So this is nothing else than the total mass off that gas. So this is the total mass off the gas divided by the volume in his occupying and we know this quantity as the mass density. So we already have the mass density here, and we also have the factor off 1/3. So the pressure is equals to 1/3 times the mass density times the average square, the last 16. Now remember that the definition off the right means queer velocity is the following. The right means squared is equal to the square it off the average square velocity. So we conclude that they are a mass velocity squared is the question the average velocity squared. Then we finally get the desired expression that says that the pressure is it goes to 1/3 times the mass density times V. R M as squared as desired.

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