show-that-for-an-ideal-gas-pfrac13-rho-v_mathrmrms2-where-p-is-the-pressure-rho-is-the-mass-density-

Question

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.

João Gabriel Alencar Caribé · Accepted Answer

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&#x27;s law, the Indio God&#x27;s Law tells us that the pressure times the volume is equal to the number off more your kings or atoms times boatmen&#x27;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&#x27;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&#x27;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.

Christopher Provencher · Answer

okay. In this problem, we were asked to express the R. M s velocity of the ruby in square velocity of a particle and a get in a gas. You asked. Express it as the square root of three p over row, where P is pressure of the gas and rose the density. Yes, these air macroscopic variables. They&#x27;re basically asked to find this microscopic, uh, quantity in terms of macroscopic variables like pressure and density. So we&#x27;re gonna need two different things. First, we can use the expression the traditional expression for the root means square velocity given to us by kinetic theory, which is square root. Three. Katie over an pretty is the temperature of the gas. M is the mass of each particle in the gas to these temperatures a macroscopic variable and is a microscopic verbal. We could say We also know that from the ideal gas law, which is a macroscopic equation, P V is equal to end, Katie began. Is the number of particles to your temperature still kay is the bolts been constant as well, So we need to combine these expressions together. First thing we can dio is we can say that uh, from the ideal gas law, you can say Katie is simply P v o ren. Okay, let&#x27;s go over here. Let&#x27;s start with our first expression. This over here, we&#x27;ll start you. We can replace the Katie by this expression involving pressure. Now to get get us one step closer, we will be left with three p times V over began over at HUD and little. And next thing you can do is we can express if we have the number of particles in the gas times the mass of the particle in the gas. This would give us the total mass in the gas being expressed that, as began, total mass of the gas. And we also know that density is simply total mass over a total volume. It&#x27;s just past per volume, and that&#x27;s where we get density in the appropriate place. And that&#x27;s it. That&#x27;s the main step. This is a combination of two expressions of microscopic one in a macroscopic. One can use that to derive this expression for, uh, roommates. Where velocity in a Connecticut&#x27;s That&#x27;s it.

Rachel Wellington · Answer

everyone, This is question number fifty from chapter thirteen. This problem asks us to show the Armas speed of molecules. And a gas is given by the square root of three p over row, where peace, pressure and rose density. Okay, so we have our initial equation for V R. M s equals the square root of three Katy over M. So we basically need to figure out how to express Katie and em in terms of Roe and peace. So we need to figure out how to convert the two. Okay, so we have Canty and our ideal gas equations of PV equals and Katie. Remember, Kay can also be expresses Arthur Ideal gas constant. So then Katie is equal to P V over and Okay, so let&#x27;s go ahead and plug that in to our equation. So instead of three Katy over and we&#x27;re gonna have the arm s equals the square root of three times Katie, we&#x27;ve just said is Peavy over and times em. Excuse me. Divided by him, which is the same. The same one over M. Okay, so now we have three p v over nm, and we&#x27;re supposed to have it in terms of pressure and row. But what we need to recognize here is that an and isn&#x27;t it equal Teo End times m Number of moles Times the mass of a molecule is equal to the massive massive molecule times the number of of molecules equals massive. Whatever you&#x27;re working with gas. Okay, so massive. The total product total the project that you&#x27;re working with. So then that can be changed too. Three square root three p V over m and we know that density equals mass over volumes Big Excuse me bigtime over volume. So that is how we get to the square root of three p over row. Because instead of obviously froze mass over volume than one over, Raoh is going to be equal. Teo volume over Mass, Okay?

João Gabriel Alencar Caribé · Answer

we begin this question by solving the ideal gas law for the pressure. It tells us that the pressure is given by the number off articles times the Bozeman constant times that temperature divided by the total volume. Now we solved the Ari massive speed for the temperature. We begin by squaring that so the square off the arm speed is given by three times Kate times t divided by the mass off each particle. Then we send the master the other side. So the mass times there are a mass speed squared is because to three times the boatman, constant times the temperature. And then we divided by three times K so that the mass times the are in mass speed squared, divided by three times the Bozeman constant. Is he close to the temperature? Then we played in his expression for the temperature into these expression for the pressure. By doing that to get the following, the pressure is given by the number off more whose times? The Bozeman constant times the temperature, which is given by the mass times. They are a massive speed squared, divided by three time skate times. The air compiled for you Now you can see that there is a cancellation happening because the boatman constant console out with the other boats, my constant. Then the pressure is given by the number of particles times the mass off each particle times the RMS speed squared divided by three times the total volume. You can see that the number of particles times the mass off each particle is actually the total mass off that gas. So these is the total mass. Then the pressure is given by the mass times the RMS speed squared divided by three times the volume. Now notice that the density is defined as the mass divided by the volume, so you can identify the density. Here we have mass divided by the volume. Before you can write the pressure, ask the dance ET finds the Ari mass speed squared, divided by three. And this is what we were trying to show. And this is how you can share that. The pressure can be given by this expression

João Gabriel Alencar Caribé · Answer

in this question we have to share that are massive speed off the mark accuse in an ideal gas can be given by these expressions. Then we begin by using the expression for the translational kinetic energy that comes from classical mechanics, which is this one. By using his expression, we get that the average inflation magnetic energy is given by the mass of the more accuse times. The average square speed divided by truth then noticed that definition off the aftermath of speed is that it is equals to the square it off the average square speed. Therefore, we can write the average square speed as the RMS speed squared. Then the average translation or kinetic energy is given by the mass of the more I accuse times, they are amassed speed squared, divided by truth. Now we work on the left hand side of this equation by using the expression that is given by the kinetic theory of gases, which is this one blocking his expression. The left hand side off these equation we got three divided by tree times, K times. Teak is equals to m times the RMS speed squared divided by tube. We can then simply fight true on both sides off this equation, and then they can serve this for the Ari Mast speed to get the following The Ari mass speed squared is equal to three times k times t divided by em. Then the are amassed. Speed is across the square, it off three times Key time steep, divided by m. Now we can use the ideal go slow to relate K times tea with our times t. So the idea of those law tells us that and time skate times t is he goes to leader when times are time steep. So K kind steep is because too little m kinds are times t divided by big em bloody India&#x27;s in tow. These equation we got the following the Ari mast speed he&#x27;s given by the square root off three Times Leader. When times are times t divided by big end times that mass note that here we have the number of my accuse time, the mass off each Moya Q. This is nothing else than the total mass off the gas and let me call these AM prime, so the arm at speed is given by three times Little AM times are time steep, divided by the total miles off the gas M prime. Now note that this is the mess pair mold off the gas. Say this is nothing else than the total mass off the gas divided by the number of moles off that gas. Then we can write these as this where it off three times Our times t divided by M because one divided by M is of course, to end divided by M prime. Due to the definition that was given by the question. Therefore, we finally get the conclusion that the Ari Mass speed can be written as three times our times d divided by m, and this is how we derived his expression.

João Gabriel Alencar Caribé · Answer

in this question, we have to show that the arm at speed can be calculated by using these equation. For that, we have to remember about three equations. One is the definition off the, um at speed. The order is expression for Deccan, eh? Tik energy according to classical mechanics And the order is the expression for the average translation or kinetic energy according to the kinetic theory off gases. So we begin by substituting the definition off the RMS speed in this equation. Sir, initially, we have an average translation or kinetic energy that is given by the expression off classical mechanics and the squared divided by truth. What we know that the Aramis speed is defined it as the square root off this value So we can say that the average squared speed is given by the IRA mass speed squared. Then we can write this expression as M v r. M s squared, divided by true. And then we conserve these expression for the r. M s. By doing that, you get the following We are a mass squared is equals to two times the translation or kinetic energy divided by the mass off. The more accuse then they are massed. Speed is given by the square root off three times the average translation or kinetic energy divided by the mass. Then we put him here the expression that comes from the kinetic theory of gases which is these one. And by doing that we get the following. The RMS speed is equal to the square it off three times three times skate times t divided by AM times True. Now you can simplify true with Chu and get the final answer to this problem, which is that the RMS speed can be given by the square root off three times in a boat. Zeman constant divided by the mass The more you keys, this is how you derived it Expression.

Problem

Estimate the percentage of the $\mathrm{O}_{2}$ m…

Problem 63 Medium Difficulty

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.

Answer

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

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