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As OP wished, assuming just ideal gas, we have for N ideal gas particles $$H(\{r_i\},\{p_i\}) = \sum_i^N (mgz_i + \frac{p_i^2}{2m})$$ We get a Boltzmann factor $$P(\{r_i\},\{p_i\}) = \Pi_i^N {e}^{-mgz_i/k_BT - \frac{p_i^2}{2m}/k_BT}$$ We can calculate the density $$\rho(r) = \frac{< \sum^N_i \delta(r_i-r) >}{< 1 >}$$, where the notation ...

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As Mikael Kuisma remarked, gas particle do accumulate at lower altitude. Consider two volumes $V_u$ and $V_l$ that are vertically thin as compared to their horizontal extend, which are separated by a distance $H$. A tube of negligible volume connects $V_l$ to $V_u$. A gas particle of mass $m$ in volume $V_u$ has a potential energy of $mgH$ as compared to a ...

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As mentioned in one of the comments, the gas particles do accumulate at the bottom of the cylinder to some extent. This is essentially why the atmosphere becomes "thinner" as the altitude increases. The exact circumstances of how the gas would behave are also dependent on the particles' kinetic energy and the thermal environment. If the thermally conductive ...

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On the phase diagram of methane, you can see that at RT (20°C), methane can only be gas (or super-critical if pressure is enough). The pressure can be calculated with the ideal gas equation $\frac{pV}{T}=nR$ You need to calculate quantity of n (in moles, given the volume, density of liquid methane, and the weight of the molecule). R is a constant, V is ...

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Background Let us assume we have a function, $f_{s}(\mathbf{x},\mathbf{v},t)$, which defines the number of particles of species $s$ in the following way: $$dN = f_{s}\left( \mathbf{x}, \mathbf{v}, t \right) \ d^{3}x \ d^{3}v$$ which tells us that $f_{s}(\mathbf{x},\mathbf{v},t)$ is the particle distribution function of species $s$ that defines a ...

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Credits given to all answers posted. They helped me figured this out. Thanks a lot. Temperature is heavily linked with Kinetic Energy. Pressure is heavily linked with number of Collisions per Time AND Kinetic Energy. Example: A gas is hot when the molecules posses high Kinetic Energy and collides with the measuring device with great force. A gas is ...

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A gas is hot when the molecules collided with your measuring device. Not quite. Gas heats your measuring device when the collisions are mostly such that the colliding gas molecule has more kinetic energy than the colliding measuring device molecule. It's instructive to think colliding molecules as sumo wrestlers: The molecule which has more ...

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To measure somethings means to compare it with an etalon or a measurement instrument, made by the help of an etalon (or the combination of etalons). To measure the pressure of a gas inside a volume one take for example a barometer and measures the pressure difference to the outer room. The measured pressure inside the volume is the result of the hitting of ...

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Pressure is a measure of force per unit area exerted on the 'measuring device', while the temperature is a measure of kinetic energy of the individual molecules of the gas. Thus, high pressure can arise when there are either many slow moving molecules with low kinetic energy colliding with the container, or a few fast moving molecules colliding with the ...

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An example of a difference where the pressure of a reasonably dilute gas depends on something else other than the kinetic energy of the particles is actually just the air on Earth. A classic exercise in statistical mechanics is to consider an ideal gas subject to gravity and find how the pressure varies with altitude. Of course, in reality the temperature ...

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Of course, they are relate to each other but that doesn't mean they are the same things. Temperature is the average kinetic energy of the molecules while pressure is the force they exert perpendicularly on any surface. Of course, more the temperature, more would be the pressure. While the former is related to the energy, the later is related to the ...

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By the Ideal gas law, $PV=nRT$, or "pressure times volume equals the number of molecules times a constant times temperature". So, all else being the same, as the temperature goes up, the pressure goes up in an exact ratio. However, all else does not have to be the same. So, for instance, if you reduce the number of molecules in a container ($n$), the ...

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The ideal gas law is the time averaged steady state of this system. Consider it on a very long timescale: if you wait several round trips, what is the average impulse imparted in $\Delta t$? This of course includes the interior transit time. Another way of seeing this is that at any instant, most of the gas particles are not pushing on the balloon surface ...

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There is not just one particle in the the box or container. There will be many particles rebounding every time on the wall under consideration. That's why we used $F_{avg}$ and not just $F$. It's because $F$ means the force applied by just one single particle and $F_{avg}$ means the total force applied by all the particles over that period of time $\Delta ... 2 You can get a rough idea from the virial theorem. This tells us that for a gravitationally bound system the kinetic energy$T$and the potential energy$V$are related by: $$2T = -V$$ or obviously: $$T = -\tfrac{1}{2}V$$ Suppose we start with our dust cloud particles at infinity with$T = V = 0$and let the system collapse until the potential energy ... 0 The ideal gas law is a cpmbonation of many other laws about gases. Some assune the pressure to be constant, others assume the quantity stays constant and others. Now those laws have been set up mostly after experiment and it people working on it noticed that the pressure$P$according to the small laws seemed to be proportional to the quantity$n$(in ... 0 In classical thermodynamics, temperature$T$is defined through ideal gas equation $$pV = nRT$$ from which we conclude that $$K.E. = \frac{3}{2}k_BT$$ is true for any ideal monatomic gas which cannot exist in real life anyways. Statistical mechanics provides postulates that is broader in context. It redefines the temperature through the second law$\$dE = ...

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