Effusion of Gas Through a Small Hole
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Effusion of Gas Through a Small Hole Mohammad Khorrami1 Received: 3 September 2020 / Accepted: 5 November 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract A vessel containing a classical non-relativistic ideal gas is studied, which has a small hole. Gas effuses through this hole, but the hole is so tiny that the gas inside the vessel remains homogeneous. The density and the temperature of the gas inside the vessel are obtained as functions of time. Keywords Cooling · Effusion · Gas · Temperature
1 Introduction If there is a hole on the side of a vessel containing a gas, gas escapes through that vessel. For an ideal gas, the escape rate is proportional to the average speed (norm of the velocity vector) of the gas molecules, the gas density (number of the molecules per volume), and the area of the hole ([1], for example). The average speed is inversely proportional to the square root of the mass of the molecules, the so-called Graham’s law of effusion ([2], for example). It is also proportional to the square root of the absolute temperature. The effusion rate has been used to determine various characteristics of gases. In [3], it is used to measure the molecular speeds. It is also used in [4] and [5] for a similar purpose. For precision measurements, however, there are cases which seem to need some corrections applied to Graham’s law ([6, 7], for example). Graham’s law is also affected by non-idealities in gas mixtures ([8], for example). As the gas effuses through the hole, the number of the molecules which are inside decreases. If the temperature of the gas inside the vessel is constant, then the time dependence of the number of the molecules inside the vessel is a simple exponential decay, and so is the time dependence of the pressure ([5, 9], for example). That time dependence is correct if the vessel is at a constant temperature (for example if it is kept at heat contact with a heat reservoir), or if the effusion is studied for short times. * Mohammad Khorrami [email protected] 1
Department of Physics, Faculty of Physics and Chemistry, Alzahra University, Tehran, Iran
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International Journal of Thermophysics
(2021) 42:14
However, if the vessel is thermally isolated, as the faster molecules escape more quickly than the slower ones ([9], for example), the average energy of the escaped molecules is larger than that of the molecules which have remained inside. So as time passes the average energy of the molecules inside the vessel, and hence the temperature of the gas inside the vessel and the average speed of the molecules inside the vessel, decreases. As a result, the differential equation governing the number of the molecules inside the vessel is coupled to another time-dependent variable, the temperature. It could be thought that if the hole is tiny, so that the escape rate is very small, the change in the temperature can be neglected. But it will be seen that even if the hole is tiny, when a substantial part of the gas has
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