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Let us say we have an ideal gas inside a cylinder closed with a piston. We put the cylinder over a heat bath at temperature $T_{1}$ and allow the gas to expand isothermally as bellow (we ignore gravity).

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We assume that initially the pressure on both sides of the piston is $P_0$.

Now, since we have an ideal gas, during each step of the process we must have $$PV=NkT,$$ and since we're considering an isothermal expansion, the equation reduces to $$PV=const.$$ This equation tells us that as the gas is expanding, the pressure inside the cylinder $P$ is decreasing, thus we will have $$P<P_{0}.$$ But the external pressure $P_{ext}$ is actually increasing (or, worst case scenario, remains at the same initial value $P_{0}$), thus $$P_{ext}\ge P_{0}.$$

So how can the piston be pushed outside if the internal pressure $P$ is less than the external pressure $P_{ext}$ during each step of the process?

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  • $\begingroup$ It is very hard to follow your equations. Do you know how to use MathJax? I believe a link to a tutorial may be available on this site. $\endgroup$
    – Bob D
    Jan 8, 2020 at 23:11
  • $\begingroup$ @BobD But I did use MathJax, you can check it by trying to edit the post. $\endgroup$
    – Hilbert
    Jan 8, 2020 at 23:13
  • $\begingroup$ Oops! Sorry. I was looking at the your edit in transition. $\endgroup$
    – Bob D
    Jan 8, 2020 at 23:14
  • $\begingroup$ @BobD I tried to make it more readable, does it look fine? $\endgroup$
    – Hilbert
    Jan 8, 2020 at 23:20
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    $\begingroup$ To get the gas to expand isothermally, you need to gradually lower the external pressure (in tandem with the internal pressure). $\endgroup$ Jan 8, 2020 at 23:20

1 Answer 1

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This equation tells us that as the gas is expanding, the pressure inside the cylinder ๐‘ƒ is decreasing...

Yes the pressure inside the cylinder is decreasing, but that is because the pressure outside the cylinder is also decreasing at the same time but the decrease in external pressure is infinitesimally greater than the decrease in internal pressure so that expansion can occur.

So how can the piston be pushed outside if the internal pressure ๐‘ƒ is less than the external pressure ๐‘ƒ๐‘’๐‘ฅ๐‘ก during each step of the process?

Because the internal pressure is not less than the external pressure. The internal pressure is infinitesimally greater than the external pressure.

Hope this helps.

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  • $\begingroup$ But the decrease in external pressure is done artificially by us, and it is not a consequence of the expansion of the gas; in other words, we have to make sure, by some means, that the external pressure is decreasing, alongside the internal one, isn'it? $\endgroup$
    – Hilbert
    Jan 9, 2020 at 6:59
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    $\begingroup$ Yes the external pressure is gradually reduced, by whatever means, allowing the gas to expand and do work. When it expands its pressure goes down in tandem with the external pressure $\endgroup$
    – Bob D
    Jan 9, 2020 at 7:24

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