Open Physics Challenge: One mole of a gas is put under weightless piston of a vertical cylinder at temprature T.the space over opens into atmosphere.Initially piston was in equilibrium.How much work should be odn by some external force to increase isothermally the volume under piston twice the volume?

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Open Physics Challenge: One mole of a gas is put under weightless piston of a vertical cylinder at temprature T.the space over opens into atmosphere.Initially piston was in equilibrium.How much work should be odn by some external force to increase isothermally the volume under piston twice the volume?

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\[pV=n_mRT\quad,\quad p=n_mRT\frac{1}{V}\]\[A=\int pdV=n_mRT\int\limits_{V_0}^{2V_0}\frac{dV}{V}=n_mRT\cdot\ln{2}\quad(n_m=1mol)\]
no try to find out what you missed
Ans is \[n_mRT(1-\ln2)\] You ignored work done against atmosphere

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\[p_0V_0=pV=n_mRT\]\[A=\int(p_0-p)dV=\int p_0dV-\int pdV=\]\[=p_0\int\limits_{V_0}^{2V_0}dV-n_mRT\int\limits_{V_0}^{2V_0}\frac{dV}{V}=p_0V_0-n_mRT\cdot\ln{2}=\]\[=n_mRT(1-\ln{2})\]Now is OK

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