$5.5.3$ For diminishing isothermally $n$ times the volume of gas in a cylinder with piston, over this piston is putted a load of mass $m$. What load should be added such that the volume of gas decreases isothermally $k$ times more?
Applying Newton's Second Law...
Initially, considering a massless piston $$P_0 = P_a \;(1)$$ When load of mass $m$ is putted over $$PS=mg+P_aS \;(2)$$ After the addition of the another load, $$P'S=(m+\Delta m)g+P_aS \;(3)$$ From Boyle-Mariotte Law $$P_0V_0 = P\frac{V_0}{n}$$ $$P_0 = \frac{P}{n} \;(4)$$ Substituting $(2)$ into $(3)$, according to $(1)$ and separating $P_a$ $$P_a = \frac{mg}{S(n-1)} \;(5)$$ Applying Boyle-Mariotte Law again $$P\frac{V_0}{n}=P'\frac{V_0}{nk}$$ $$P=\frac{P'}{k} \;(6)$$ Substituting $(2)$ and $(3)$ into $(6)$ and developing algebraically $$\left(\frac{mg}{S}+P_a\right)(k-1)=\frac{\Delta mg}{S} \;(7)$$ Putting $(5)$ into $(7)$ and separating $\Delta m$ $$\boxed{\Delta m=m\frac{(k-1)n}{(n-1)}}$$
BSc. Luis Daniel Fernández Quintana
Physics Department (FCNE)
Universidad de Oriente, Cuba