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A cylinder contains 0.50 mol of an ideal...

A cylinder contains 0.50 mol of an ideal gas at temperature of 310 K. as the gas expands isothermally from an initial volume of 0.31 `m^(3)` to a final volume of 0.45`m^(3)`, find the amount of heat that must be added to the gas in order to maintain a constant temperature.

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Since the process is isothermal, the internal energy of an ideal gas does not change, i.e., `DeltaU=0`. The relation `DeltaU=Q-W` gives `Q=W`, where W is positive (work is done by the system). Now, we know that the gas gains heat from the surrounding (positive Q). as the work done W is given by `int_(V_(i))^(V_(f))PdV`, we obtain W as
`W=int_(V_(i))^(V_(f))PdV=int_(V_(i))^(V_(f))(nRT)/(V)dV=nRT int_(V_(i))^(V_(f))(dV)/(V)=nRT" In "V|_(V_(i))^(V_(f))=nRT" In "((V_(f))/(V_(i)))`
Substituting `n=0.50mol,R=8.31J//(mol" "K),V_(f)=0.45m^(3) and V_(i)=0.31m^(3)`, we obatain Q=W=480J.
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