538J of work must be done to compress a gas to half its initial volume at consta

Question

538J of work must be done to compress a gas to half its initial volume at constant temperature. How much work must be done to compress the gas by a factor of 9.00, starting from its initial volume?

Explanation / Answer

isothermal process> PV = RT = k (say) temp const work done dW = PdV = k (dV/V) integrating from V1 to V2 W = k log (e) (V2/V1) ---------------------------------- 538 = k log (e) [(V1/2) / V1] = k log (e) [(1/2)] 538= - k log (e) (2) -------- (1) this compression is by factor of 2 or V2/V1 = 1/2 ------------------------- for compression by factor of 9 or V2/V1 = 1/9 W = - k log (e) (9) -------- (2) =============== (2) / (1) W = 538 * log (e) (9) / log (e) (2) W = 538 * log (e) (3^2) / log (e) (2) W = 538 * 2* log (e) (3) / log (e) (2) W = 1705.41 Joule work needed to compress by factor of 9 goes to log-scale

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