A cylinder is closed by a piston connected to a spring of aconstant 4000 N/m. Wi

Question

A cylinder is closed by a piston connected to a spring of aconstant 4000 N/m. With the spring relaxed, the cylinder is filledwith 5 L of gas at a pressure of 1 atm and temperature of 25degrees C. The piston has a cross-sectional area of .0200 meterssquared and negligible mass. How high will the piston risewhen temperature of the gas is increased by 250 degrees C ?? Answershould be around 36.2 cm Thank you. A cylinder is closed by a piston connected to a spring of aconstant 4000 N/m. With the spring relaxed, the cylinder is filledwith 5 L of gas at a pressure of 1 atm and temperature of 25degrees C. The piston has a cross-sectional area of .0200 meterssquared and negligible mass. How high will the piston risewhen temperature of the gas is increased by 250 degrees C ?? Answershould be around 36.2 cm Thank you.

Explanation / Answer

Force balance P_atm*Area + k*x =P_gas*Area           P_atm is atmospheric pressure P_gas = nRT/V V = Area*height = Area*(initial height + x) So P_atm*Area + k*x = nRT*Area/ [Area*(initial height +x)] = nRT/(initial height + x) initial height you find from the initial volume initial height = V/Area = (5 L*1e-3 m3/ 1L) / [ 0.02 m2] = 0.25meters n = PV/RT = (1 atm*5 L)/(0.0821 L*atm/mol/K * 298K) = 0.204366912moles P_atm = 1 atm = 101325 Pa P_atm*Area + k*x = nRT/(initial height + x) k*x2 + (initial height*k + P_atm*Area)x +P_atm*initial height*Area= nRT x2 + (initial height*k + P_atm*Area)/k *x +(P_atm*initial height*Area - nRT)/k = 0 x2 + bx + c = 0 b = initial height*k +P_atm*Area)/k               Note use P_atm = 101325 Pa = 101325 N/m2    = (0.25 m*4000 N/m + 101325 Pa*0.02m2)/ (4000 N/m)    = 0.756625 m c = (P_atm*initial height*Area - nRT)/k    = (101325 Pa*0.25 m*0.02m2 - 0.204366912 moles*8.314J/mol/K*523 K)/ (4000 N/m)    = -0.0955019257 m2 x2 + bx + c = x2 +(0.756625 m)x -0.0955019257 m2 = 0            -0.756625 m         ±   [(0.756625 m)2 - 4*1*(-0.0955019257m2) ]1/2 x =--------------------------------------------------------------------------------                                                               2 x = 0.11 m final height = initial height + x = 0.25 + 0.11 = 0.36 meters~ 36 cm

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