Isothermal vs. adabatc comresson 1. One and a half moles of a datomc gas at temerature 5 C are comressed sothermally from a volume of 0.015 m to a volume of 0.0015 m. a. Sketch the rocess on a dagram and show what corresonds to the work done on the gas. b. Calculate the work done on the gas. (Ignore vbratonal energy of the molecules) c. How much heat flowed nto the gas? d. What s the fnal ressure of the gas?
(Isothermal vs. adabatc comresson) 2. The same system undergoes an adabatc comresson wth the same ntal and fnal volumes. a. Sketch the rocess on the same dagram you used for the sothermal rocess. b. Is the fnal temerature hgher, lower or the same as n the sothermal comresson? Exlan. c. Fnd the fnal temerature of the gas. d. Accordng to your dagram, s the fnal ressure greater, lesser or the same as n the sothermal case? Exlan why (.e., what s the energy flow n each case?). Calculate the fnal ressure. e. Based on your dagram, would the amount of work done on the gas be larger, smaller, or the same as n the sothermal case? What about the heat flow? Exlan your answers brefly. f. Comute the work done on the gas for ths rocess. [Hnt: There s a long way to do ths, the ntegral of d, and a short way, usng the frst law of thermodynamcs.]
Isothermal vs. adabatc comresson 1. One and a half moles of a datomc gas at temerature 5 C are comressed sothermally from a volume of 0.015 m to a volume of 0.0015 m. a. Sketch the rocess on a dagram and show what corresonds to the work done on the gas. W fnal b. Calculate the work done on the gas. ntal f f d f W d nrt ln nrt J 0.0015 m W 1.5 moles8.1 27 K 5 Kln 88 J mol K 0.015 m Snce W>0 when work s done by the gas, then the work done on the gas s Won = +88 J. c. How much heat flowed nto the gas? Frst law of thermodynamcs: U Q W Snce T s constant, U 0, thus QW 88 J. (Q < 0, thus ths s absorbed heat) d. What are the ntal and fnal ressure of the gas? J 1.5 moles 8.1 08 K nrt mol K 0.015 m f 5 2.56 10 Pa J 1.5 moles 8.1 08 K nrt mol K 0.0015 m f 2.56 10 Pa
(Isothermal vs. adabatc comresson) 2. The same system undergoes an adabatc comresson wth the same ntal and fnal volumes as n art 1. a. Sketch the rocess on the same dagram you used for the sothermal rocess. adabatc fnal sothermal ntal b. Is the fnal temerature hgher, lower or the same as n the sothermal comresson? Exlan. Snce Q = 0, W = ΔU. W < 0 (volume decrease), so nternal energy ncreases, and that, for an deal gas, means a temerature ncrease. c. Fnd the fnal temerature of the gas. 1 For an adabatc rocess, T constant, wth Therefore, T T 1 1 f f 1 1.1 0.015 m Tf T 08 K 77 K f 0.0015 m 7 1. for a datomc deal gas. 5 d. Accordng to your dagram, s the fnal ressure greater, lesser or the same as n the sothermal case? Exlan why (.e., what s the energy flow n each case?). Calculate the fnal ressure. For an adabatc rocess, constant. Therefore,
(Isothermal vs. adabatc comresson) f f 1. 5 0.015 m 6 f 2.5610 Pa 6.10 Pa f 0.0015 m Ths fnal ressure after the adabatc rocess s greater than the fnal ressure after the sothermal rocess obtaned n art d of secton 1. e. Based on your dagram, would the amount of work done on the gas be larger, smaller, or the same as n the sothermal case? What about the heat flow? Exlan your answers brefly. Work s the area under the curve. The work s thus larger for the adabatc rocess. The heat flow s zero for the adabatc case, so the heat flow s larger for the sothermal case. f. Comute the work done on the gas for ths rocess. The short way: We can answer the queston usng the frst law, U Q W. For an adabatc rocess, Q = 0. 5 For a datomc deal gas, U nrt. 2 Therefore, 5 5 J W U nrt 1.5 moles8.1 77 K 08 K 1.510 J 2 2 mol K Agan, we need to be careful wth the sgn: the work n the exresson above s done by the gas, so the work done on the gas s W W 1.5 10 J on The long way: f We can also do the ntegral W d. For any ont along the adabatc rocess, constant at the fnal value, for nstance. Then, constant. We can evaluate the f f.
(Isothermal vs. adabatc comresson) Usng ths exresson for ressure n the ntegral for work, we obtan: f f d 1 1 f f W d ff 1 1 1 f 6 1. 6. 10 Pa 0.0015 m 1 1 W 0. 0. 0. 0.0015 m 0.015 m Agan, Won W 1.5 10 J 1.5 10 J Recall that for the sothermal rocess, the work done on the gas was 88 J. The work done on the gas s greater n the adabatc rocess.