Instructor's Solutions Manual to Accompany Atkins' Physical by P. W. Atkins

By P. W. Atkins

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For an isothermal expansion in a van der Waals gas dUm = (a/Vm )2 . Hence Um = −a(1/Vm,2 − 1/Vm,1 ). 14(a). This formula corresponds to what one would expect for a real gas. As the molecules get closer and closer the molar volume gets smaller and smaller and the energy of attraction gets larger and larger. 23 shows that the Joule–Thomson coefficient can be expressed in terms of the parameters representing the attractive and repulsive interactions in a real gas. If the attractive forces predominate then expanding the gas will reduce its energy and hence its temperature.

P RT 2a =− + 3 ∂V T (V − b)2 V R ∂p ; = ∂T V V −b then, dp = R V −b dT + RT 2a − V3 (V − b)2 dV ∂V is more readily evaluated with the use ∂T p Because the van der Waals equation is a cubic in V , of the permuter. ∂p R ∂T V ∂V V −b =− =− ∂p ∂T p + V2a3 − (VRT ∂V T −b)2 RV 3 (V − b) RT V 3 − 2a(V − b)2 = For path a T2 ,V2 T1 ,V1 dp = = T2 T1 V2 R RT2 2a − + 3 dT + 2 V1 − b (V − b) V V1 dV RT2 RT2 R (T2 − T1 ) + − −a V1 − b (V2 − b) (V1 − b) =− RT2 RT1 + −a V1 − b V 2 − b 1 V22 − 1 V22 − 1 V12 1 V12 For path b T2 ,V2 T1 ,V1 dp = = V2 V1 − T2 RT1 R 2a dV + + dT 2 3 (V − b) V T1 V2 − b RT1 RT1 − −a V 2 − b V1 − b =− RT2 RT1 + −a V1 − b V 2 − b 1 V22 1 − 1 V22 V12 − + R (T2 − T1 ) V2 − b 1 V12 Thus, they are the same and dp satisfies the condition of an exact differential, namely, that its integral between limits is independent of path.

We can conclude that, if the proposal involves collecting heat from environmentally cool ground water and using the energy to heat a home or to perform work, the proposal cannot succeed no matter what level of sophisticated technology is applied. Should the “ground” water be collected from deep within the Earth so that Th > Tc , the resultant geothermal pump is feasible. 11, must be high to compete with fossil fuels because high installation costs must be recovered during the lifetime of the apparatus.

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