Question

We isolate $n$ moles of an ideal gas with heat capacity $C_{P m}$ and then heat the gas and allow it to expand from a state $P_1, V_1, T_1$ to a final state $P_2, V_2, T_2$. Find a general formula for the $\Delta S$ of this process by treating the process as having two steps: an isothermal expansion step and a separate isobaric heating step. Show that the solution does not depend on which step (the expansion or the heating) is carried out first.

    We isolate $n$ moles of an ideal gas with heat capacity $C_{P m}$ and then heat the gas and allow it to expand from a state $P_1, V_1, T_1$ to a final state $P_2, V_2, T_2$. Find a general formula for the $\Delta S$ of this process by treating the process as having two steps: an isothermal expansion step and a separate isobaric heating step. Show that the solution does not depend on which step (the expansion or the heating) is carried out first.
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Physical Chemistry : Thermodynamics, Statistical Mechanics & Kinetics
Physical Chemistry : Thermodynamics, Statistical Mechanics & Kinetics
Andrew Cooksy 1st Edition
Chapter 9, Problem 22 ↓
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We isolate $n$ moles of an ideal gas with heat capacity $C_{P m}$ and then heat the gas and allow it to expand from a state $P_1, V_1, T_1$ to a final state $P_2, V_2, T_2$. Find a general formula for the $\Delta S$ of this process by treating the process as having two steps: an isothermal expansion step and a separate isobaric heating step. Show that the solution does not depend on which step (the expansion or the heating) is carried out first.
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Transcript

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00:01 In this question we are given as delta s is equal to n cv log t2 upon t1 plus nr log v2 upon v1 so c p is equal to cv plus r so this is given and the value of row is given as a ratio of cp upon cv.
00:41 So this is the ratio of molar capacity at pressure respect to volume.
00:52 So now we know that du is equal to dq minus dw and by the second law of thermodynamics that change in entry.
01:04 Is equal to change in heat with respect to temperature so we have this d -q is equal to t -d -s now solving for ds and we have this d -s is equal to d -s is equal to where d -q is given as d -u plus d -w so we will get as d -u from this equation 1, this is d .u plus d .w.
01:41 Divided by t.
01:43 Now, delta s will be equal to integration at the initial stage of the entropy and the final stage of the entropy as d .s.
02:00 So, further, d .s will be as equal to initial and final of the entropy of.
02:09 D u plus d w upon t hence delta s is equal to u initial and u final then top initial and final d u upon t plus integration with work initial and work final as d w upon t now we know that d .u is equal to n .c .v.
02:57 D .t.
03:00 And d .w.
03:02 Is equal to p.
03:05 D .v.
03:07 Is equal to nrt upon v.
03:13 D .v.
03:13 So here we can put this equation as delta s is equal to t1 and t final n c v d t upon t plus v initial and v final of nr t upon t.
03:46 So t is canceling out...
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