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| ### Statement |
| ### Statement |
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| $5.6.14.$ [Insert the problem statement] |
| $5.6.14.One mole of gas participating in the process, the graph of which is shown in |
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| the figure, passes through states 1, 2, and 3 sequentially. The internal energy |
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| of the gas is proportional to the temperature (U = cT). Find the amount of |
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| heat absorbed by the gas in this process. |
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| ### Solution |
| ### Solution |
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| Assuming we're working with ideal gases\ |
| Assuming we're working with ideal gases\ |
| We have process 1-2 (isochoric), $W_1-2=0$, and for first law of thermodynamics\ |
| We have process 1-2 (isochoric), $W_{12}=0$, and for first law of thermodynamics\ |
| $Q_1-2=Delta U_1-2$\ |
| $Q_{12}=\Delta U_{12}$\ |
| $Q_1-2=c(T_2 - T_1)$\ |
| $Q_{12}=c(T_2 - T_1)$\ |
| and for process 2-3 (isobaric), we'll have\ |
| and for process 2-3 (isobaric), we'll have\ |
| $Q_2-3=Delta U_2-3 + W_2-3$ , where $W_2-3=P_2(V_2 -V_1)$\ |
| $Q_{23}=\Delta U_{23} + W_{23}$ , where $W_{23}=P_2(V_2 -V_1)$, $P_2=P_3$\ |
| $Q_2-3=c(T_3 - T_2) + P_2(V_2 -V_1)\ |
| $Q_{23}=c(T_3 - T_2) + P_2(V_2 -V_1)$\ |
| The amount of heat absorbed by the gas in this whole process is $Q_a=Q_1-2 +Q_2-3$\ |
| The amount of heat absorbed by the gas in this whole process is\ |
| $Q_a=c(T_2 -T1) + c(T_3 -T_2)+P_2(V_2 -V_1)$\ |
| $Q_a=Q_{12 }+Q_{23}$\ |
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| $Q_a=c(T_2 -T_1) + c(T_3 -T_2)+P_2(V_2 -V_1)$\ |
| $Q_a=c(T_3 -T_1)+P_2(V_2 -V_1)$\ |
| $Q_a=c(T_3 -T_1)+P_2(V_2 -V_1)$\ |
| by the ideal gas law, we have\ |
| by the ideal gas law, we have\ |
| $P_1V_1=RT_1$, and $P_2V_2=RT_3$, from this, we get\ |
| $P_1V_1=RT_1$, and $P_2V_2=RT_3$, from this, we get\ |
| $T_3 -T_1=\frac{P_2V_2 -P_1V_1}{R}$\ |
| $T_3 -T_1=\frac{P_2V_2 -P_1V_1}{R}$\ |
| and substituting this into Q_a, we get\ |
| and substituting this into Q_a, we get\ |
| $Q_a=\frac{c(P_2V_2 -P_1V_1}{R} + P_2(V_2 -V_1)$ |
| $Q_a=\frac{c(P_2V_2 -P_1V_1)}{R} + P_2(V_2 -V_1)$ |
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| #### Answer |
| #### Answer |
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| $Q_a=\frac{c(P_2V_2 -P_1V_1)}{R} +P_2(V_2 -V_1)$ |
| [Insert a concise answer or boxed result] |
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