Thermodynamics and Thermochemistry 1 Question 36
39. A reversible cyclic process for an ideal gas is shown below. Here, $p, V$ and $T$ are pressure, volume and temperature, respectively. The thermodynamic parameters $q, w, H$ and $U$ are heat, work, enthalpy and internal energy, respectively.
(2018 Adv.)
The correct options is (are)
(a) $q_{A C}=\Delta U_{B C}$ and $w_{A B}=p_{2}\left(V_{2}-V_{1}\right)$
(b) $w_{B C}=p_{2}\left(V_{2}-V_{1}\right)$ and $q_{B C}=\Delta H_{A C}$
(c) $\Delta H_{C A}<\Delta U_{C A}$ and $q_{A C}=\Delta U_{B C}$
(d) $q_{B C}=\Delta H_{A C}$ and $\Delta H_{C A}>\Delta U_{C A}$
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Solution:
- In the given curve $A C$ represents isochoric process as volume at both the points is same i.e., $V_{1}$
Similarly, $A B$ represents isothermal process (as both the points are at $T_{1}$ temperature) and $B C$ represents isobaric process as both the points are at $p_{2}$ pressure.
Now (i) for option (a)
$$ q_{A C}=\Delta U_{\mathrm{BC}}=n C_{V}\left(T_{2}-T_{1}\right) $$
where, $n=$ number of moles
$C_{v}=$ specific heat capacity at constant volume However, $W_{A B} \neq p_{2}\left(V_{2}-V_{1}\right)$ instead
$$ W_{A B}=n R T_{1} \ln \left(\frac{V_{2}}{V_{1}}\right) $$
So, this option is incorrect.
(ii) For option (b)
$$ q_{B C}=\Delta H_{A C}=n C_{p}\left(T_{2}-T_{1}\right) $$
where, $C_{p}=$ specific heat capacity at constant pressure
Likewise,
$$ W_{B C}=-p_{2}\left(V_{1}-V_{2}\right) $$
Hence, this option is correct.
(iii) For option (c)
$$ \begin{array}{lc} \text { as } & n C_{p}\left(T_{2}-T_{1}\right)<n C_{V}\left(T_{2}-T_{1}\right) \ \text { so } & \Delta H_{C A}<\Delta U_{C A} \ \text { and } & q_{A C}=\Delta U_{B C} \end{array} $$
Hence, this option is also correct.
(iv) For option (d)
Although $q_{B C}=\Delta H_{A C}$
but $\Delta H_{C A} \ngtr \Delta U_{C A}$
Hence, this option is incorrect.
