Thermodynamics and Thermochemistry - Result Question 39
####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)
$$ q _{A C}=\Delta U _{BC}=n C _V\left(T _2-T _1\right) $$
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.
