Thermodynamics and Thermochemistry - Result Question 44
####44. The reversible expansion ob an ideal gas under adiabatic and isothermal conditions is shown in the figure. Which of the following statement(s) is (are) correct?
(2012)
(a) $T _1=T _2$
(b) $T _3>T _1$
(c) $W _{\text {isothermal }}>W _{\text {adiabatic }}$
(d) $\Delta U _{\text {isothermal }}>\Delta U _{\text {adiabatic }}$
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Solution:
- (a) Since, change of state $\left(p _1, V _1, T _1\right)$ to $\left(p _2, V _2, T _2\right)$ is isothermal therefore, $T _1=T _2$.
(b) Since, change of state $\left(p _1, V _1, T _1\right)$ to $\left(p _3, V _3, T _3\right)$ is an adiabatic expansion it brings about cooling of gas, therefore, $T _3<T _1$.
(c) Work done is the area under the curve of $p-V$ diagram. As obvious from the given diagram, magnitude of area under the isothermal curve is greater than the same under adiabatic curve, hence $W _{\text {isothermal }}>W _{\text {adiabatic }}$
(d) $\Delta U=n C _v \Delta T$
In isothermal process, $\Delta U=0$ as $\Delta T=0$
In adiabatic process, $\Delta U=n C _v\left(T _3-T _1\right)<0$ as $T _3<T _1$.
$$ \Rightarrow \quad \Delta U _{\text {isothermal }}>\Delta U _{\text {adiabatic }} $$
NOTE Here only magnitudes of work is being considered otherwise both works have negative sign.
