One mole of a monoatomic ideal gas goes through a thermodynamic cycle, as shown in the volume versus temperature ($V-T$) diagram. The correct statement($s$) is/are :
[ $R$ is the gas constant]
$(1)$ Work done in this thermodynamic cycle $(1 \rightarrow 2 \rightarrow 3 \rightarrow 4 \rightarrow 1)$ is $| W |=\frac{1}{2} RT _0$
$(2)$ The ratio of heat transfer during processes $1 \rightarrow 2$ and $2 \rightarrow 3$ is $\left|\frac{ Q _{1 \rightarrow 2}}{ Q _{2 \rightarrow 3}}\right|=\frac{5}{3}$
$(3)$ The above thermodynamic cycle exhibits only isochoric and adiabatic processes.
$(4)$ The ratio of heat transfer during processes $1 \rightarrow 2$ and $3 \rightarrow 4$ is $\left|\frac{Q_{U \rightarrow 2}}{Q_{3 \rightarrow 4}}\right|=\frac{1}{2}$
IIT 2019, Advanced
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Process $1 \rightarrow 2$ is isobaric with $P =\frac{ RT _0}{ V _0}$
Process $2 \rightarrow 3$ is isochoric with $V =2 V _0$
Process $3 \rightarrow 4$ is isobaric with $P =\frac{ RT _0}{2 V _0}$
Process $4 \rightarrow 1$ is isochoric with $V = V _0$
Work in cycle $=\frac{ RT _0}{ V _0} \cdot V _0-\frac{ RT _0}{2 V _0} \cdot V _0=\frac{ RT _0}{2}$
$Q _{1-2}= nC _{ p } \Delta T = n \cdot \frac{5 R }{2} \cdot T _0$
$Q _{2-3}= nC _{ V } \Delta T = n \cdot \frac{3 R }{2} \cdot T _0$
$\therefore\left|\frac{ Q _{1-2}}{ Q _{2-3}}\right|=\frac{5}{3}$
$Q _{3-4}= nC _{ p } \Delta T = n \cdot \frac{5 R }{2} \cdot \frac{ T _0}{2}$
$\therefore\left|\frac{ Q _{1-2}}{ Q _{3-4}}\right|=2$
Ans.$1, 2$
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