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One mole of ideal gas taken through a cycle process with alternate isothermal and adiabatic curves. In $P-V$ diagram $AB, CD, EF$ are isothermal curves at the absolute temperature $T_1, T_2$ and $T_3$ respectively and $BC, DE$ and $FA$ are adiabatic curves respectively. If $\frac{{{V_B}}}{{{V_A}}} = 2,\,\frac{{{V_D}}}{{{V_C}}} = 2$ then for cycle is shown in figure four statements are being made given below. (Figure is not drawn on scale)
Statement $1$ : Ratio of volumes $\frac{{{V_E}}}{{{V_F}}} = 4$
Statement $2$ : Magnitude of work done in isothermal compression $EF$ is $2RT_3\ ln\ (2)$
Statement $3$ : Ratio of heat supplied to gas in the process $AB$ to heat rejected by gas in process $EF$ is $\frac{{{T_1}}}{{2{T_3}}}$
Statement $4$ : Net work done by gas in the cycle $ABCDEFA$ is $(T_1 + T_2 - 2T_3) R\ ln\ (2)$
Find the number of correct statement $(s)$ given for the cyclic process followed by gas
$1$
$2$
$3$
$4$
Solution
$\frac{V_{B}}{V_{A}}=2, \frac{V_{D}}{V_{C}}=2$
from a diabatic process equation $\frac{V_{E}}{V_{F}}=\frac{V_{B}}{V_{A}} \times \frac{V_{D}}{V_{C}}=2 \times 2=4$
magnitude of work done in $EF$ is
$\left|\mathrm{RT}_{3} ln \left(\frac{\mathrm{V}_{\mathrm{F}}}{\mathrm{V}_{\mathrm{E}}}\right)\right|=2 \mathrm{RT}_{3} l(2) \cdot \quad$ Ratio $\quad$ of $\quad$ heat
supplied and rejected is equal to ratio of their work done.
