An ideal gas exists in a state with pressure , volume . It is isothermally expanded to 4 times of its initial volume (), then isobarically compressed to its original volume. Finally the system is heated isochorically to bring it to its initial state. The

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Q.
An ideal gas exists in a state with pressure $P_{0}$ , volume $V_{0}$ . It is isothermally expanded to 4 times of its initial volume ( $V_{0}$ ), then isobarically compressed to its original volume. Finally the system is heated isochorically to bring it to its initial state. The amount of heat exchanged in this process is: [2025]

1 $P_{0} V_{0} (2 \ln 2 - 0.75)$

2 $P_{0} V_{0} (\ln 2 - 0.75)$

3 $P_{0} V_{0} (\ln 2 - 0.25)$

4 $P_{0} V_{0} (2 \ln 2 - 0.25)$

Ans.
(1)

$Q = △ U + W$

For cyclic process Q = W

$W = W_{1} + W_{2} + W_{3}$

   $= P_{0} V_{0} \ln \frac{4 V_{0}}{V_{0}} + \frac{P_{0}}{4} (V_{0} - 4 V_{0}) + 0$

   $= P_{0} V_{0} \ln 4 - \frac{3}{4} P_{0} V_{0}$

   $= P_{0} V_{0} (2 \ln 2 - 0.75)$

Solution

1 $P_{0} V_{0} (2 \ln 2 - 0.75)$

2 $P_{0} V_{0} (\ln 2 - 0.75)$

3 $P_{0} V_{0} (\ln 2 - 0.25)$

4 $P_{0} V_{0} (2 \ln 2 - 0.25)$

Ans.
(1)

$Q = △ U + W$

For cyclic process Q = W

$W = W_{1} + W_{2} + W_{3}$

   $= P_{0} V_{0} \ln \frac{4 V_{0}}{V_{0}} + \frac{P_{0}}{4} (V_{0} - 4 V_{0}) + 0$

   $= P_{0} V_{0} \ln 4 - \frac{3}{4} P_{0} V_{0}$

   $= P_{0} V_{0} (2 \ln 2 - 0.75)$

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Solution

1 P0V0(2 ln 2–0.75)

2 P0V0(ln 2–0.75)

3 P0V0(ln 2–0.25)

4 P0V0(2 ln 2–0.25)

Ans. (1) Q=△U+W For cyclic process Q = W W=W1+W2+W3 =P0V0 ln 4V0V0+P04(V0–4V0)+0 =P0V0 ln 4–34P0V0 =P0V0(2 ln 2–0.75)

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1 $P_{0} V_{0} (2 \ln 2 - 0.75)$

2 $P_{0} V_{0} (\ln 2 - 0.75)$

3 $P_{0} V_{0} (\ln 2 - 0.25)$

4 $P_{0} V_{0} (2 \ln 2 - 0.25)$

Ans.
(1)

$Q = △ U + W$

For cyclic process Q = W

$W = W_{1} + W_{2} + W_{3}$

$= P_{0} V_{0} \ln \frac{4 V_{0}}{V_{0}} + \frac{P_{0}}{4} (V_{0} - 4 V_{0}) + 0$

$= P_{0} V_{0} \ln 4 - \frac{3}{4} P_{0} V_{0}$

$= P_{0} V_{0} (2 \ln 2 - 0.75)$