Name: Second Law of Thermodynamics
Uploaded: 2019-07-19
Duration: 8 min 37 s
Description: The statement of the second law of thermodynamics is defined in detail. The video also explains the quantity called entropy and helps to understand the law i...

Question

Figure shows three isotherms at temperature $T_{1} = 4000 K, T_{2} = 2000 K, T_{3} = 1000 K$ . When one mole of an ideal monatomic gas is taken through the paths $A B, B C, C D$ and $D A$ , find the change in internal energy $Δ U$ , the work done by the gas $W$ and the heat $Q$ absorbed by the gas in each path. Also, find these quantities for complete cycle $A B C D A$ .

Accepted Answer

Given that :-

$T_{1} = 4000 K, T_{2} = 2000 K, T_{3} = 1000 K$ .

We have to find the change in internal energy $Δ U$ , the work done by the gas $W$ and the heat $Q$ absorbed by the gas in each path and quantities for complete cycle $Δ U$ .

When one mole of an ideal gas undergoes a change in temperature $Δ T$ , the change in its internal energy equals $C_{v} Δ T$ , where $C_{V}$ is the molar specific heat of the gas at constant volume. For monatomic gas $C_{V} = (3 / 2) R$ .

Thus, the change of internal energy is given by

Process $Δ U$ $W$ $Q$

$A \to B$
$= n C_{y} (T_{B} - T_{A})$

$= 1 \times \frac{3}{2} \times (4000 - 2000)$

$= 3000 R$

$= P (V_{B} - V_{A})$

$= R (T_{B} - T_{A})$

$= R (4000 - 2000)$

$= 2000 R$

$= Δ U + W$

$= 3000 R + 2000 R$

$= 5000 R$

$B \to C$
$= n C_{ν} (T_{C} - T_{B})$

$= 1 \times \frac{3}{2} \times (2000 - 4000)$

$= - 3000 R$
Zero
$= Δ U + W$

$= - 3000 R + 0$

$= - 3000 R$

$C \to D$
$= n C_{ν} (T_{D} - T_{C})$

$= 1 \times \frac{3}{2} \times (1000 - 2000)$

$= - 1500 R$

$= P (V_{D} - V_{C})$

$= R (T_{D} - T_{C})$

$= R (1000 - 2000)$

$= - 1000 R$

$= Δ U + W$

$= - 1500 R - 1000 R$

$= - 2500 R$

$D \to A$
$= n C_{v} (T_{A} - T_{D})$

$= 1 \times \frac{3}{2} \times (2000 - 1000)$

$= 1500 R$
Zero
$= Δ U + W$

$= 1500 R + 0$

$= 1500 R$

Total
$= 3000 R + (- 3000 R) + (- 1500 R) + 1500 R = 0$

$= 2000 R + 0$

$+ (- 1000 R) + 0$

$= 1000 R$

$= 5000 R +$

$(- 3000 R) +$

$(- 2500 R) + 1500$

$R = 1000 R$

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