Show that during an adiabatic expansion of an ideal monatomic gas the temperature $T$ and volume $V$ are given by

$T{V}^{\frac{2}{3}}$ = constant

Calculate the efficiency of the cycle.

The work done during the isothermal expansion A → B is 540 J. Calculate the thermal energy that leaves the gas during one cycle.

Calculate the ratio $\frac{V_C}{V_B}$ where V_C is the volume of the gas at C and V_B is the volume at B.

Calculate the change in the entropy of the gas during the change A to B.

Explain, by reference to the second law of thermodynamics, why a real engine operating between the temperatures of 620 K and 340 K cannot have an efficiency greater than the answer to (b)(i).

substitution of $P=\frac{nRT}{V}$ in $P_X{V}_X^{\frac{5}{3}}=P_Y{V}_Y^{\frac{5}{3}}$ ✔

manipulation to get result ✔

e « = 1 −$\frac{T_c}{T_h}$ = 1 − $\frac{340}{620}$ » = 0.45 ✔

heat into gas «is along AB» and equals

Q_in «= ΔU + W = 0 + 540» = 540 «J» ✔

heat out is (1−e) Q_{in =} (1−0.45) × 540 = 297 «J» ≈ 3.0 × 10² «J» ✔

Award [2] for bald correct answer.

$T_B{V}_B^{\frac{2}{3}}=T_C{V}_C^{\frac{2}{3}}\Rightarrow \frac{V_C}{V_B}={\left(\frac{T_B}{T_C}\right)}^{\frac{3}{2}}$  ✔

$\frac{V_C}{V_B}={\left(\frac{620}{340}\right)}^{\frac{3}{2}}=2.5$ ✔

Award [2] for bald correct answer.

 ΔS «= $\frac{Q}{T}$ = $\frac{540}{620}$»= 0.87 «JK⁻¹» ✔

the Carnot cycle has the maximum efficiency «for heat engines operating between two given temperatures »✔

real engine can not work at Carnot cycle/ideal cycle ✔

the second law of thermodynamics says that it is impossible to convert all the input heat into mechanical work ✔

a real engine would have additional losses due to friction etc ✔

The algebraic manipulation required for this question was well mastered by only the better-prepared candidates. Many candidates tried to find the required formula via randomly selected equations from the data booklet.

The efficiency of the Carnot cycle was well calculated by most of the candidates, but the thermal energy that leaves the gas was well calculated only by the best candidates. Many candidates were able to establish that the heat added to the gas was 540 J - but struggled to then link this with the efficiency to determine the thermal energy leaving the gas. In iii) the better candidates used the formula given at i) to appropriately calculate the ratio but many were not able to manipulate the expressions to achieve the desired outcome. The reciprocal of 2.5 (0.4) was a common error in the final result and so was the incorrect use of the formula for isobaric expansion. The change in the entropy was well calculated by many candidates with ECF being very prominent from part biii). A complete and proper explanation in ii) was well formulated only by the better candidates. However, many were able to recognize that the Carnot engine was ideal and not real. Many answers referred to heat being lost to the environment but not to “additional losses” that make the engine less than its ideal capacity.

The efficiency of the Carnot cycle was well calculated by most of the candidates, but the thermal energy that leaves the gas was well calculated only by the best candidates. Many candidates were able to establish that the heat added to the gas was 540 J - but struggled to then link this with the efficiency to determine the thermal energy leaving the gas. In iii) the better candidates used the formula given at i) to appropriately calculate the ratio but many were not able to manipulate the expressions to achieve the desired outcome. The reciprocal of 2.5 (0.4) was a common error in the final result and so was the incorrect use of the formula for isobaric expansion. The change in the entropy was well calculated by many candidates with ECF being very prominent from part biii). A complete and proper explanation in ii) was well formulated only by the better candidates. However, many were able to recognize that the Carnot engine was ideal and not real. Many answers referred to heat being lost to the environment but not to “additional losses” that make the engine less than its ideal capacity.

The efficiency of the Carnot cycle was well calculated by most of the candidates, but the thermal energy that leaves the gas was well calculated only by the best candidates. Many candidates were able to establish that the heat added to the gas was 540 J - but struggled to then link this with the efficiency to determine the thermal energy leaving the gas. In iii) the better candidates used the formula given at i) to appropriately calculate the ratio but many were not able to manipulate the expressions to achieve the desired outcome. The reciprocal of 2.5 (0.4) was a common error in the final result and so was the incorrect use of the formula for isobaric expansion. The change in the entropy was well calculated by many candidates with ECF being very prominent from part biii). A complete and proper explanation in ii) was well formulated only by the better candidates. However, many were able to recognize that the Carnot engine was ideal and not real. Many answers referred to heat being lost to the environment but not to “additional losses” that make the engine less than its ideal capacity.

The efficiency of the Carnot cycle was well calculated by most of the candidates, but the thermal energy that leaves the gas was well calculated only by the best candidates. Many candidates were able to establish that the heat added to the gas was 540 J - but struggled to then link this with the efficiency to determine the thermal energy leaving the gas. In iii) the better candidates used the formula given at i) to appropriately calculate the ratio but many were not able to manipulate the expressions to achieve the desired outcome. The reciprocal of 2.5 (0.4) was a common error in the final result and so was the incorrect use of the formula for isobaric expansion. The change in the entropy was well calculated by many candidates with ECF being very prominent from part biii). A complete and proper explanation in ii) was well formulated only by the better candidates. However, many were able to recognize that the Carnot engine was ideal and not real. Many answers referred to heat being lost to the environment but not to “additional losses” that make the engine less than its ideal capacity.

The efficiency of the Carnot cycle was well calculated by most of the candidates, but the thermal energy that leaves the gas was well calculated only by the best candidates. Many candidates were able to establish that the heat added to the gas was 540 J - but struggled to then link this with the efficiency to determine the thermal energy leaving the gas. In iii) the better candidates used the formula given at i) to appropriately calculate the ratio but many were not able to manipulate the expressions to achieve the desired outcome. The reciprocal of 2.5 (0.4) was a common error in the final result and so was the incorrect use of the formula for isobaric expansion. The change in the entropy was well calculated by many candidates with ECF being very prominent from part biii). A complete and proper explanation in ii) was well formulated only by the better candidates. However, many were able to recognize that the Carnot engine was ideal and not real. Many answers referred to heat being lost to the environment but not to “additional losses” that make the engine less than its ideal capacity.