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Date	May 2019	Marks available	2	Reference code	19M.3.SL.TZ2.10
Level	Standard level	Paper	Paper 3	Time zone	2
Command term	Calculate	Question number	10	Adapted from	N/A

Question

A heat pump is modelled by the cycle A→B→C→A.

The heat pump transfers thermal energy to the interior of a building during processes C→A and A→B and absorbs thermal energy from the environment during process B→C. The working substance is an ideal gas.

Show that the work done on the gas for the isothermal process C→A is approximately 440 J.

[2]

Calculate the change in internal energy of the gas for the process A→B.

[2]

bi.

Calculate the temperature at A if the temperature at B is −40°C.

[1]

bii.

Determine, using the first law of thermodynamics, the total thermal energy transferred to the building during the processes C→A and A→B.

[3]

Suggest why this cycle is not a suitable model for a working heat pump.

[2]

Markscheme

evidence of work done equals area between AC and the Volume axis ✓

reasonable method to estimate area giving a value 425 to 450 J ✓

Answer 440 J is given, check for valid working.

Examples of acceptable methods for MP2:

- estimates 17 to18 small squares x 25 J per square = 425 to 450 J.
- 250 J for area below BC plus a triangle of dimensions 5 × 3, 3 × 5, or 4 × 4 small square edges giving 250 J + 187.5 J or 250 J + 200 J.

Accurate integration value is 438 J - if method seen award [2].

«use of $U = \frac{3}{2}nRT$ and $pV = nRT$ to give»

$\Delta U = \frac{3}{2}\Delta pV$ ✔

« $= \frac{3}{2} \times - 2.5 \times {10^5} \times 1 \times {10^{ - 3}}$ »

=«–»375«J» ✔

Another method is possible: eg realisation that ΔU for BC has same magnitude, so ΔU = 3/2 PΔV.

bi.

T_A = 816«K» OR 543«°C»✔

bii.

for CA ΔU = 0 so Q = W = −440 «J» ✔

for AB W = 0 so Q = ΔU = −375 «J» ✔

815 «J» transferred to the building ✔

Must use the first law of thermodynamics for MP1 and MP2.

the temperature changes in the cycle are too large ✔

the cycle takes too long «because it contains an isothermal stage» ✔

energy/power output would be too small ✔

Examiners report

At SL, Correct answers were rare and very few candidates used the fact the work done was area under the curve, and even fewer could estimate this area. At HL, the question was better answered. Candidates used a range of methods to estimate the area including counting the squares, approximating the area using geometrical shapes and on a few occasions using integral calculus.

Not very many candidates seem to know the generalised formula ΔU =1.5(P2V2 -P1V1) however many correct answers were seen.

bi.

The temperature at A was found correctly by most candidates.

bii.

The main problem here was deciding whether each Q was positive or negative. But the question was quite well answered.

Because the question was about a heat pump rather than a heat engine very few answers were correct. Only a very small number of candidates mentioned the fact that the isothermal change would take an impracticably long time.

Syllabus sections

Option B: Engineering physics » Option B: Engineering physics (Core topics) » B.2 – Thermodynamics

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16N.3.SL.TZ0.10c:
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17M.3.SL.TZ2.7a.i:
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17N.3.SL.TZ0.8c:
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17M.3.SL.TZ2.7a.iii:
Show that the thermal energy removed from the gas for the change BC is approximately 330 J.
18M.3.SL.TZ2.7d.i:
Sketch, on the pV diagram, the complete cycle of changes for the gas, labelling the changes clearly. The expansion shown in (a) and (b) is drawn for you.
16N.3.SL.TZ0.10a:
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17N.3.SL.TZ0.10c:
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18M.3.SL.TZ1.7a:
Show that the pressure at B is about 5 × 10⁵ Pa.
17M.3.SL.TZ1.6a:
State what is meant by an adiabatic process.
17M.3.SL.TZ1.6b:
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18N.3.SL.TZ0.7c.ii:
State, without calculation, during which change (A → B, B → C or C → A) the entropy of the gas decreases.
18N.3.SL.TZ0.7c.i:
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18N.3.SL.TZ0.7a.ii:
Show that at C the temperature is 254 K.
18M.3.SL.TZ2.7a:
Show that the final volume of the gas is about 53 m³.
18N.3.SL.TZ0.7a.i:
Show that at C the pressure is 1.00 × 10⁶Pa.
18M.3.SL.TZ2.7c:
Determine the thermal energy which enters the gas during this expansion.
16N.3.SL.TZ0.10b:
Explain, with a reason, why a real nuclear power plant operating between the stated temperatures cannot reach the efficiency calculated in (a).
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17M.3.SL.TZ1.6e:
State a reason why a Carnot cycle is of little use for a practical heat engine.
19N.3.SL.TZ0.6b(iii): state and explain whether the second law of thermodynamics is violated.
18M.3.SL.TZ1.7b.ii:
For the process BC, calculate, in J, the change in the internal energy of the gas.
19N.3.SL.TZ0.6b(i):
determine the thermal energy removed from the system.
18M.3.SL.TZ1.7b.i:
For the process BC, calculate, in J, the work done by the gas.
18M.3.SL.TZ1.7c.ii:
Determine, without any calculation, whether the net work done by the engine during one full cycle would increase or decrease.
18M.3.SL.TZ2.7d.ii:
Outline the change in entropy of the gas during the cooling at constant volume.
18N.3.SL.TZ0.7b:
Show that the thermal energy transferred from the gas during the change B → C is 238 J.
19N.3.SL.TZ0.6a(ii):
Calculate the ratio $\frac{V_{A}}{V_{C}}$ .
17M.3.SL.TZ1.6c.ii:
The volume at B is 2.30 × 10^–3 $\,$ m³. Determine the pressure at B.
17N.3.SL.TZ0.8d:
Using your sketched graph in (b), identify the feature that shows that net work is done by the gas in this three-step cycle.
18M.3.SL.TZ2.7e:
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18M.3.SL.TZ1.7d:
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18M.3.SL.TZ2.7b:
Calculate, in J, the work done by the gas during this expansion.
18M.3.SL.TZ1.7b.iii:
For the process BC, calculate, in J, the thermal energy transferred to the gas.
17M.3.SL.TZ2.7a.ii:
Show that the temperature of the gas at C is 386 K.
20N.3.SL.TZ0.9b(i):
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20N.3.SL.TZ0.9a(i):
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20N.3.SL.TZ0.9a(ii):
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17M.3.SL.TZ1.6d.ii:
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17N.3.SL.TZ0.8a:
Show that the volume of the gas at the end of the adiabatic expansion is approximately 5.3 x 10^–3 m³.
19N.3.SL.TZ0.6b(ii):
explain why the entropy of the gas decreases.
17M.3.SL.TZ1.6d.i:
Show that ${P_B}V_B^{\frac{5}{3}} = nR{T_C}V_C^{\frac{2}{3}}$
19N.3.SL.TZ0.6a(i):
Show that the pressure at B is about 130 kPa.
19M.3.SL.TZ2.10bii: Calculate the temperature at A if the temperature at B is −40°C.
19M.3.SL.TZ2.10c: Determine, using the first law of thermodynamics, the total thermal energy transferred to the...
19M.3.SL.TZ2.10d: Suggest why this cycle is not a suitable model for a working heat pump.
17M.3.SL.TZ2.7a.iv:
Determine the efficiency of the heat engine.
17M.3.SL.TZ1.6c.i:
Determine the temperature of the gas at A.
17M.3.SL.TZ2.7b:
State and explain at which point in the cycle ABCA the entropy of the gas is the largest.
19M.3.SL.TZ2.10a:
Show that the work done on the gas for the isothermal process C→A is approximately 440 J.

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Option B: Engineering physics » Option B: Engineering physics (Core topics)

Option B: Engineering physics

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Markscheme

Examiners report

Syllabus sections

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