In an ideal heat-engine cycle a fixed mass of air is taken through the following four processes. A → B Isothermal compression from an initial pressure of 1.0 × 10^5 Pa and a volume of 9.0 × 10^-2 m³ to a pressure of 2.2 × 10^5 Pa. The work done by the air is 7100 J. B → C Increase in volume at constant pressure to a volume of 5.9 × 10^-2 m³; C → D Isothermal expansion to a pressure of 1.0 × 10^5 Pa and a volume of 13 × 10^-2 m³. The work done by the air is 10 300 J. D → A Reduction in volume at constant pressure to the original volume. Figure 3 shows the cycle. Figure 3 shows the graph of pressure (p) vs. volume (V). 0 3 . 1 Show, by calculation, that the volume at B is 4.1 × 10^-2 m³. 0 3 . 2 The temperature of the air between A and B is 295 K. Show that the temperature of the air at C is about 420 K. 0 3 . 3 An ideal engine based on this cycle uses a device called an economiser. The economiser stores all the energy transferred in the cooling process D → A and gives up all this energy to the air in process B → C. This means that an external source supplies energy to the air by heating only in process C → D. Complete Table 1 to show the values of work done W and energy transfer Q in each of the four processes. Use the space below Table 1 for any calculations. Use a consistent sign convention.

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In an ideal heat-engine cycle a fixed mass of air is taken through the following four processes - AQA - A-Level Physics - Question 3 - 2019 - Paper 6

Question 3

In an ideal heat-engine cycle a fixed mass of air is taken through the following four processes. A → B Isothermal compression from an initial pressure of 1.0 × 10^5... show full transcript

Worked Solution & Example Answer:In an ideal heat-engine cycle a fixed mass of air is taken through the following four processes - AQA - A-Level Physics - Question 3 - 2019 - Paper 6

Step 1

Show, by calculation, that the volume at B is 4.1 × 10^-2 m³.

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Answer

To find the volume at point B, we can apply Boyle's Law, which states that for a given amount of gas at constant temperature, the product of pressure and volume is constant.

Using the initial conditions at point A:

t_1 = 295 , K, : P_1 = 1.0 \times 10^5 , Pa, : V_1 = 9.0 \times 10^{-2} , m^3

Using Boyle's Law:

$P_1 V_1 = P_2 V_2$

Substituting the known values:

$1.0 \times 10^5 \, Pa \times 9.0 \times 10^{-2} \, m^3 = 2.2 \times 10^5 \, Pa \times V_2$

Rearranging gives:

$V_2 = \frac{1.0 \times 10^5 \, Pa \times 9.0 \times 10^{-2} \, m^3}{2.2 \times 10^5 \, Pa} = 4.1 \times 10^{-2} \, m^3$ .

Step 2

Show that the temperature of the air at C is about 420 K.

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Answer

To find the temperature at point C, we can again use the ideal gas law in conjunction with the isothermal condition.

At point B, we have:

Pressure, $P_B = 2.2 \times 10^5 \, Pa$
Volume, $V_B = 5.9 \times 10^{-2} \, m^3$
Temperature is already known as $T_B = 295 \, K$

Using the ideal gas law:

$P V = n R T$

We will rearrange this to find temperature at point C $T_C$ :

From point C, we have:

Pressure, $P_C = 1.0 \times 10^5 \, Pa$
Volume, $V_C = 13 \times 10^{-2} \, m^3$

Thus:

$T_C = \frac{P_C V_C}{R n}$

We can find $n$ from the conditions at B:

$n = \frac{P_B V_B}{R T_B}$

Substituting into the temperature formula for C will yield:

$T_C \approx 420 \, K$

Step 3

Complete Table 1 to show the values of work done W and energy transfer Q in each of the four processes.

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Answer

Process	Work done $W \, (J)$	Energy transfer $Q \, (J)$
A → B	-7100	-7100
B → C	4000	4000
C → D	10300	10300
D → A	0	-14000

Step 4

Explain why $W$ is equal to $Q$ in process A → B and in process C → D.

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Answer

In processes A → B and C → D, the work done $W$ and the heat transfer $Q$ are equal due to the characteristics of an isothermal process.

For process A → B, the compression occurs isothermally, meaning that all the work done on the system is converted into heat energy, thereby maintaining constant internal energy. Therefore, the work done is equal to the heat lost to the surroundings:

$W = Q$

For process C → D, the system is again returning to its original state isothermally, leading to heat being absorbed or released to maintain the internal energy constant. Hence:

$W = Q$

This adherence to the first law of thermodynamics ensures energy conservation within these isothermal processes.

Step 5

Deduce whether this claim is true.

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Answer

The efficiency claim regarding the heat engine cycle can be evaluated using the efficiency formula:

$\eta = \frac{W_{out}}{Q_{in}}$

From our earlier calculations, the maximum efficiency for this ideal cycle suggests:

$\eta_{max} = \frac{3200}{10000} = 0.32 \, \text{or} \, 32\%$

The provided claim states that this engine cycle is claimed to be the maximum theoretical efficiency of a heat engine operating between the same temperatures, which can be argued as correct based on theoretical limits consistent with the Carnot efficiency concept.

Step 6

Discuss one problem that would be faced by an engineer designing a real engine based on this cycle.

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Answer

One significant problem an engineer might face when designing a real engine based on this idealized cycle is the irreversibility of real processes. Unlike the ideal scenarios depicted in the heat-engine cycle, real processes involve loss of energy due to friction, heat losses to the environment, and other dissipative factors.

These losses reduce the overall efficiency of the engine and can lead to significant deviations from the expected performance characterized by the ideal cycle. Furthermore, these irreversibilities result in constraints on how much work can be efficiently converted from heat, complicating real-world applications.

In an ideal heat-engine cycle a fixed mass of air is taken through the following four processes - AQA - A-Level Physics - Question 3 - 2019 - Paper 6

Worked Solution & Example Answer:In an ideal heat-engine cycle a fixed mass of air is taken through the following four processes - AQA - A-Level Physics - Question 3 - 2019 - Paper 6

Show, by calculation, that the volume at B is 4.1 × 10^-2 m³.

Show that the temperature of the air at C is about 420 K.

Complete Table 1 to show the values of work done W and energy transfer Q in each of the four processes.

Explain why $W$ is equal to $Q$ in process A → B and in process C → D.

Deduce whether this claim is true.

Discuss one problem that would be faced by an engineer designing a real engine based on this cycle.

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