For the thermodynamic cycle shown in Fig. 20-3, find (a) the...

For the thermodynamic cycle shown in Fig. $20-3$, find $(a)$ the net work output of the gas during the cycle and $(b)$ the net heat flow into the gas per cycle. Method 1 (a) From Problem 20.13, the net work done is $1.0 \mathrm{~J}-0.50 \mathrm{~J}=0.5$ J. Method 2 The net work done is equal to the area enclosed by the $P-V$ diagram: Work $=$ Area $A B C D A=\left(2.0 \times 10^{5} \mathrm{~N} / \mathrm{m}^{2}\right)\left(2.5 \times 10^{-6} \mathrm{~m}^{3}\right)=0.50 \mathrm{~J}$ (b) Suppose the cycle starts at point- $A$. The gas returns to this point at the end of the cycle, so there is no difference in the gas at its start and end points. For one complete cycle, $\Delta U$ is therefore zero. We have then, if the first law is applied to a complete cycle, $$ \Delta Q=\Delta U+\Delta W=0+0.50 \mathrm{~J}=0.50 \mathrm{~J}=0.12 \text { cal } $$

For the thermodynamic cycle shown in Fig. $20-3$, find $(a)$ the net work output of the gas during the cycle and $(b)$ the net heat flow into the gas per cycle.
Method 1
(a) From Problem 20.13, the net work done is $1.0 \mathrm{~J}-0.50 \mathrm{~J}=0.5$ J.
Method 2
The net work done is equal to the area enclosed by the $P-V$ diagram:
Work $=$ Area $A B C D A=\left(2.0 \times 10^{5} \mathrm{~N} / \mathrm{m}^{2}\right)\left(2.5 \times 10^{-6} \mathrm{~m}^{3}\right)=0.50 \mathrm{~J}$
(b) Suppose the cycle starts at point- $A$. The gas returns to this point at the end of the cycle, so there is no difference in the gas at its start and end points. For one complete cycle, $\Delta U$ is therefore zero. We have then, if the first law is applied to a complete cycle,
$$
\Delta Q=\Delta U+\Delta W=0+0.50 \mathrm{~J}=0.50 \mathrm{~J}=0.12 \text { cal }
$$

Key Concepts

Thermodynamic Cycle

A thermodynamic cycle consists of a series of processes that return a system to its initial state. In such cycles, although the system undergoes various changes in pressure, volume, and temperature, the net change in internal energy over one complete cycle is zero. This property is essential in evaluating the performance of engines and refrigerators where energy conversion is key.

Pressure-Volume (P-V) Diagram

A P-V diagram is a graphical representation of the relationship between pressure and volume during the various processes in a thermodynamic cycle. It is a crucial tool because the area enclosed by a closed cycle on this diagram represents the net work output or input of the cycle, providing a visual and quantitative method to assess work done.

Work Done in a Cycle

The net work done by a system during a thermodynamic cycle is determined by the area enclosed by its path on a P-V diagram. This area calculation is a fundamental concept in thermodynamics as it links the visually observed path of the cycle to the energy transferred by work, independent of the specific processes involved.

First Law of Thermodynamics

The first law of thermodynamics, which is essentially the law of energy conservation, relates the change in internal energy of a system to the heat added minus the work done by the system. For a complete cycle, this law is applied by noting that the net change in internal energy is zero, thereby making the net heat flow equal to the net work done.

Heat Flow

Heat flow in the context of a thermodynamic cycle is the amount of energy transferred as heat into or out of the system. According to the first law of thermodynamics, because the internal energy change over a complete cycle is zero, the net heat flow into the system is directly equal to the net work done during the cycle.

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