Comprehensive Guide to Calculating Work in a Thermodynamic Cycle

Characterization of the Ideal Gas Thermodynamic Cycle

The fundamental basis for solving the provided problem involves the detailed analysis of a thermodynamic cycle executed by an ideal gas, as represented on a Pressure-Volume (P−VP-V) diagram. A thermodynamic cycle is a sequence of processes that returns the system to its original state. In this specific case, the cycle is composed of four distinct stages: first, an isobaric expansion from state A to state B; second, an isochoric heating process from state B to state C; third, an isobaric compression from state C to state D; and finally, an isochoric cooling process from state D back to state A.

Methodological Principles of Thermodynamic Work Calculation

To calculate the total work performed during a complete thermodynamic cycle, one must utilize the principle that the work (WW) done by a gas during a process is equal to the area under its path on a Pressure-Volume diagram. For a full cycle, the net work performed is equivalent to the enclosed area within the boundaries of the cycle established by the transitions between the four states. This area represents the net energy transferred to the surroundings in the form of work over one complete repetition of the cycle.

Qualitative Analysis of Cycle Transitions

The cycle is broken down into segments to identify which processes contribute to the total work. The transition from A to B is defined as a horizontal line in the P−VP-V diagram, indicating an isobaric expansion where pressure remains constant while volume increases. Conversely, the transition from B to C is a vertical line known as an isochoric process. Because the change in volume is zero (ΔV=0\Delta V = 0), no work is performed during this stage. The transition from C to D is another horizontal line, representing isobaric compression where the pressure remains constant but the volume decreases. Finally, the segment from D back to A is another isochoric vertical line, which, like the transition from B to C, contributes zero work to the total cycle.

Formulas for Work in Isobaric Processes

The work done during the isobaric segments of the cycle is calculated using the product of the constant pressure and the change in volume (ΔV\Delta V). The specific formula for work is given as W=P×(Vfinal−Vinitial)W = P \times (V_{\text{final}} - V_{\text{initial}}). For the expansion phase from A to B, the work is denoted as WA→B=PAB×(VB−VA)W_{A \rightarrow B} = P_{AB} \times (V_B - V_A), where PABP_{AB} (also referred to as P1P_1) is the constant pressure. For the compression phase from C to D, the work is denoted as WC→D=PCD×(VD−VC)W_{C \rightarrow D} = P_{CD} \times (V_D - V_C), where PCDP_{CD} (referred to as P2P_2) is the constant pressure maintained during this segment.

Comprehensive Step-by-Step Calculation with Specific Values

To arrive at a quantitative solution, the following variable assignments are derived from the transcript. For the expansion from point A to point B, the pressure is P1=20 N/m2P_1 = 20\,N/m^2 and the volume increases from VA=4 m3V_A = 4\,m^3 to VB=32 m3V_B = 32\,m^3. The work calculation for this segment is as follows:

WA→B=20×(32−4)=20×28=560 JW_{A \rightarrow B} = 20 \times (32 - 4) = 20 \times 28 = 560\,J

For the compression from point C to point D, the pressure is P2=5 N/m2P_2 = 5\,N/m^2 and the volume decreases from VC=32 m3V_C = 32\,m^3 (since state C follows state B) to VD=4 m3V_D = 4\,m^3 (since state D precedes state A). The work calculation for this segment results in a negative value due to compression:

WC→D=5×(4−32)=5×(−28)=−140 JW_{C \rightarrow D} = 5 \times (4 - 32) = 5 \times (-28) = -140\,J

Final Determination of Net Cycle Work

The total work realized during the full cycle is the algebraic sum of the work done in each individual segment. Since the isochoric processes (B→CB \rightarrow C and D→AD \rightarrow A) do not contribute to the work (W=0W = 0), the calculation relies solely on the isobaric segments:

WTotal=WA→B+WC→DW_{\text{Total}} = W_{A \rightarrow B} + W_{C \rightarrow D}

WTotal=560+(−140)=420 JW_{\text{Total}} = 560 + (-140) = 420\,J

The final calculated total work performed during the complete cycle is 420 J420\,J. This numerical result corresponds to Option D in the provided multiple-choice question. This thorough analysis demonstrates the efficacy of using a P−VP-V diagram to visual and calculate energy transitions within thermodynamic systems.