A thermodynamic system is taken through the cycle ABCD as shown in the figure. Heat rejected by the gas during the cycle is:

(a) PV
(b) 2PV
(c) 4PV
(d) 1/2 PV

To determine the heat rejected by the gas during the thermodynamic cycle ABCD, we need to understand the concept of heat exchange in thermodynamic processes. The heat rejected by the gas is related to the work done by the gas and the change in internal energy over the course of the cycle. We can approach this using the first law of thermodynamics, which is:
ΔQ=ΔU+W\Delta Q = \Delta U + W
Where:
• ΔQ\Delta Q is the heat added to the system (positive if heat is added, negative if heat is rejected),
• ΔU\Delta U is the change in internal energy,
• WW is the work done by the system (positive if work is done by the system).
In a cyclic process, the net change in internal energy ΔU\Delta U over a full cycle is zero, as the system returns to its initial state. Thus, the heat exchanged ΔQ\Delta Q over the cycle is equal to the work done by the system. If the system undergoes a process where it expels heat, the heat rejected corresponds to the work done by the system during the cycle.
Given information:
From the problem description, we assume that the cycle ABCD involves some form of expansion or compression of the gas.
To determine the heat rejected, we need to focus on the phase of the cycle where the gas is cooling down or losing energy (which corresponds to heat rejection). In thermodynamic processes such as an isothermal expansion or compression, the amount of heat rejected is usually calculated based on the work done and the properties of the process.
Step-by-step approach:
1. Isothermal processes: If any part of the cycle is isothermal (constant temperature), then the work done by the gas can be found using W=PΔVW = P \Delta V for isothermal expansion or compression. If the gas is expanding, it is doing work, and heat is required. If the gas is compressed, work is done on the system, and heat is released.
2. Calculation of heat rejected: If the system goes through an isothermal expansion followed by an isothermal compression, the heat rejected by the system will be proportional to the volume changes during the cycle. In some cases, such as if the cycle is a specific idealized process (like an ideal gas in a Carnot cycle), the heat rejected could be a fraction of the total work done in the cycle.
Answer Explanation:
Without seeing the exact figure or further information about the path of the cycle (i.e., whether it includes isothermal, adiabatic, or other processes), we can provide a general answer based on the common behavior of thermodynamic cycles.
The heat rejected is typically proportional to the work done during the compression phase of the cycle. Based on the standard answers and assuming an idealized process, the heat rejected is usually a fraction of the total work done during the cycle.
Conclusion:
Based on the provided options and common results from thermodynamic cycles, the correct answer is likely (d) 12PV\frac{1}{2} PV, which corresponds to the heat rejected by the gas during the cycle, assuming the system undergoes an idealized process.