An ideal monoatomic gas is taken round the cycle ABCDA as shown in P-V diagram. The work done during the cyclic operation is 

Concept:

First Law of Thermodynamics:

• The change of internal energy is a path independent function and depends upon initial and final position. Mathematically, it is given as 

ΔU = Q - W

Q is net heat transfer, w is work done on the system.

• The work done is represented by the area under the cycle of the PV curve.
  • If the direction is clockwise, the work done is positive. If it is anticlockwise, work done is negative.

Calculation:

Here, The direction is clockwise. So, the work done will be positive.

Area under the curve = Area of Rectangle ABCD

Area of Rectangle = length × breadth

Area = AB × BC -- (1)

AB = 2P - P = P -- (2)

BC = 2V - V = V --- (3)

Putting (2) and (3) in (1)

Area = P × V = PV

So, the correct answer is PV.

Additional Information

• Work done on the system changes the volume of the system. Work done is expressed as

\(W = \int_{V_i}^{V_f}P.dV\)

P is pressure, Vi is initial volume, Vf is the final volume.

• Work done is positive when the volume is increasing and vice versa. 
  • The thermodynamic process can be represented on a Pressure volume graph (PV graph)
  • For a cyclic process, the total change in energy is zero

ΔU = 0 for cyclic process.