Understanding Physics Topics is essential for solving complex problems in many fields, including engineering and medicine.
Is Positive or Negative Work on a PV Diagram?
Indicator diagram or pV-diagram is drawn by plotting the volume V along the horizontal axis and the pressure p along the vertical axis [Fig.]. The utilities of such a diagram are:
i) A point (say, A) represents fixed equilibrium values of p and V. Then, following the equation of state, the temperature will also have an equilibrium value. So, this expresses an equilibrium state of a system.
A point ≡ an equilibrium state.
ii) The two points A and B represent two equilibrium states. So, any line joining them, with an arrow, like 1 or 2, represents a process A → B
A line ≡ a process.
The same two points may be joined by different lines, like 1 and 2. This means that, different processes can be performed between the same two states. The line 3 in Fig. denotes a process B → A.
iii) Work done in a finite process from volume V1 to volume V2 is,
W = \(\int_{V_1}^{V_2} p d V\)
In integral calculus, we know that an integral has a graphical representation. Using that concept, it can be inferred that the work done W in a process is represented by the area under the line denoting the process and above the V – axis of the pV-diagram. For example, in Fig., work done in the process A → B(1) ≡ area A1BCDA. In general, on a pV diagram,
an area ≡ an amount of work done.
The process B → A (3) represents a contraction; so W is negative. Therefore, an area under a line going from a higher to a lower volume represents a negative amount of work.
Clearly, an isochoric process (volume = constant) is represented by a vertical line (say, AB) on a pV-diagram [Fig.]. It encloses no area under it; so, work done, W = 0.
Again, an isobaric process (pressure = constant) is represented by a horizontal line (say, CD). The area under this is p(V2 – V1) ; so, the work done in this process, W = p(V2 – V1).
If a system, after any number of individual process, comes back to its initial state, it is said to have gone through a cyclic process or cycle. It is represented by a closed curve, like ABCA, on a pV-diagram [Fig.]
Clearly, the initial state ≡ the final state ≡ state A. The work done in such a cycle is
W = WAB + WBC + WCA
= area ABB’A’A – area BB’C’CV – area CC’A’AC
= area ABCA
Thus, the work done in a cycle = area enclosed by the cycle on the pV-diagram. This work is positive for a clockwise cycle, like ABCA, but is a negative for an anticlock cycle, for example if the process was ACBA. Positive work indicates that work is done by the system and negative work represents work done on the system.
Numerical Examples
Example 1.
The volume of a gas changes from 20 l to 10 l under a pressure of 106 dyn ᐧ cm-2. What will be the heat evolved? [HS 2000]
Solution:
Work done on the gas,
W = p(V2 – V1) = 106(20 – 10) × 103
= 1010erg = 103J
∴ Heat evolved, Q = \(\frac{W}{J}\) = \(\frac{10^3}{4.2}\)
= 238.1 cal.
Example 2.
Find out the amount of work done in the cyclic process of Fig. [Orissa PMT ’04]
Solution:
Work done in the cycle ABCD = area ABCD = AB ᐧ BC = (3V1 – V1)(2p1 – p1) = 2p1V1.