The pressure and volume of a gas are changed along a path ABCA in the figure. The vertical divisions on the graph represent 4.0 x 10^5 Pa, and the horizontal divisions represent 4.5 x 10^-3 m3.
The diagram is a upright rectangle divided into small squares (8 up and 6 across). The horizontal 6 squares represents volume and the vertical 8 squares respresents pressure. Point A is 3 up and 2 across. Point B is 7 up and 2 across. Point C is 7 up and 5 across.
Determine the work done (including algebraic sign) in each segment of the path.
(a) A to B
(b) B to C
(c) C to A
The work done BY the fluid is the integral under the PdV curve in each case. The dependence of P upon V will depend upon whether the volume change is isothermal, adiabatic, or something in between. You should be able to figure that relationship out from the location of the points
To determine the work done in each segment of the path (A to B, B to C, and C to A), we need to calculate the area under the pressure-volume (P-V) curve for each segment.
First, let's determine the relationship between pressure (P) and volume (V) for each segment based on the given points A, B, and C on the graph.
Point A is located at 3 units up and 2 units across on the graph, which means the pressure is 3 * 4.0 x 10^5 Pa and the volume is 2 * 4.5 x 10^-3 m^3. Therefore, the initial point A corresponds to (P1, V1) = (3 * 4.0 x 10^5 Pa, 2 * 4.5 x 10^-3 m^3).
Similarly, point B is located at 7 units up and 2 units across on the graph, which gives us (P2, V2) = (7 * 4.0 x 10^5 Pa, 2 * 4.5 x 10^-3 m^3).
Lastly, point C is at 7 units up and 5 units across, giving us (P3, V3) = (7 * 4.0 x 10^5 Pa, 5 * 4.5 x 10^-3 m^3).
Now, we can calculate the work done in each segment using the formula:
Work = ∫PdV
(a) A to B:
The work done from point A to point B can be calculated by integrating the area under the curve between V1 and V2, while considering the relationship between P and V for this segment. Based on the given points, we can observe that the volume is constant because both points have the same x-coordinate. Therefore, no work is done in this segment.
(b) B to C:
To calculate the work done from point B to point C, we need to integrate the area under the curve between V2 and V3, while considering the relationship between P and V for this segment. Since we don't have any specific information about the relationship between P and V for this segment, we cannot determine the work done unless additional information is provided.
(c) C to A:
Similarly, to calculate the work done from point C back to point A, we need to integrate the area under the curve between V3 and V1, while considering the relationship between P and V for this segment. Without specific information about this relationship, we cannot determine the work done unless additional information is given.
In summary, we can determine that no work is done in the segment from A to B. However, without knowing the relationship between P and V for the other segments, we cannot calculate the work done for segments B to C and C to A.