in finding the work done from the p-v graph,, work done is:- (PdV+VdP),, then why we need to take the area covered by the graph with volume axis,, why not with pressure???????

When analyzing a p-V (pressure-volume) graph, it's essential to understand how work is defined in thermodynamics. The work done by or on a system during a process can indeed be represented in terms of pressure and volume changes. However, the area under the curve in a p-V diagram specifically relates to the work done during a process, and this is where the distinction between using pressure or volume becomes crucial.

Understanding Work in Thermodynamics

In thermodynamics, the work done by a gas during expansion or compression is given by the integral of pressure with respect to volume. Mathematically, this is expressed as:

W = ∫ P dV

This equation indicates that the work done (W) is the area under the curve of the p-V graph, where pressure (P) is plotted on the y-axis and volume (V) on the x-axis. This area represents the total work done by the system as it moves from one state to another.

Why Area Under the Volume Axis Matters

The reason we focus on the area under the curve with respect to the volume axis is that work is fundamentally defined as a force applied over a distance. In the context of a gas, the "force" is related to pressure, and the "distance" corresponds to the change in volume. When a gas expands, it pushes against the external pressure, and the work done is a result of this expansion.

• Pressure as Force: Pressure is defined as force per unit area. When a gas expands, it exerts a force on the walls of its container, which is directly related to the pressure.
• Volume as Distance: The change in volume (dV) represents the distance over which this force is applied. Thus, the work done is the product of pressure and the change in volume.

Visualizing the Concept

Imagine a piston in a cylinder filled with gas. When the gas expands, it pushes the piston outward. The work done by the gas can be visualized as the area of the rectangle or shape formed under the curve on the p-V graph. If you were to calculate the area using pressure as the base (on the x-axis), it wouldn't accurately represent the physical work done because it doesn't account for the distance (volume change) over which that pressure acts.

Clarifying the Equation

The equation you mentioned, (PdV + VdP), is a differential form that arises from the first law of thermodynamics and can be used in more complex scenarios involving changes in both pressure and volume. However, when calculating work done in a straightforward process, the integral of pressure with respect to volume (∫ P dV) is the most direct and relevant approach.

Practical Example

Consider a gas expanding isothermally (at constant temperature) in a piston. If the pressure decreases as the volume increases, the work done by the gas can be calculated by finding the area under the curve on the p-V diagram. If you were to look at it from a pressure perspective, you would miss the essential relationship between the force exerted by the gas and the distance over which it acts.

In summary, while both pressure and volume are crucial in understanding thermodynamic processes, the area under the p-V curve directly correlates to the work done by the gas. This is why we focus on the volume axis when determining work in a p-V graph. The relationship between pressure and volume is what ultimately defines the work done in these processes.