Step 1: Understanding the Question:
The objective is to identify the shape of a specific thermodynamic graph for an ideal gas under isothermal (constant temperature) conditions.
We are specifically looking at how the product of pressure and volume (\(PV\)) changes as the pressure (\(P\)) itself changes.
Step 2: Detailed Explanation:
According to the Ideal Gas Law, the relationship between pressure, volume, and temperature is given by \(PV = nRT\).
In the problem, it is stated that the temperature (\(T\)) is constant. Furthermore, for a fixed mass of gas, the number of moles (\(n\)) and the universal gas constant (\(R\)) are also constants.
Therefore, the entire right-hand side of the equation (\(nRT\)) is a constant value. Let's call this constant \(k\). So, \(PV = k\).
This is a statement of Boyle's Law: for an ideal gas at a fixed temperature, the product of pressure and volume is always constant.
Now, if we plot a graph where the Y-axis represents the product \(PV\) and the X-axis represents the pressure \(P\), we are plotting the function \(y = k\).
In coordinate geometry, the equation \(y = \text{constant}\) represents a horizontal straight line. This line has a slope of zero.
This means that no matter how much you increase or decrease the pressure \(P\), the value of \(PV\) remains exactly the same on the graph.
Consequently, the resulting graph is a straight line that runs parallel to the X-axis, which is the P-axis in this context.
It is important to note that for real gases, this line would deviate at very high pressures due to intermolecular forces, but for an "ideal gas," it remains a perfect straight line.
Step 3: Final Answer:
The graph of \(PV\) versus \(P\) for an ideal gas at constant temperature is a straight line parallel to the P-axis.