Which of the following graphs correctly represents the variation of `beta=-(dV//dP)/V`
with P for an ideal gas at constant temperature?

Options

A.
B.
C.
D.

Explanation

To solve the problem of determining how the parameter \( \beta = -\frac{dV}{dP} \cdot \frac{1}{V} \) varies with pressure \( P \) for an ideal gas at constant temperature, we can follow these steps:

### Step 1: Understand the Ideal Gas Law
The ideal gas law is given by the equation:
\[ PV = nRT \]
where \( P \) is the pressure, \( V \) is the volume, \( n \) is the number of moles, \( R \) is the universal gas constant, and \( T \) is the temperature.

### Step 2: Differentiate the Ideal Gas Law
At constant temperature, we can differentiate the ideal gas law with respect to pressure \( P \). The differentiation gives us:
\[ P \frac{dV}{dP} + V = 0 \]
This is because \( nRT \) is constant when temperature \( T \) is constant.

### Step 3: Solve for \(\frac{dV}{dP}\)
Rearranging the equation from Step 2, we can isolate \( \frac{dV}{dP} \):
\[ P \frac{dV}{dP} = -V \]
\[ \frac{dV}{dP} = -\frac{V}{P} \]

### Step 4: Substitute into the Expression for \(\beta\)
Now, we substitute \( \frac{dV}{dP} \) into the definition of \( \beta \):
\[ \beta = -\frac{dV}{dP} \cdot \frac{1}{V} \]
Substituting \( \frac{dV}{dP} = -\frac{V}{P} \):
\[ \beta = -\left(-\frac{V}{P}\right) \cdot \frac{1}{V} \]
This simplifies to:
\[ \beta = \frac{1}{P} \]

### Step 5: Analyze the Relationship
The relationship \( \beta = \frac{1}{P} \) indicates that \( \beta \) is inversely proportional to \( P \). This means as pressure \( P \) increases, \( \beta \) decreases.

### Step 6: Graphical Representation
The graph of \( \beta \) versus \( P \) will be a hyperbola, showing that \( \beta \) decreases as \( P \) increases. Specifically, it will approach zero as \( P \) increases indefinitely.

### Conclusion
Thus, the correct graph representing the variation of \( \beta \) with \( P \) for an ideal gas at constant temperature is a hyperbolic graph, indicating that \( \beta \) is inversely proportional to \( P \).

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