Question

Isothermal bulk modulus of a gas at a pressure \( P \) is:

• A. \( \gamma \)
• B. \( \gamma P \)
• C. \( P \)
• D. \( \frac{\gamma}{P} \)

Hint:

For isothermal processes, the bulk modulus is directly proportional to the pressure of the gas. This is derived from the relationship between pressure and volume during isothermal compression or expansion.

Answer 1

The Correct Option is C

The isothermal bulk modulus \( K_T \) is defined as the change in pressure \( P \) with respect to the change in volume \( V \) at a constant temperature. For an ideal gas, the isothermal bulk modulus is given by: \[ K_T = P \] Thus, the correct answer is \( P \).

Answer 2

Step 1: Understand what isothermal bulk modulus means.
The bulk modulus \( B \) of a material is a measure of its resistance to uniform compression. It is defined as:
\[ B = -V \frac{dP}{dV} \]
For an **isothermal** process (constant temperature), the ideal gas law \( PV = \text{constant} \) applies.
Differentiating both sides with respect to volume \( V \):
\[ \frac{d(PV)}{dV} = 0 \Rightarrow P + V \frac{dP}{dV} = 0 \Rightarrow \frac{dP}{dV} = -\frac{P}{V} \]
Now substitute into the bulk modulus formula:
\[ B = -V \cdot \left(-\frac{P}{V}\right) = P \]

Final Answer: \( \boxed{P} \)