Assertiobn : Ratio of isothermal bulk modulus and adiabatic bulk modulus for a monoatomic gas at a given pressure is `3/5`.
This ratio is equal to `gamma =C_(p)/C_(v)`.

Options

A. (a) If both Assertion and Reason are true and the Reason is correctn explanation of the Assertion.
B. (b) If both Assertion and Reason are true but Reason is not the correct explanation of Assertion.
C. If Assertion is true, but the Reason is false.
D. If Assertion is false but the Reason is true.

Explanation

To solve the problem, we need to analyze the relationship between the isothermal bulk modulus (K_iso) and the adiabatic bulk modulus (K_adi) for a monoatomic gas and establish the ratio between them.

### Step-by-step Solution:

1. **Understanding Bulk Modulus**:
- The bulk modulus (K) is defined as the measure of a substance's resistance to uniform compression. It is given by the formula:
\[ K = -V \frac{dP}{dV} \]
- Where \( V \) is the volume, \( P \) is the pressure, and \( dP/dV \) is the rate of change of pressure with respect to volume.

2. **Isothermal Process**:
- For an isothermal process (constant temperature), the relationship between pressure and volume is given by:
\[ PV = \text{constant} \]
- The bulk modulus for an isothermal process can be derived as:
\[ K_{\text{iso}} = P \]
- This is because, during an isothermal process, the change in pressure with respect to volume leads to a constant pressure value.

3. **Adiabatic Process**:
- For an adiabatic process (no heat exchange), the relationship is given by:
\[ PV^\gamma = \text{constant} \]
- The bulk modulus for an adiabatic process can be expressed as:
\[ K_{\text{adi}} = \gamma P \]
- Here, \( \gamma \) (gamma) is the ratio of specific heats \( C_p/C_v \).

4. **Finding the Ratio**:
- Now, we can find the ratio of the isothermal bulk modulus to the adiabatic bulk modulus:
\[ \frac{K_{\text{iso}}}{K_{\text{adi}}} = \frac{P}{\gamma P} = \frac{1}{\gamma} \]

5. **For a Monoatomic Gas**:
- For a monoatomic gas, the value of \( \gamma \) is given by:
\[ \gamma = \frac{C_p}{C_v} = \frac{5}{3} \]
- Therefore, the ratio becomes:
\[ \frac{K_{\text{iso}}}{K_{\text{adi}}} = \frac{1}{\frac{5}{3}} = \frac{3}{5} \]

6. **Conclusion**:
- The assertion states that the ratio of isothermal bulk modulus to adiabatic bulk modulus for a monoatomic gas at a given pressure is \( \frac{3}{5} \), which we have confirmed.
- However, the assertion claims that this ratio is equal to \( \gamma \), which is incorrect. The ratio is actually \( \frac{1}{\gamma} \).

### Final Answer:
- The assertion is true, but the reasoning is incorrect. The correct conclusion is that the ratio of isothermal bulk modulus to adiabatic bulk modulus is \( \frac{3}{5} \), but it is equal to \( \frac{1}{\gamma} \), not \( \gamma \).