In a standing wave pattern obtained in an open tube filled with Iodine, due to vibrations of frequency 800 cycle/sec the dist between first node, eleventh node is found to be 1 m when the temperature of iodine vapour is `352^@C`. If the temperature is `127^@C` , the distance between consecutive nodes is (in centimeters) (approximately).

Answer

To solve the problem step by step, we will follow the principles of wave motion and the relationship between temperature and the speed of sound in a medium.

### Step 1: Understand the relationship between wave speed, frequency, and wavelength
The speed of sound in a medium is given by the equation:
\[ v = f \lambda \]
where:
- \( v \) = speed of sound,
- \( f \) = frequency,
- \( \lambda \) = wavelength.

### Step 2: Relate wave speed to temperature
The speed of sound in a gas is directly proportional to the square root of the absolute temperature (in Kelvin):
\[ v \propto \sqrt{T} \]
Thus, we can write:
\[ \frac{v_1}{v_2} = \sqrt{\frac{T_1}{T_2}} \]

### Step 3: Convert temperatures to Kelvin
Convert the given temperatures from Celsius to Kelvin:
- For \( T_1 = 352^\circ C \):
\[ T_1 = 352 + 273 = 625 \, K \]
- For \( T_2 = 127^\circ C \):
\[ T_2 = 127 + 273 = 400 \, K \]

### Step 4: Calculate the ratio of wave speeds
Using the relationship derived in Step 2:
\[ \frac{v_1}{v_2} = \sqrt{\frac{625}{400}} \]
Calculating the right side:
\[ \frac{v_1}{v_2} = \sqrt{1.5625} = 1.25 \]

### Step 5: Relate the distances between nodes
The distance between nodes in a standing wave is half the wavelength:
- The distance between the first node and the eleventh node is given as 1 meter, which corresponds to 10 wavelengths (since there are 10 half-wavelengths between 11 nodes).
Thus, the wavelength \( \lambda_1 \) can be calculated as:
\[ \text{Distance between nodes} = \frac{1 \, m}{10} = 0.1 \, m \]

### Step 6: Calculate the new wavelength at the new temperature
Using the ratio of speeds and the relationship between wavelength and speed, we can find the new wavelength \( \lambda_2 \):
\[ \frac{\lambda_1}{\lambda_2} = \frac{v_1}{v_2} \]
Thus:
\[ \lambda_2 = \lambda_1 \cdot \frac{v_2}{v_1} = 0.1 \cdot \frac{1}{1.25} = 0.08 \, m \]

### Step 7: Convert the new wavelength to the distance between consecutive nodes
Since the distance between consecutive nodes is half the wavelength:
\[ \text{Distance between consecutive nodes} = \frac{\lambda_2}{2} = \frac{0.08}{2} = 0.04 \, m \]

### Step 8: Convert meters to centimeters
Finally, convert the distance from meters to centimeters:
\[ 0.04 \, m = 0.04 \times 100 = 4 \, cm \]

### Final Answer
The distance between consecutive nodes at a temperature of \( 127^\circ C \) is approximately **4 cm**.

---