The atomic masses of He and Ne are 4 and 20 amu respectively . The value of the de Broglie wavelength of He gas at`-73.^(@)C` is ''M'' times that of the de Broglie wavelength of Ne at `727.^(@)C.` M is

Answer

To solve the problem, we need to determine the value of \( M \) which represents the ratio of the de Broglie wavelengths of helium (He) gas at \(-73^\circ C\) and neon (Ne) gas at \(727^\circ C\).

### Step-by-step Solution:

1. **Convert Temperatures to Kelvin:**
- For Helium at \(-73^\circ C\):
\[ T_{He} = 273 - 73 = 200 \, K \]
- For Neon at \(727^\circ C\):
\[ T_{Ne} = 727 + 273 = 1000 \, K \]

2. **Write the Formula for de Broglie Wavelength:**
The de Broglie wavelength (\( \lambda \)) is given by the formula:
\[ \lambda = \frac{h}{\sqrt{3 m k T}} \]
where:
- \( h \) = Planck's constant
- \( m \) = mass of the gas particle
- \( k \) = Boltzmann's constant
- \( T \) = temperature in Kelvin

3. **Calculate the de Broglie Wavelength for He:**
Using the formula for He:
\[ \lambda_{He} = \frac{h}{\sqrt{3 \cdot m_{He} \cdot k \cdot T_{He}}} \]
where \( m_{He} = 4 \, \text{amu} \) and \( T_{He} = 200 \, K \).

4. **Calculate the de Broglie Wavelength for Ne:**
Using the formula for Ne:
\[ \lambda_{Ne} = \frac{h}{\sqrt{3 \cdot m_{Ne} \cdot k \cdot T_{Ne}}} \]
where \( m_{Ne} = 20 \, \text{amu} \) and \( T_{Ne} = 1000 \, K \).

5. **Set Up the Ratio of Wavelengths:**
According to the problem, we have:
\[ \lambda_{He} = M \cdot \lambda_{Ne} \]
Substituting the expressions for the wavelengths:
\[ \frac{h}{\sqrt{3 \cdot 4 \cdot k \cdot 200}} = M \cdot \frac{h}{\sqrt{3 \cdot 20 \cdot k \cdot 1000}} \]

6. **Cancel Out Common Terms:**
Since \( h \) and \( \sqrt{3} \) and \( k \) are common in both sides, we can simplify:
\[ \frac{1}{\sqrt{4 \cdot 200}} = M \cdot \frac{1}{\sqrt{20 \cdot 1000}} \]

7. **Cross Multiply:**
\[ \sqrt{20 \cdot 1000} = M \cdot \sqrt{4 \cdot 200} \]

8. **Square Both Sides:**
\[ 20 \cdot 1000 = M^2 \cdot (4 \cdot 200) \]

9. **Calculate the Values:**
- Left side:
\[ 20 \cdot 1000 = 20000 \]
- Right side:
\[ 4 \cdot 200 = 800 \]
Thus, we have:
\[ 20000 = M^2 \cdot 800 \]

10. **Solve for \( M^2 \):**
\[ M^2 = \frac{20000}{800} = 25 \]

11. **Find \( M \):**
\[ M = \sqrt{25} = 5 \]

### Final Answer:
\[ M = 5 \]