Fundamental frequency of a organ pipe filled with `N_(2)` is `1000 Hz`. Find the fundamental frequency if `N_(2)` is replaced by `H_(2)`.

Explanation

To solve the problem of finding the fundamental frequency of an organ pipe filled with hydrogen (H₂) after it was initially filled with nitrogen (N₂) at a frequency of 1000 Hz, we can follow these steps:

### Step 1: Understand the relationship between frequency and the properties of the gas
The fundamental frequency (f) of an organ pipe is given by the formula:
\[ f = \frac{V}{2L} \]
where \( V \) is the velocity of sound in the gas and \( L \) is the length of the organ pipe.

### Step 2: Determine the velocity of sound in the gas
The velocity of sound in a gas can be expressed as:
\[ V = \sqrt{\frac{\gamma R T}{M}} \]
where:
- \( \gamma \) is the ratio of specific heats (constant for diatomic gases like N₂ and H₂),
- \( R \) is the universal gas constant (constant),
- \( T \) is the absolute temperature (assumed constant for both gases),
- \( M \) is the molecular weight of the gas.

### Step 3: Establish the relationship between frequencies of different gases
Since both nitrogen and hydrogen are diatomic gases, we can compare their frequencies. The fundamental frequency is inversely proportional to the square root of the molecular weight:
\[ \frac{f_1}{f_2} = \sqrt{\frac{M_2}{M_1}} \]
where:
- \( f_1 \) is the frequency with nitrogen,
- \( f_2 \) is the frequency with hydrogen,
- \( M_1 \) is the molecular weight of nitrogen (N₂ = 28 g/mol),
- \( M_2 \) is the molecular weight of hydrogen (H₂ = 2 g/mol).

### Step 4: Substitute the values into the equation
Using the molecular weights:
\[ \frac{f_1}{f_2} = \sqrt{\frac{2}{28}} = \sqrt{\frac{1}{14}} \]
This can be rearranged to find \( f_2 \):
\[ f_2 = f_1 \cdot \sqrt{14} \]

### Step 5: Calculate the new frequency
Given that \( f_1 = 1000 \, \text{Hz} \):
\[ f_2 = 1000 \cdot \sqrt{14} \]

### Step 6: Final calculation
Calculating \( \sqrt{14} \approx 3.74 \):
\[ f_2 \approx 1000 \cdot 3.74 \approx 3740 \, \text{Hz} \]

### Conclusion
The fundamental frequency of the organ pipe when filled with hydrogen is approximately \( 3740 \, \text{Hz} \).

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