Fundamental frequency of a organ pipe filled with `N_2` is 500 Hz. the fundamental frequency if `N_2` is replaced by `H_2` is

Options

A. `1000sqrt14 Hz `
B. `500sqrt14 Hz `
C. 500 Hz
D. 250 Hz

Answer

To solve the problem, we need to find the fundamental frequency of an organ pipe when the gas inside it is changed from nitrogen (N₂) to hydrogen (H₂). We know the fundamental frequency of the organ pipe with nitrogen is 500 Hz.

### Step-by-Step Solution:

1. **Understanding the relationship between frequency and speed of sound**:
The fundamental frequency (f) of an organ pipe is directly proportional to the speed of sound (v) in the gas contained in it. This can be expressed as:
\[ f \propto v \]

2. **Speed of sound in different gases**:
The speed of sound in a gas can be expressed using the formula:
\[ v = \sqrt{\frac{8RT}{M}} \]
where \( R \) is the universal gas constant, \( T \) is the temperature, and \( M \) is the molar mass of the gas.

3. **Setting up the ratio of frequencies**:
Let \( f_N \) be the fundamental frequency with nitrogen and \( f_H \) be the fundamental frequency with hydrogen. We can set up the following ratio:
\[ \frac{f_H}{f_N} = \frac{v_H}{v_N} \]

4. **Substituting the speed of sound expressions**:
Substituting the expressions for the speed of sound in hydrogen and nitrogen, we get:
\[ \frac{f_H}{500} = \frac{\sqrt{\frac{8RT}{M_H}}}{\sqrt{\frac{8RT}{M_N}}} \]
Since \( R \) and \( T \) are constant for both gases, they cancel out:
\[ \frac{f_H}{500} = \sqrt{\frac{M_N}{M_H}} \]

5. **Inserting the molar masses**:
The molar mass of nitrogen (N₂) is approximately 28 g/mol and the molar mass of hydrogen (H₂) is approximately 2 g/mol. Thus, we have:
\[ \frac{f_H}{500} = \sqrt{\frac{28}{2}} = \sqrt{14} \]

6. **Calculating the fundamental frequency with hydrogen**:
Now we can solve for \( f_H \):
\[ f_H = 500 \times \sqrt{14} \]

7. **Final calculation**:
To find the numerical value:
\[ \sqrt{14} \approx 3.74 \]
Therefore,
\[ f_H \approx 500 \times 3.74 \approx 1870 \text{ Hz} \]

### Final Answer:
The fundamental frequency of the organ pipe when filled with hydrogen is approximately **1870 Hz**.