organ pipe filled with oxygen gas at 47 c resonates in its fundamental mode at frequency 300 hz if it is now filled with nitrogen gas at which temperature will it resonate at thw same frequency in the fundamental mode

To determine the temperature at which an organ pipe filled with nitrogen gas will resonate at the same frequency of 300 Hz as it did with oxygen gas at 47 °C, we need to understand how the speed of sound in a gas is affected by its temperature and molecular composition. The frequency of a sound wave in a pipe is related to the speed of sound in the gas and the length of the pipe.

Understanding the Basics

The fundamental frequency (first harmonic) of a pipe is given by the formula:

• f = v / λ

Where:

• f is the frequency (in Hz),
• v is the speed of sound in the gas (in m/s), and
• λ is the wavelength (in meters).

For a pipe open at both ends, the wavelength is related to the length of the pipe (L) by:

• λ = 2L

Thus, we can express the frequency in terms of the speed of sound and the length of the pipe:

• f = v / (2L)

Speed of Sound in Gases

The speed of sound in a gas can be calculated using the formula:

• v = √(γRT/M)

Where:

• γ is the adiabatic index (ratio of specific heats),
• R is the universal gas constant (approximately 8.314 J/(mol·K)),
• T is the absolute temperature in Kelvin, and
• M is the molar mass of the gas in kg/mol.

Calculating for Oxygen

For oxygen (O₂), the molar mass is about 0.032 kg/mol, and the adiabatic index (γ) is approximately 1.4. First, we convert the temperature from Celsius to Kelvin:

• T₁ = 47 °C + 273.15 = 320.15 K

Now we can calculate the speed of sound in oxygen:

• v₁ = √(1.4 × 8.314 × 320.15 / 0.032)

Calculating this gives:

• v₁ ≈ 316.5 m/s

Finding the Temperature for Nitrogen

Now, we need to find the temperature at which nitrogen (N₂) will produce the same frequency. The molar mass of nitrogen is about 0.028 kg/mol, and its adiabatic index is also approximately 1.4. We set the speed of sound in nitrogen equal to the speed of sound in oxygen:

• v₂ = v₁ = 316.5 m/s

Using the speed of sound formula for nitrogen:

• 316.5 = √(1.4 × 8.314 × T₂ / 0.028)

Squaring both sides gives:

• 100145.25 = (1.4 × 8.314 × T₂ / 0.028)

Now, solving for T₂:

• T₂ = (100145.25 × 0.028) / (1.4 × 8.314)

Calculating this yields:

• T₂ ≈ 120.6 K

Finally, converting this back to Celsius:

• T₂ = 120.6 K - 273.15 ≈ -152.55 °C

Summary

To resonate at the same frequency of 300 Hz in the fundamental mode, the organ pipe filled with nitrogen gas would need to be at approximately -152.55 °C. This significant drop in temperature highlights how the properties of gases can dramatically influence sound propagation.