compare velocities of sound in hydrogen and carbon dioxide.the ratio of specific heat of hydrogen and carbon dioxide is 1.4 and 1.3

To compare the velocities of sound in hydrogen and carbon dioxide, we can use the formula for the speed of sound in a gas, which is given by:

Speed of Sound Formula

The speed of sound (v) in a gas can be calculated using the equation:

v = √(γRT/M)

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

Specific Heats and Molar Masses

In this case, we know the ratios of specific heats for hydrogen (γ₁ = 1.4) and carbon dioxide (γ₂ = 1.3). We also need to consider the molar masses:

• The molar mass of hydrogen (H₂) is approximately 0.002 kg/mol.
• The molar mass of carbon dioxide (CO₂) is approximately 0.044 kg/mol.

Calculating the Speed of Sound

Now, let’s calculate the speed of sound in both gases. For simplicity, we can assume the temperature (T) is constant for both gases, so it will cancel out when we compare the two speeds.

For hydrogen:

v₁ = √(γ₁RT/M₁) = √(1.4 * R * T / 0.002)

For carbon dioxide:

v₂ = √(γ₂RT/M₂) = √(1.3 * R * T / 0.044)

Finding the Ratio of Velocities

To find the ratio of the speeds of sound in hydrogen to that in carbon dioxide, we can set up the following equation:

v₁/v₂ = √((γ₁/M₁) / (γ₂/M₂))

Substituting the values:

v₁/v₂ = √((1.4 / 0.002) / (1.3 / 0.044))

Now, simplifying this:

• Calculate the first part: 1.4 / 0.002 = 700.
• Calculate the second part: 1.3 / 0.044 ≈ 29.545.

Now, substituting these values back into the ratio:

v₁/v₂ = √(700 / 29.545) ≈ √(23.7) ≈ 4.87.

Conclusion

This means that the speed of sound in hydrogen is approximately 4.87 times faster than in carbon dioxide under the same conditions. This significant difference can be attributed to hydrogen's lower molar mass and higher specific heat ratio, which allows sound waves to propagate more quickly through it compared to carbon dioxide.