The velocity of sound in an ideal gas is given by the equation:
v = √(γ * R * T)
Where: v = velocity of sound γ = adiabatic index (specific heat ratio) R = ideal gas constant T = temperature
The adiabatic index, γ, depends on the type of gas. For a monatomic ideal gas, such as helium or argon, γ is approximately 5/3 (1.67). For a diatomic ideal gas, such as nitrogen or oxygen, γ is approximately 7/5 (1.4).
Since the problem does not specify the type of gas, let's assume it is a monatomic ideal gas with γ = 5/3.
Given: Initial temperature, T1 = 10°C = 10 + 273.15 Kelvin Initial velocity, v1 = 344 m/s Final temperature, T2 = 35°C = 35 + 273.15 Kelvin
Using the equation v = √(γ * R * T), we can calculate the final velocity, v2, as follows:
v2 = √(γ * R * T2)
Substituting the known values:
v2 = √((5/3) * R * T2)
To find the ratio of v2 to v1, we can divide v2 by v1:
(v2/v1) = √((5/3) * R * T2) / v1
Now, we can substitute the values and calculate:
(v2/v1) = √((5/3) * R * (35 + 273.15)) / 344
Using the value of the ideal gas constant, R = 8.314 J/(mol*K), we get:
(v2/v1) = √((5/3) * 8.314 * (35 + 273.15)) / 344
Calculating this expression gives us:
(v2/v1) ≈ 1.0807
Therefore, the velocity of sound when the temperature changes from 10°C to 35°C is approximately 1.0807 times the initial velocity of 344 m/s.
To find the final velocity, we can multiply the ratio by the initial velocity:
v2 = (v2/v1) * v1 v2 ≈ 1.0807 * 344 v2 ≈ 372.35 m/s
Thus, the velocity of sound when the temperature changes to 35°C is approximately 372.35 m/s.