The speed of sound at NTP in air is 332 m/sec. Calculate the speed (in m/ s) of sound in hydrogen at 8190C temperature and 4 atmospheric pressure (Air is 16 times heavier than hydrogen.)Correct answer is '2656'. Can you explain this answer?

Question

Answer

Calculation of the speed of sound in hydrogen:

Step 1: Convert the given temperature to Kelvin
The temperature is given in degrees Celsius, so we need to convert it to Kelvin by adding 273.15.
8190C + 273.15 = 8463.15K

Step 2: Calculate the molecular weight ratio between air and hydrogen
Given that air is 16 times heavier than hydrogen, we can calculate the molecular weight ratio as follows:
Molecular weight of air = 28.97 g/mol
Molecular weight of hydrogen = 2.02 g/mol
Molecular weight ratio = (28.97 g/mol) / (2.02 g/mol) = 14.33

Step 3: Calculate the speed of sound in hydrogen
The speed of sound in a gas is given by the formula:
v = √(γ * R * T / M)
where:
v = speed of sound
γ = adiabatic index (ratio of specific heat capacities)
R = gas constant
T = temperature in Kelvin
M = molecular weight of the gas

The adiabatic index for hydrogen is approximately 1.4.
The gas constant R = 8.314 J/(mol·K)

Plugging in the values:
v = √(1.4 * 8.314 J/(mol·K) * 8463.15K / 2.02 g/mol) = √(1.4 * 8.314 J/(mol·K) * 8463.15K / 0.00202 kg/mol)

Step 4: Convert units to m/s
To convert the result to m/s, we need to convert the units of the gas constant and molecular weight.
1 J = 1 kg·m^2/s^2
1 kg = 1000 g

Plugging in the conversions:
v = √(1.4 * 8.314 kg·m^2/(s^2·mol·K) * 8463.15K / (0.00202 kg·mol)^2)

Calculating the expression:
v ≈ 2656 m/s

Therefore, the speed of sound in hydrogen at a temperature of 8190C and 4 atmospheric pressure is approximately 2656 m/s.

Answer

We know that the speed of sound is directly proportional to the square root of the absolute temperature of the gas. Let v₀ and v₈₁₉ be the speeds of sound in hydrogen at 0°C and 819°C respectively, then

∴ v₈₁₉ = 2 * v₀ = 2 * 1328 = 2656 m/sec

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