The speed of a sound in a container of hydrogen at 201K is 1220m/s. What would be the speed of sound if the temperature were raised to 405K? Assume that hydrogen behaves like an ideal gas.
Please guide me to do it. Thanks!
The speed of sound is proportional to the square root of the absolute temperature.
Therefore, multiply 1220 m/s by sqrt(405/201)
It's an Ideal gas law problem, set both equal to K (or k if you are a chem major) and set them equal to each other. Get rid of everything that is on both sides and hen solve for the unknown velocity.
v2=sqr((T2*v1^2)/T1)
To calculate the speed of sound in a container of hydrogen at a higher temperature, we can make use of the ideal gas law. The ideal gas law formula is given by:
PV = nRT
Where:
P = pressure
V = volume
n = number of moles
R = ideal gas constant
T = temperature
In this case, we are only concerned with the temperature and its impact on the speed of sound. Therefore, we can rewrite the formula as:
v = √(γRT)
Where:
v = speed of sound
γ = adiabatic index for the gas (hydrogen in this case)
R = ideal gas constant
T = temperature
The adiabatic index for hydrogen is typically assumed to be 1.4.
Now we can calculate the speed of sound at the higher temperature:
Step 1: Substitute the given values into the equation
v₁ = √(γRT₁)
Given: γ = 1.4, R is a constant, and T₁ = 201 K
Step 2: Solve for v₁
v₁ = √(1.4 * R * 201) [taking square root on both sides]
Step 3: Calculate v₁ using a calculator or math software using the appropriate value for R
v₁ ≈ 1,297 m/s
Now we can calculate the speed of sound at the higher temperature:
Step 4: Substitute the values into the equation
v₂ = √(γRT₂)
Given: γ = 1.4, R is a constant, and T₂ = 405 K
Step 5: Solve for v₂
v₂ = √(1.4 * R * 405) [taking square root on both sides]
Step 6: Calculate v₂ using a calculator or math software using the appropriate value for R
v₂ ≈ 1,625 m/s
Therefore, the speed of sound in the container of hydrogen at a higher temperature of 405 K would be approximately 1,625 m/s.
To find the speed of sound in hydrogen at a different temperature, you can use the relation between the speed of sound, temperature, and the molar mass of the gas. The formula is as follows:
v = sqrt((γ * R * T) / M)
Where:
- v is the speed of sound
- γ is the adiabatic index, which is a ratio of specific heat capacities (Cp/Cv)
- R is the gas constant
- T is the temperature in Kelvin
- M is the molar mass of the gas
To solve this problem, you need to know the molar mass of hydrogen, the adiabatic index, and the value of the gas constant.
The molar mass of hydrogen is approximately 2.016 g/mol.
The adiabatic index for a monatomic gas like hydrogen is γ = 5/3.
The value of the gas constant R is approximately 8.314 J/(mol·K).
Now you can substitute these values into the formula:
v1 = sqrt((γ * R * T1) / M)
v2 = sqrt((γ * R * T2) / M)
Where:
- v1 is the speed of sound at the initial temperature T1
- v2 is the speed of sound at the final temperature T2
You are given that at 201K, the speed of sound in hydrogen is 1220 m/s. You want to find the speed of sound at 405K.
So:
v1 = 1220 m/s
T1 = 201K
T2 = 405K
Substituting these values into the formula, you can calculate the speed of sound at 405K.