1. The c of a certain gas at 25˚C is 411 m/sec.
Determine the molar mass of the gas.
2. The c of a sample of O2 is 575 m/sec.
Determine the temperature of the gas.
To determine the molar mass of the gas in question 1 and the temperature of the gas in question 2, we can use the ideal gas law. The ideal gas law is expressed as PV = nRT, where P is the pressure, V is the volume, n is the number of moles, R is the ideal gas constant, and T is the temperature in kelvin.
To solve question 1:
Step 1: We are given the speed of sound (c) and the temperature (25˚C). First, convert the temperature to kelvin by adding 273 to it:
T = 25˚C + 273 = 298 K
Step 2: Use the speed of sound to calculate the speed of molecules:
c = √(γRT/M)
where γ is the adiabatic index (specific heat ratio), R is the gas constant, and M is the molar mass.
Rearrange the equation to solve for the molar mass (M):
M = γRT/c^2
The adiabatic index (γ) depends on the gas in question. For example, for a diatomic gas like oxygen (O2), γ is approximately 1.4.
Step 3: Plug in the known values to calculate the molar mass:
M = (1.4 * 8.314 J/(mol·K) * 298 K)/(411 m/s)^2 ≈ 0.0329 kg/mol
Therefore, the molar mass of the gas in question 1 is approximately 0.0329 kg/mol.
To solve question 2:
Step 1: We are given the speed of sound (c) and need to find the temperature (T). No additional information is provided, so we need to make an assumption.
The speed of sound in a gas is dependent on its temperature. Assuming an ideal gas, we can use the formula:
c = √(γRT/M)
However, since we don't have the molar mass (M) or any other data to solve for temperature directly, we can't determine the temperature accurately without additional information.
However, we can make an estimation using the molar mass of the gas. Assuming the gas in question is oxygen (O2), we can use the calculated molar mass from question 1, which was approximately 0.0329 kg/mol.
Step 2: Rearrange the equation:
T = (c^2 * M) / (γR)
Step 3: Plug in the known values to estimate the temperature:
T = (575 m/s)^2 * 0.0329 kg/mol / (1.4 * 8.314 J/(mol·K)) ≈ 7261 K
Therefore, the estimated temperature of the gas in question 2 is approximately 7261 K.