To determine the speed of sound in the gas under the given conditions, we can use the relationship between the speed of sound, the properties of the gas, and the pressure and temperature conditions. The speed of sound in an ideal gas can be calculated using the formula:
Formula for Speed of Sound
The speed of sound (c) in an ideal gas is given by:
c = √(γ * R * T)
Where:
- γ (gamma) is the adiabatic index (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.
Step 1: Determine the Properties of the Gas
First, we need to identify the gas we are dealing with. Since the problem does not specify, we can assume it is a common ideal gas like air, which has a γ value of approximately 1.4. The molecular weight of air is about 29 g/mol, which we can use to find R for this specific gas:
R = 8.314 J/(mol·K) / 0.029 kg/mol ≈ 287 J/(kg·K)
Step 2: Calculate the Temperature
Next, we need to find the temperature in Kelvin. The problem states that the gas is at atmospheric temperature, which we can assume to be around 20°C (or 293 K).
Step 3: Calculate the Speed of Sound
Now we can substitute the values into the speed of sound formula:
c = √(γ * R * T)
c = √(1.4 * 287 J/(kg·K) * 293 K)
c = √(1.4 * 287 * 293)
c = √(117,000.6) ≈ 342.5 m/s
Final Result
The speed of sound in the gas at atmospheric temperature is approximately 342.5 m/s. This value is typical for air at room temperature, confirming our assumptions and calculations.
Understanding the Context
When the piston was pulled out, the gas expanded adiabatically, which means no heat was exchanged with the surroundings. This expansion caused the pressure to drop from 1 atm to 0.355 atm, demonstrating the principles of thermodynamics in action. The speed of sound is influenced by the temperature and the properties of the gas, which is why we used the ideal gas law and the specific heat ratio to arrive at our answer.
