Question

In which medium will sound travel fastest for a given temperature: air, helium, or argon?

Short answer

Answer: For a given temperature, sound will travel the fastest in helium.

Step-by-step solution

Recall the formula for the speed of sound in a medium

The speed of sound in a medium can be calculated using the following formula: v = sqrt(γ * R * T / M) where v is the speed of sound, γ is the adiabatic index (ratio of specific heats), R is the universal gas constant (8.314 J/mol-K), T is the temperature in kelvin, and M is the molar mass of the medium in kg/mol.

Find the values for the adiabatic index and molar mass of each gas

We need to gather the adiabatic index and molar mass for air, helium, and argon. You may look up these values in a reference material or textbook. Here are the values: - For air: γ = 1.4, M = 0.029 kg/mol - For helium: γ = 1.67, M = 0.004 kg/mol - For argon: γ = 1.67, M = 0.040 kg/mol

Calculate the speed of sound in each gas at a given temperature

Using the formula from Step 1 and the values from Step 2, we will now calculate the speed of sound in each gas for a given temperature T. Remember to keep the temperature T the same for all three gases. For example, let's assume a temperature of 300K. For air: v_air = sqrt(1.4 * 8.314 * 300 / 0.029) ≈ 343 m/s For helium: v_helium = sqrt(1.67 * 8.314 * 300 / 0.004) ≈ 965 m/s For argon: v_argon = sqrt(1.67 * 8.314 * 300 / 0.040) ≈ 323 m/s

Determine in which medium sound travels the fastest

Comparing the calculated speeds of sound, we can see that: v_helium > v_air > v_argon Thus, for a given temperature, sound will travel the fastest in helium.