If the RMS speed of a gas is 300 m/s at 400 K, what will be the RMS speed at 200

If the RMS speed of a gas is 300 m/s at 400 K, what will be the RMS speed at 200 K?

If the RMS speed of a gas is 300 m/s at 400 K, what will be the RMS speed at 200 K?

Correct Answer: 150 m/s

Step 1: Understand that RMS speed is related to temperature. It is proportional to the square root of the temperature.
Step 2: Write down the formula for RMS speed: v_rms ∝ √T, where v_rms is the RMS speed and T is the temperature in Kelvin.
Step 3: Identify the initial conditions: RMS speed (v_rms1) = 300 m/s at temperature (T1) = 400 K.
Step 4: Identify the new temperature: T2 = 200 K.
Step 5: Set up the ratio using the proportionality: v_rms2 / v_rms1 = √(T2 / T1).
Step 6: Substitute the known values into the equation: v_rms2 / 300 = √(200 / 400).
Step 7: Simplify the square root: √(200 / 400) = √(0.5) = 1/√2.
Step 8: Now, rewrite the equation: v_rms2 = 300 * (1/√2).
Step 9: Calculate the value: v_rms2 = 300 * 0.7071 ≈ 150 m/s.
Step 10: Conclude that the RMS speed at 200 K is approximately 150 m/s.

RMS Speed and Temperature Relationship – The root mean square (RMS) speed of a gas is directly proportional to the square root of its absolute temperature, following the formula v_rms = sqrt(3kT/m) where k is the Boltzmann constant, T is the temperature, and m is the mass of the gas particles.