If the temperature of a gas is doubled, how does its RMS speed change?

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Question: If the temperature of a gas is doubled, how does its RMS speed change?

Options:

Correct Answer: Increases by a factor of sqrt(2)

Solution:

The RMS speed is proportional to the square root of the temperature. If the temperature is doubled, the RMS speed increases by a factor of sqrt(2).

If the temperature of a gas is doubled, how does its RMS speed change?

If the temperature of a gas is doubled, how does its RMS speed change?

Correct Answer: RMS speed increases by a factor of sqrt(2)

Step 1: Understand that RMS speed stands for Root Mean Square speed, which is a measure of the average speed of gas particles.
Step 2: Know that the RMS speed of a gas is related to its temperature. The formula is: RMS speed = sqrt(3 * k * T / m), where T is the temperature.
Step 3: Recognize that if the temperature (T) is doubled, we can express this as T' = 2T.
Step 4: Substitute the new temperature into the RMS speed formula: RMS speed' = sqrt(3 * k * (2T) / m).
Step 5: Simplify the equation: RMS speed' = sqrt(2) * sqrt(3 * k * T / m).
Step 6: Notice that sqrt(3 * k * T / m) is the original RMS speed. Therefore, RMS speed' = sqrt(2) * (original RMS speed).
Step 7: Conclude that when the temperature is doubled, the RMS speed increases by a factor of sqrt(2).

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