If the temperature of a gas is halved, what happens to its RMS speed?

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Question: If the temperature of a gas is halved, what happens to its RMS speed?

Options:

Correct Answer: Decreases by sqrt(2)

Solution:

RMS speed is directly proportional to the square root of temperature. Halving the temperature results in a decrease in RMS speed by sqrt(2).

If the temperature of a gas is halved, what happens to its RMS speed?

If the temperature of a gas is halved, what happens to its RMS speed?

Step 1: Understand that RMS speed refers to the root mean square speed of gas particles.
Step 2: Know that the RMS speed is directly related to the temperature of the gas.
Step 3: Recall the formula for RMS speed: v_rms = sqrt(3kT/m), where k is the Boltzmann constant, T is the temperature, and m is the mass of the gas particles.
Step 4: If the temperature (T) is halved, we can express this as T' = T/2.
Step 5: Substitute T' into the RMS speed formula: v_rms' = sqrt(3k(T/2)/m).
Step 6: Simplify the equation: v_rms' = sqrt(3kT/m * 1/2) = sqrt(1/2) * v_rms.
Step 7: Recognize that sqrt(1/2) is the same as 1/sqrt(2), which means the new RMS speed is v_rms' = v_rms/sqrt(2).
Step 8: Conclude that halving the temperature results in the RMS speed decreasing by a factor of sqrt(2).

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, 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.