Physics Question:
If the absolute temperature of a gas is doubled, what happens to the root-mean-square (rms) speed of the molecules?
Options:
- A) Nothing happens to the rms speed.
- B) The new rms speed is 4 times the original rms speed.
- C) The new rms speed is 2 times the original rms speed.
- D) The new rms speed is approximately 1.414 times the original rms speed.
- E) The new rms speed is (1/2) times the original rms speed.
Correct Answer:
D) The new rms speed is approximately 1.414 times the original rms speed.
Concept Explanation:
The root-mean-square (rms) speed of gas molecules is given by the formula:
vrms = √(3kT / m)
- k is the Boltzmann constant
- T is the absolute temperature (in Kelvin)
- m is the mass of one gas molecule
As per the equation, vrms ∝ √T. This means the rms speed is proportional to the square root of the temperature.
Step-by-Step Derivation:
If the original temperature is T, then the new temperature is 2T.
We now calculate the new rms speed:
vrms,new = √(3k × 2T / m) = √2 × √(3kT / m) = √2 × vrms
Since √2 ≈ 1.414,
vrms,new ≈ 1.414 × vrms
Conclusion:
When the absolute temperature of a gas is doubled, the rms speed of the molecules increases by a factor of √2, or approximately 1.414 times the original value.