If the absolute temperature of a gas is doubled, what happens to the root-mean-square speed of the molecules?
New RMS speed is 1.414 times original rms speed
v = sqrt(3RT/M)
If the absolute temperature of a gas is doubled, the root-mean-square (RMS) speed of the molecules will increase.
The RMS speed of the molecules in a gas is directly proportional to the square root of the temperature. Mathematically, it can be expressed as:
v_rms = √(3kT/m)
Where:
- v_rms is the root-mean-square speed of the molecules
- k is the Boltzmann constant (1.38 × 10⁻²³ J/K)
- T is the temperature in Kelvin
- m is the molar mass of the gas
If the absolute temperature T is doubled, the RMS speed will increase because the square root of the temperature is also doubled. Therefore, the molecules will move at a higher average speed.
To understand what happens to the root-mean-square (RMS) speed of gas molecules when the absolute temperature is doubled, we first need to understand the relationship between temperature and the RMS speed of gas molecules.
The RMS speed of gas molecules is directly proportional to the square root of the temperature. This relationship is described by the root-mean-square velocity formula:
v_rms = sqrt(3kT/m)
Where:
- v_rms is the root-mean-square speed
- k is the Boltzmann constant (1.38 × 10^-23 J/K)
- T is the absolute temperature
- m is the molar mass of the gas molecules
Now, let's see how doubling the absolute temperature affects the RMS speed.
If the absolute temperature (T) is doubled, we can substitute 2T into the formula:
v_rms = sqrt(3k * 2T / m)
Since the square root of 2 is approximately 1.414, we can simplify the formula:
v_rms = 1.414 * sqrt(3kT/m)
Now we can observe that the RMS speed becomes approximately 1.414 times faster when the absolute temperature is doubled. Therefore, doubling the absolute temperature doubles the RMS speed of gas molecules.