Step 1: Understanding the Question:
The Root Mean Square (RMS) speed is a measure of the speed of particles in a gas. The problem asks for the change in this speed when the temperature is significantly increased.
Step 2: Key Formula or Approach:
The formula for the RMS speed of a gas molecule is:
\[ v_{rms} = \sqrt{\frac{3RT}{M}} \]
where \(R\) is the gas constant, \(T\) is the absolute temperature in Kelvin, and \(M\) is the molar mass. For a specific gas like Oxygen, \(R\) and \(M\) are constant, so:
\[ v \propto \sqrt{T} \]
Step 3: Detailed Explanation:
Step A: Convert the initial temperature to Kelvin.
\(T_1 = 27^\circ C = 27 + 273 = 300 K\).
Step B: Convert the final temperature to Kelvin.
\(T_2 = 927^\circ C = 927 + 273 = 1200 K\).
Step C: Express the ratio of final speed (\(v_2\)) to initial speed (\(v_1 = v\)):
\[ \frac{v_2}{v} = \sqrt{\frac{T_2}{T_1}} \]
Step D: Substitute the temperatures into the ratio:
\[ \frac{v_2}{v} = \sqrt{\frac{1200}{300}} \]
Step E: Simplify the fraction under the square root:
\[ 1200 / 300 = 4 \]
Step F: Calculate the square root:
\[ \sqrt{4} = 2 \]
This means \(v_2 = 2v\).
According to kinetic theory, temperature is a direct measure of the average kinetic energy of the molecules. When the absolute temperature increases by a factor of 4, the average kinetic energy also increases by 4 times. Since kinetic energy is proportional to the square of velocity (\(K.E. \propto v^2\)), the velocity itself must increase by the square root of 4, which is 2.
Step 4: Final Answer:
The RMS speed of the oxygen molecules at the higher temperature will be \(2v\).