The earth retains its atmosphere, due to

Options

A. The earth is sphere
B. the earth has population
C. the escape velocity of molecules is less than the mean speed of molecules
D. the escape velocity of molecules is more than the mean speed of molecules

Answer

To understand why the Earth retains its atmosphere, we need to analyze the relationship between the velocities of gas molecules and the escape velocity of the Earth. Here’s a step-by-step solution:

### Step 1: Understanding Escape Velocity
Escape velocity is the minimum velocity that an object must have to break free from the gravitational pull of a planet without any additional propulsion. For Earth, this velocity is approximately 11.2 km/s.

### Step 2: Root Mean Square Velocity (V_rms)
The root mean square velocity (V_rms) is a measure of the average speed of gas molecules in a given temperature. It can be calculated using the formula:
\[ V_{rms} = \sqrt{\frac{3kT}{m}} \]
where \( k \) is the Boltzmann constant, \( T \) is the temperature in Kelvin, and \( m \) is the mass of the gas molecule.

### Step 3: Comparing V_rms with Escape Velocity
For the Earth to retain its atmosphere, the average speed of the gas molecules (V_rms) must be less than the escape velocity. If V_rms is less than the escape velocity, the gas molecules do not have enough energy to escape the gravitational pull of the Earth.

### Step 4: Conclusion
Since the root mean square velocity of the gas molecules in the Earth's atmosphere is less than the escape velocity, these molecules cannot escape into space. Therefore, the Earth retains its atmosphere.

### Final Answer
The Earth retains its atmosphere because the root mean square velocity of the gas molecules is less than the escape velocity.

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