The molecules of air in the room that you are sitting are all experiencing the force of gravity tending to bring them down. The molecules are also frequently and randomly undergoing collisions, which tend to oppose the effect of fall under gravity. The density of air is nearly uniform throughout the room because

Options

A. The mass of the molecules are very small
B. The gravitational potential energy mgh is much lesser than the average thermal energy kT
C. The gravitational potential energy mgh is much greater than the average thermal energy kT
D. mgh is nearly of the same magnitude askt, which results in the cancellation of the two opposing factors

Answer

To solve the problem regarding the uniform density of air in a room, we need to analyze the forces acting on the air molecules and their energies.

### Step-by-Step Solution:

1. **Understanding the Forces on Air Molecules**:
- Air molecules in a room are subject to the force of gravity, which pulls them downwards. This force can be expressed as \( F = mg \), where \( m \) is the mass of a molecule and \( g \) is the acceleration due to gravity.

2. **Collisions and Random Motion**:
- The air molecules are in constant random motion and frequently collide with each other. These collisions impart kinetic energy to the molecules, which counteracts the gravitational force.

3. **Energy Considerations**:
- The gravitational potential energy of a molecule at height \( h \) is given by \( U = mgh \).
- The average thermal kinetic energy of the molecules can be expressed as \( K = \frac{3}{2}kT \), where \( k \) is the Boltzmann constant and \( T \) is the temperature in Kelvin.

4. **Comparing Energies**:
- For the density of air to be nearly uniform, the average thermal kinetic energy must be comparable to or greater than the gravitational potential energy. If the average thermal energy is much greater than the gravitational potential energy, the random motion due to thermal energy will dominate, leading to a uniform distribution of air molecules throughout the room.

5. **Conclusion**:
- Therefore, the correct reasoning is that the gravitational potential energy \( mgh \) is much lesser than the average thermal kinetic energy \( kT \). This results in the air molecules being able to move freely and uniformly throughout the room, leading to a nearly uniform density of air.

### Final Answer:
The density of air is nearly uniform throughout the room because the gravitational potential energy \( mgh \) is much lesser than the average thermal kinetic energy \( kT \).

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