Explain the patterns of temperature and air pressure throughout the atmosphere.

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1. **Atmospheric Structure and Temperature Patterns**

- The atmosphere is divided into layers, each with a unique temperature gradient:
     
     - **Troposphere:**  
       Temperature decreases with altitude at an average lapse rate of about $$ 6.5^\circ \text{C/km} $$. In this layer, weather phenomena occur due to the mixing of air masses.
     
     - **Stratosphere:**  
       Temperature initially levels off and then increases with altitude. This warming is caused by the absorption of ultraviolet radiation by ozone.
     
     - **Mesosphere:**  
       Temperature decreases with altitude again. It is the coldest layer in the atmosphere where meteors burn upon entry.
     
     - **Thermosphere:**  
       Temperature rises significantly with altitude because of the absorption of high-energy solar radiation.

2. **Air Pressure Patterns**

- Air pressure decreases with altitude due to the gravitational pull on the mass of the atmosphere. At the surface, the weight of the overlying air is greatest.
     
   - The hydrostatic equilibrium condition is represented by:
     $$
     \frac{dp}{dz} = -\rho g
     $$
     where:
     - $$ p $$ is the air pressure,
     - $$ z $$ is the altitude,
     - $$ \rho $$ is the air density, and
     - $$ g $$ is the acceleration due to gravity.
     
   - Incorporating the ideal gas law $$ p = \rho R T $$ (where $$ R $$ is the specific gas constant and $$ T $$ is the temperature), one may derive the barometric formula:
     $$
     p = p_0 e^{-\frac{Mgz}{RT}}
     $$
     Here, $$ p_0 $$ is the sea-level pressure and $$ M $$ is the molar mass of air. This equation shows the exponential decrease of pressure with an increase in altitude.

3. **Summary of Patterns**

- **Temperature:**  
     Varies based on the atmospheric layer. It generally decreases with altitude in the troposphere, increases in the stratosphere, decreases in the mesosphere, and increases again in the thermosphere.
     
   - **Air Pressure:**  
     Continuously decreases with altitude, primarily due to the reduction in the weight of the air above as one moves higher in the atmosphere.
Temperature decreases with altitude in the troposphere, levels off in the stratosphere, decreases again in the mesosphere, and increases in the thermosphere. Air pressure decreases with altitude due to the weight of the overlying air.

Explain the patterns of temperature and air pressure throughout the atmosphere.

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