Assume a parcel of air on the surface with a temperature of 29 degrees Celsius with a relative humidity of 50%. If the parcel rises at what altitude should clouds form?

To determine at what altitude clouds should form, we need to understand the concept of the dew point temperature. The dew point temperature is the temperature at which air becomes saturated, meaning it cannot hold any more water vapor. When the temperature drops to or below the dew point, condensation occurs, and clouds or fog can form.

Here's how you can calculate the dew point temperature and find the altitude where clouds should form:

1. Convert the air temperature from Celsius to Kelvin by adding 273.15.
- 29 degrees Celsius + 273.15 = 302.15 Kelvin

2. Calculate the dew point temperature using the given relative humidity. You can use the Magnus-Tetens formula, which is one of the methods commonly used to estimate the dew point temperature:
a. Calculate the saturation vapor pressure (es) for the given air temperature. You can use the formula:
- es = 6.112 * e^[(17.67 * T) / (T + 243.5)]
- T is the air temperature in Celsius
b. Calculate the actual vapor pressure (e) using the relative humidity:
- e = (H / 100) * es
- H is the relative humidity
c. Use the actual vapor pressure (e) to calculate the dew point temperature (Td) using the inverse of the saturation vapor pressure equation:
- Td = (243.5 * log(e / 6.112)) / (17.67 - log(e / 6.112))

Let's plug in the values:
- T = 29 degrees Celsius
- H = 50%

Calculations:
- es = 6.112 * e^[(17.67 * 29) / (29 + 243.5)]
- e = (50 / 100) * es
- Td = (243.5 * log(e / 6.112)) / (17.67 - log(e / 6.112))

The resulting dew point temperature (Td) will be in Celsius.

3. Once you have the dew point temperature, you can determine the altitude where clouds should form. As air rises, it cools due to decreasing atmospheric pressure. If the temperature of the rising air reaches the dew point temperature, clouds will form. The rate at which air temperature decreases with height is called the lapse rate and typically varies between 5 and 10 degrees Celsius per kilometer.

- Calculate the lapse rate by subtracting the dew point temperature from the air temperature:
- Lapse Rate = T - Td

- Divide the lapse rate by the temperature change per kilometer (typically 5-10 degrees Celsius) to find the altitude where clouds should form:
- Altitude = (Lapse Rate / Temperature Change per Kilometer)

For example, let's assume a temperature change rate of 10 degrees Celsius per kilometer:
- Lapse Rate = 29 - Td
- Altitude = (Lapse Rate / 10)

By following these steps, you can calculate the altitude at which clouds should form for a given air temperature and relative humidity.