If the difference between the dry bulb temperature and wet bulb temperature is 0°C, what is the relative humidity?

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If the difference between the wet and dry bulbs is zero degrees Celsius, then the relative humidity is 100%.

Relative humidity is the amount of water vapor in the air. Relative humidity is expressed as a percentage. The percentage represents the ratio between the amount of water vapor that is actually in the air at a certain temperature and the maximum amount of water that could be in the air at that temperature.

A psychrometer is the instrument that is used to determine the relative humidity. A psychrometer is made of two thermometers that are attached to a dowel by strings. One of the thermometers has a wick at the base of its bulb. The wick is dipped into water. The thermometer on the psychrometer that contains the wet wick is called the "wet bulb". The thermometer on the psychrometer without the wick is called the "dry bulb".  The thermometers are then whirled around the dowel by their strings. The thermometers are whirled until the wet bulb is dry. When there is little water vapor present, the water on the wet bulb is able to evaporate. This results in a greater difference between the wet and dry bulbs of the psychrometer. When the air is saturated with water vapor, then the water on the wet bulb cannot evaporate. Therefore, there is little difference in the temperature readings of the dry and wet bulbs of the psychrometer. In other words, the relative humidity is 100%.

 

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The relative humidity is 100%.  The device you are describing is called a psychrometer, two thermometers, one with a wick on the end of it. 
The two thermometers are attached to a sling, or a stick, devised for the scientist to wet the wick, producing a "wet" thermometer.  The scientist then swings the device for a few minutes, long enough for the air to cause the water in the wick of the wet thermometer to evaporate.  The evaporation produces a cooling effect.  The scientist subtracts the wet bulb reading from the dry bulb reading.  The greater the number of degrees difference, the less water vapor is in the air, which means the relative humidity is lower.  In other words, there is more room for the liquid water molecules to evaporate into gaseous water molecules.  So the relative humidity is lower. 

In this case, however, there was NO difference between the readings of the wet bulb and the dry bulb thermometers.  This means there is absolutely no room at this temperature for water to evaporate, indicating the humidity of the air is saturated, or full.  So the relative humidity is 100%.

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