Question

A particular fog consists of 10000 droplets of water per \(\mathrm{cm}^{3}\). The average diameter of the drops is \(1.5 \mu \mathrm{m}\). Compare the mass of water in the liquid phase to that in the gaseous form if the temperature is \(35^{\circ} \mathrm{C}\) and the relative humidity is \(100 \%\).

Short answer

The mass of water in the liquid phase is much greater than in the gaseous form.

Step-by-step solution

Calculate Volume of a Single Water Droplet

Given that the average diameter of a water droplet is \(1.5 \mu \mathrm{m}\), we can find the radius \(r\) by dividing by 2: \(r = 0.75 \mu \mathrm{m} = 0.75 \times 10^{-6} \mathrm{m}\). The volume \(V\) of a single spherical droplet is calculated using the formula for the volume of a sphere: \[ V = \frac{4}{3} \pi r^3 \]. Substituting the radius into the formula gives: \[ V = \frac{4}{3} \pi (0.75 \times 10^{-6})^3 \]

Calculate Total Volume of Water in Liquid Phase

There are 10,000 droplets per cubic centimeter, which is equivalent to \(10,000 \times 10^6\) droplets per cubic meter. So, the total volume of all droplets in 1 \( \mathrm{m}^3 \) is \(10,000 \times 10^6 \times V \), where \(V\) is the volume of a single droplet calculated in Step 1.

Calculate Mass of Water in Liquid Phase

The density \(\rho\) of water is \(1000 \mathrm{kg/m^3}\). Thus, the mass \(m\) of water in liquid form is given by \(m = \rho \times \text{Total Volume} \). Substitute the total volume from Step 2 to find the mass of water in liquid form.

Calculate Mass of Water Vapor in Gaseous Form

At \(35^{\circ}\mathrm{C}\) and 100% relative humidity, the air holds the maximum amount of water vapor possible. This is equivalent to the saturation vapor pressure. The density of water vapor \(\rho_v\) can be calculated using the ideal gas law \(\rho_v = \frac{P}{RT}\), where \(P\) is the saturation pressure (determined from tables), \(R\) is the specific gas constant for water vapor \(461.5 \mathrm{J/(kg \, K)}\), and \(T\) is the temperature in Kelvin. Calculate \(\rho_v\) and multiply by the volume \(1 \mathrm{m^3}\) to find the mass of water in gaseous form.

Compare Masses

Compare the masses of water in the liquid phase (from Step 3) and the gaseous phase (from Step 4) to determine which is greater.

Key concepts

Fog Composition

Fog is a fascinating weather phenomenon primarily composed of tiny water droplets suspended in the air. Unlike steam, fog is visible, and its cloud-like appearance is due to the scattering of light by these droplets.
The composition of fog can vary, but it's predominantly made up of:

• Water Droplets: These are the tiny spheres of liquid water that make fog appear dense and cloudy.
• Aerosols: Particles present in the air that can help water droplets form more easily.

Fog forms when air near the ground cools to the point of condensation, leading water vapor to change into liquid droplets. The specific condition necessary for fog formation is high relative humidity, ideally reaching 100%.
At this point, the air contains the maximum amount of water vapor it can hold, and further cooling will turn the vapor into droplets, contributing to the fog's density. Understanding fog composition helps us analyze various weather conditions and predict the occurrence of fog in different environments.

Water Droplets

Water droplets, fundamental to the makeup of fog, are tiny but numerous. Each droplet is essentially a small sphere of water that has formed through the condensation of water vapor.
Here's a closer look at their characteristics:

• Size: In the context of fog, droplets average about 1 to 10 micrometers in diameter. In our specific exercise, they are 1.5 micrometers in diameter, showing how minute they really are.
• Density: There can be around 10,000 water droplets per cubic centimeter, creating the dense "cloud" like fog we see.

The volume of a single droplet is critical for determining the total mass of water in fog. This is calculated by using the formula for the volume of a sphere, \( V = \frac{4}{3} \pi r^3 \), where \( r \) is the radius. The radius is half of the diameter, so our tiny droplets have a radius of 0.75 micrometers.
Knowing these details about water droplets tells us more about their impactful role in environmental phenomena like fog.

Mass of Water

When measuring the mass of water in different phases, such as liquid and vapor, various factors need to be considered, especially in environmental chemistry. One must first calculate the volume and, consequently, the mass of liquid water in fog.
Given the high number of water droplets per cubic meter, the total volume of the water can be determined by multiplying the number of droplets by the volume of a single droplet.
This step is critical since the mass of the water in liquid form is found by multiplying the total volume by the density of water, which is 1000 kg/m³. This calculation gives us insight into the tangible content of water in fog.
In contrast, the mass of water as vapor is calculated using atmospheric pressure and temperature data. With the ideal gas law, one can measure the density of water vapor. The mass is found by multiplying this density by the volume considered, typically 1 m³ in standard conditions.
Comparing these masses reveals how much water is divided between the liquid and gaseous states in the fog, shedding light on its transient nature and its dependency on environmental temperature, pressure, and humidity levels.