Question

Purification of water for drinking using UV light is a viable way to provide potable water in many areas of the world. Experimentally, the decrease in UV light of wavelength \(250 \mathrm{nm}\) follows the empirical relation \(I / I_{0}=e^{-s^{\prime} l}\) where \(l\) is the distance that the light passed through the water and \(\varepsilon^{\prime}\) is an effective absorption coefficient. \(\varepsilon^{\prime}=0.070 \mathrm{cm}^{-1}\) for pure water and \(0.30 \mathrm{cm}^{-1}\) for water exiting a wastewater treatment plant. What distance corresponds to a decrease in \(I\) of \(15 \%\) from its incident value for (a) pure water and (b) waste water?

Short answer

The distance required for a decrease in intensity of 15% is approximately 2.07 cm for pure water and 0.46 cm for wastewater.

Step-by-step solution

Find the decrease in intensity

The intensity \(I\) decreases by 15% from its initial value \(I_0\). So, we can write the proportion as follows: \(\frac{I}{I_0} = 1 - 0.15 = 0.85\)

Use the empirical relation to solve for distance in pure water

We substitute the given values into the empirical relation to solve for the distance in pure water: \[0.85 = e^{-\varepsilon^{\prime} l}\] Using \(\varepsilon^{\prime} = 0.070\,cm^{-1}\) for pure water, we get: \[0.85 = e^{-0.070\,l}\] Now we'll take the natural log of both sides: \[\ln 0.85 = -0.070\,l\] Then, solve for \(l\): \[l = \frac{\ln 0.85}{-0.070} \approx 2.07\,cm\] So, the distance required for a decrease in intensity of 15% in pure water is approximately 2.07 cm.

Use the empirical relation to solve for distance in wastewater

For wastewater, we follow the same method as in Step 2 but substitute the value of \(\varepsilon^{\prime} = 0.30\,cm^{-1}\) instead: \[0.85 = e^{-0.30\,l}\] Taking the natural log of both sides: \[\ln 0.85 = -0.30\,l\] And solving for \(l\): \[l=\frac{\ln 0.85}{-0.30}\approx 0.46\,cm\] So, the distance required for a decrease in intensity of 15% in wastewater is approximately 0.46 cm. To summarize: (a) The distance required for a decrease in intensity of 15% in pure water is approximately 2.07 cm. (b) The distance required for a decrease in intensity of 15% in wastewater is approximately 0.46 cm.

Key concepts

Beer-Lambert Law

The Beer-Lambert Law is fundamental to understanding how light interacts with absorbing materials such as water in the UV purification process. This scientific law states that there is a linear relationship between the absorbance of a substance and the concentration of that substance, as well as the path length the light travels through it. In essence, it helps predict how much light is absorbed by the water and indirectly, how effective the purification process is at different depths.

In water purification, UV light can be absorbed by various contaminants. The Beer-Lambert Law is used to calculate the amount of light absorbed by water at different distances, which correlates to the decrease in UV light intensity. Simply put, if we know the intensity of UV light before and after it passes through a certain volume of water, we can deduce information about the substances present within the water.

Empirical Absorption Coefficient

The empirical absorption coefficient, represented in the formula by \(\varepsilon^\prime\), is a measure of a substance's ability to absorb light at a specific wavelength. In the context of UV water purification, this coefficient tells us how effectively the water absorbs UV light. It differs for pure water and contaminated water, such as wastewater, because the presence of additional substances changes the water's absorption properties.

The higher the empirical absorption coefficient, the shorter the distance UV light can travel before it is significantly weakened. As seen in the exercise, wastewater with \(\varepsilon^\prime = 0.30\,cm^{-1}\) absorbs UV light much more than pure water with \(\varepsilon^\prime = 0.070\,cm^{-1}\). This principle is critical for designing purification systems, as it dictates the distance UV lights should be placed apart to achieve effective purification.

Intensity of Light

Intensity of light refers to the power of the light wave per unit area. In the context of UV water purification, it is crucial to consider the initial intensity of the UV light and how it decreases as it travels through the water. The initial intensity, \(I_0\), is the starting point that we compare subsequent measurements against, to understand how much the intensity has dropped.

As UV light penetrates the water, it loses some of its intensity due to absorption by the water and its impurities. The remaining intensity, \(I\), is what's measured after the light has traveled a certain distance. A fundamental aspect of this process is understanding that the UV light must maintain a sufficient intensity to effectively deactivate or destroy harmful organisms in the water.

Exponential Decay

Exponential decay is the mathematical principle describing the reduction of light intensity as it passes through a medium, like water in the purification process. This decay follows an exponential function, which very quickly decreases from the initial value. In the formula \(I / I_{0}=e^{-\varepsilon^\prime l}\), the term \(e^{-\varepsilon^\prime l}\) signifies the exponential decay of the light intensity as it travels the distance \(l\) through the water.

Understanding exponential decay is important for accurately calculating how far UV light can effectively travel in a purification setting. Since the decay is not linear, a small increase in distance or absorption coefficient can have a large effect on the remaining light intensity. As the exercise demonstrates, even a relatively short distance can result in a significant decrease in light intensity, which is vital knowledge for effectively utilizing UV light in water purification.