An impurity in water has an extinction coefficient of 3.45⨉10³ M^-1 cm^-1 at 280 nm, its absorption maximum (A Closer Look, p. 576). Below 50 ppb, the impurity is not a problem for human health. Given that most spectrometers cannot detect absorbances less than 0.0001 with good reliability, is measuring the absorbance of a water sample at 280 nm a good way to detect concentrations of the impurity above the 50-ppb threshold?

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Hi everyone. Let's take a look at this problem. The maximum allowable limit for a cadmium is 0.2 parts per billion per day. The spectral photo metric quantitative of cadmium and drinking water can be performed using a complex ation reaction cadmium to makes a deep greenish yellow complex with a 1-1 mole ratio with allies are in red. S. The extinction coefficient of the complex is the following value. At 422 nanometers, given that most spectral photo meters cannot detect absorbent less than 4220.1 with good reliability. Is this method a good way to quantify cadmium at its allowable limit and drinking water assume a cell path length of one centimeter. So our goal here is to determine if this is a good method or not. Okay, so what we can do is we can calculate the concentration of the complex at the absorbent value given once we get that concentration will know the mill arat. E and from the mill aren? T We can convert it to parts per billion and see if this method is going to be good or not. So let's go ahead and start there by first solving for our concentration. The equation that we can use to solve for our concentration is absorbency is equal to are Moeller absorptive. Itty times our path length times concentration. So we're interested in solving for concentration. So let's go ahead and divide both sides of our equation to isolate that. And so we're going to get our concentration is equal to the following. This should be that value. Let me make that correction here. Okay, there we go. So let's go ahead and plug in because we have all these values. So we're told that our absorbency is 0.0001. Okay, the coefficient Is going to be 2.24 times 10 to the - leaders times more. Inverse times centimeters in verse. Multiplied by our path length of one centimeter. Okay, so once we do that calculation, we're going to get a concentration of 4.46 times 10 to the negative eight m. Okay, we know that the mole ratio between the complex and cadmium two is 1 to 1. So that means our concentration for our cadmium too is going to equal the following. What we just solved for since we have a 1-1 ratio. Okay, so now what we're going to do is convert this two parts per billion. Okay, so we're going to express this value in parts Pavilion and we know that concentration orm alaric E is most per leader. So this cadmium to let's express that as a fraction. And so our cadmium too is what we want to convert to parts per billion. So we'll take this value 4. Times 10 to the -8 moles per liter. That's our concentration. Okay. And we want to go from moles per liter. two parts per billion. And so to go from moles two g we're going to use the molar mass of our cadmium too. So in one mole We have the molar mass 112. g. Okay, so our moles are going to cancel each other and we're left with grams. And then we need to go from grams to micrograms. Okay, so and one microgram, There's 10 to the -6 g. Okay, so our grams cancel and we're left with micrograms per leader. This is the units of parts per billion. Okay, so one part per billion is equal to microgram per leader. Okay. And that's the units that were left with an R Equation below. So once we solve this, we're going to get a final answer of 5.01 parts per billion. The detectable concentration of cadmium to add absorbent value of .0001 is 5.01 which is above the allowable limit of cadmium which is .2 parts per billion, which is the value that was given in the problem. So that means this method is not able to detect cadmium at its allowable limit. So our final answer for this problem is going to be no. This method is not a good way to quantify cadmium at its allowable limit. Okay, so that's it for this problem. I hope this was helpful

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Key Concepts

Extinction Coefficient

Absorbance and Beer-Lambert Law

Detection Limit