By definition, 2% by mass means 2 grams of  in every 100 grams of solution. In order to solve this problem, we may assume we are dealing with 100 grams of solution. We can use the density as a conversion factor to determine the volume of solution we are dealing with:

In order to calculate the number of moles of _, we need to convert 2 grams of  to moles using it's molecular weight. This is because we are assuming we are dealing 100 grams of a 2%  solution. 

In order to calculate molarity, we need to plug the moles and volumes calculated into the following equation:

A calibration curve is a graph that gives the relationship between a measured quantity and an analytical signal. In this question the analytical signal is absorbance and the measured quantity is the concentration. To answer this problem, simply determine where the absorbance value 1.5 is on the y-axis and find what point the x-axis crosses this absorbance.

In chemistry, percentage yield is a term used to quantify the efficiency of a chemical reaction. In order to calculate this quantity, the value for the actual yield and theoretical yield is needed. The actual yield is the amount of product obtained during a chemical reaction. The theoretical yield is the maximum amount of product possible to be obtain by a chemical reaction which is based on the amount of reactants used. Below is the formula needed to calculate the percentage yield.

 is the limiting reactant because based on the equation,  and  react in a 1:1 mole ratio. However, there is 0.0040 moles more of  than  present. Therefore the amount of  produced is limited by the amount of  present in the reaction. Based on the equation, for every 1 mole of  reacted, 1 mole of  was produced:

Therefore,

In chemistry, percentage yield is a term used to quantify the efficiency of a chemical reaction. In order to calculate this quantity, the value for the actual yield and theoretical yield is needed. The actual yield is the amount of product obtained during a chemical reaction. The theoretical yield is the maximum amount of product possible to be obtain by a chemical reaction which is based on the amount of reactants used. Below is the formula needed to calculate the percentage yield.

There are four steps to finding the standard deviation of random variables. First, calculate the mean of the random variables. Second, for each value in the group (45, 40, 25, and 12), subtract the mean from each and multiply the result by the probability of that outcome occurring. Third, add the four results together. Fourth, find the square root of the result.

The mean for any set of random variables is additive in the sense that

The difference is also additive, so we have 

This means the mean of  is . 

The variance is additive when the random variables are independent, which they are in this case. But it's additive in the sense that for any real numbers  (even when negative), we have

.

So for this difference, we have

.

So the mean and variance are  and , respectively. In addition to that,  is normally distributed because the sum or difference of any set of independent normal random variables is also normally distributed.

For this problem we need to determine the equation of our standard curve. This can be done by creating a graph from the data points and determining the slope.

Assuming that a sample of zero concentration will also have zero absorbance, we can find the equation for the line generated by finding the slope.

Pick two points on the line to find the slope. We will use (0,0) and (40,0.405).

Use this equation and the absorbance given in the question to find the concentration of the unknown sample.

The slope of the line, , can be calculated by the following equation:

The subscripts are simply the points you chose in the order you arbitrarily chose.The most important thing to do is plug the values into equation in the order you chose them.

To determine the slope of the line we need to use two points on the graph. Assuming the two points chosen correspond to  and , plugging these values into the equation gives:

Therefore the slope of the line is equal to 

For a general chemical equation , where A, B, C, and D are elements and the Greek letters are their coefficients, we have the reaction quotient equation:

We can find the reaction quotient equation for our reaction by substituting the variables.

Notice that the concentration of  is in the denominator and is squared, so doubling the concentration of  changes the reaction quotient by a factor of one-fourth.