Problem 29 - Chemistry for Allied Health

Question

A \(2.50 \mathrm{M}\) solution undergoes a chemical reaction. After \(3.00\) minutes, the concentration of the solution is \(2.15 \mathrm{M}\). What is the rate of the reaction in \(\mathrm{M} / \mathrm{s}\) ?

Step-by-Step Solution

Answer

The rate of the reaction is \(1.94 \times 10^{-3} \mathrm{M/s}\).

Step 1: Determine Initial and Final Concentrations

The initial concentration of the solution is given as \(2.50 \mathrm{M}\), and the final concentration after the reaction is \(2.15 \mathrm{M}\).

Step 2: Calculate the Change in Concentration

Subtract the final concentration from the initial concentration to find the change in concentration: \[ \Delta [A] = [A]_{initial} - [A]_{final} = 2.50 \mathrm{M} - 2.15 \mathrm{M} = 0.35 \mathrm{M} \]

Step 3: Convert Minutes to Seconds

Since the rate of reaction is usually expressed in \(\mathrm{M/s}\), convert the time from minutes to seconds: \[ 3.00 \, \text{minutes} = 3.00 \, \times 60 \, \text{seconds} = 180.00 \, \text{seconds} \]

Step 4: Calculate the Rate of Reaction

Using the formula for the rate of a reaction, rate \(R\), given by \[ R = \frac{\Delta [A]}{\Delta t} \]Substitute the values:\[ R = \frac{0.35 \mathrm{M}}{180.00 \mathrm{s}} \]This simplifies to:\[ R = 0.001944 \mathrm{M/s} \]

Step 5: Round the Result

Round the result to an appropriate number of significant figures based on the given data, typically 3 significant figures. Hence, the rate of reaction is:\[ R = 1.94 \times 10^{-3} \mathrm{M/s} \]

Key Concepts

Change in Concentration

In any chemical reaction, the change in concentration is a direct indication of how much the concentration of a reactant or product has changed over time. To calculate this change, you typically subtract the final concentration of a substance from its initial concentration. In our exercise, the initial concentration is given as \(2.50\,\mathrm{M}\), while the final concentration is \(2.15\,\mathrm{M}\). By subtracting these values, \(\Delta [A] = 2.50 - 2.15 = 0.35\,\mathrm{M}\), we determine that the concentration has decreased by \(0.35\,\mathrm{M}\).
This calculation is crucial because it allows us to analyze the progression of the reaction and calculate the rate of reaction. The rate at which this change occurs can provide insights into the efficiency and speed of the reaction.

Conversion of Time Units

Before calculating the rate of a reaction, the time interval over which the reaction occurs must be accurately represented. Often, the problem provides time in minutes, but for the purpose of calculating reaction rates, time needs to be converted to seconds. This is because reaction rates are typically expressed in terms of \(\text{M/s}\) (molarity per second).
Conversion is straightforward: multiply the number of minutes by 60 to find the equivalent in seconds. For example, in the exercise, the reaction occurs over \(3.00\,\text{minutes}\). Converting this to seconds, we multiply \(3.00\,\text{minutes}\) by \(60\), which equals \(180.00\,\text{seconds}\).
By doing this, we ensure our calculations are consistent with standard units, thereby avoiding errors in the reaction rate calculation.

Rate of Reaction Formula

The rate of a chemical reaction is a measure of how fast a reactant is consumed or a product is formed. To calculate the rate of reaction, we use the formula:\[ R = \frac{\Delta [A]}{\Delta t} \]where \(\Delta [A]\) is the change in concentration, and \(\Delta t\) is the change in time.
Inserting the known values: \(\Delta [A] = 0.35\,\mathrm{M}\) and \(\Delta t = 180.00\,\text{seconds}\), the rate \(R\) becomes \(\frac{0.35}{180.00} = 0.001944\,\mathrm{M/s}\).
This formula is pivotal in stoichiometric calculations because it helps us predict how quickly a reaction will reach completion. A slower rate means a longer time to complete the reaction, whereas a faster rate indicates a quicker completion.

Significant Figures Rounding

Significant figures are essential in ensuring that the precision of our calculated values reflects the precision of the initial data. When rounding, consider the least number of significant figures in the given data. In this case, our concentrations and time values are all reported to three significant figures. Thus, our final calculated reaction rate should also reflect three significant figures.
From our previous calculation, the rate is \(0.001944\,\mathrm{M/s}\). Rounding this to three significant figures gives \(1.94 \times 10^{-3}\,\mathrm{M/s}\).
It is important to maintain this precision to ensure that the results of the calculation are both accurate and reliable, reflecting the true conditions and constraints of the original data.