Problem 128 - General Chemistry: Principles and Modern Applications

Question

It is often difficult to determine the concentration of a species in solution, particularly if it is a biological species that takes part in complex reaction pathways. One way to do this is through a dilution experiment with labeled molecules. Instead of molecules, however, we will use fish. An angler wants to know the number of fish in a particular pond, and so puts an indelible mark on 100 fish and adds them to the pond's existing population. After waiting for the fish to spread throughout the pond, the angler starts fishing, eventually catching 18 fish. Of these, five are marked. What is the total number of fish in the pond?

Step-by-Step Solution

Answer

The total number of fish in the pond is estimated to be 360.

Step 1: Identify Known Variables

The angler marked and released 100 fish into the pond. Then, 18 fish were caught in total, out of which 5 were marked ones.

Step 2: Apply the Estimation Formula

To find the total estimated fish population in the pond, use the formula: (Number of marked individuals in first sample * Total number of individuals in second sample) / Number of marked individuals in second sample. Substitute the known values into the formula.

Step 3: Calculation

Substitute the known values into the formula. So, (100 * 18) / 5 = 360. Hence, the estimated total fish population in the pond is 360.

Key Concepts

Concentration Determination

Determining the concentration of a species, such as fish in a pond, involves understanding the relationship between the labeled or marked individuals and the total population. In contexts such as ecological studies or biology, it's crucial to know how to accurately estimate population sizes. Instead of determining the concentration in a chemical sense, we assess the proportion of marked individuals to make educated guesses about the entire group.

This approach, while simple, relies heavily on the assumptions that the marked fish mix uniformly with the rest of the population. It assumes an even distribution, meaning each fish has an equal chance of being caught. The idea is similar to concentration determination in chemical solutions, where labeled molecules help to measure the concentration of a substance.

Estimation Formula

The estimation formula is a crucial part of population studies, helping us estimate the total population from a small sample. This particular scenario uses the formula:

• Estimated Population = (Number of marked individuals in first sample * Total number of individuals in second sample) / Number of marked individuals in second sample

The formula is derived from the concept of proportions. If you know the proportion of marked fish in a small catch, you can use that proportion to estimate the total population. This helps in various fields, from ecology to conservation efforts, where knowing the population size of an organism is vital.

To correctly use the formula, ensure you have a good sample size and that the marked individuals have mixed well with the rest of the population. Otherwise, the estimate might be off.

Marked Individuals Technique

The marked individuals technique involves capturing a number of organisms in the population, marking them in a harmless way, and releasing them back so they can reintegrate with the group.

This method has been used effectively in wildlife studies to estimate animal populations without disturbing them extensively. It's essential to ensure that the marking has no significant effect on the organisms' survival or behavior. In our fish pond example, marking 100 fish helps us keep track of a known portion of the population.

• Advantages include the non-invasive nature of marking and the ability to track movements and behavior of individuals.
• Disadvantages might include the potential for marked individuals to not mix evenly or avoid recapture.

The key to success with this technique is ensuring that the method of marking is durable and visible during firsthand observation.