Which of the following statements use exact numbers? (a) The little girl weighs \(74.3\) lb. (b) The boiling point of ethyl alcohol is \(78^{\circ} \mathrm{C}\). (c) The square root of 144 is 12 .

Scientific constants are fundamental numbers in chemistry that have precise values. These numbers are essential in executing calculations and understanding the principles of chemistry. For instance, the boiling point of a substance such as ethyl alcohol, is a specific value that doesn't change under standard conditions. In our exercise, the boiling point of ethyl alcohol is mentioned as \(78^{\text{o}}C\). This is a scientific constant because it represents a well-defined and unchanging value that can be reliably used in calculations and chemical understanding.

It's important for students to recognize these constants as they not only aid in problem-solving but also serve as benchmarks for identifying abnormal conditions in experiments. For example, if ethyl alcohol were to boil at a different temperature under standard conditions, it could indicate the presence of impurities or measurement errors. Identifying scientific constants as exact numbers can foster precision and certainty in both laboratory and theoretical work.

Measurement and approximation play a critical role in chemistry. Most values we deal with in the laboratory are approximations to some degree, owing to the limitations of our measurement tools. When a little girl's weight is logged as \(74.3\) lb, this number represents an approximation. Even though sophisticated equipment can measure quite accurately, there's always a degree of uncertainty associated with every measurement.

The key to understanding measurement in chemistry lies in recognizing the significance of significant figures and the precision of the instruments used. When a number is expressed with a decimal point, as in the case of the weight measurement, it indicates that the value has been determined to a certain degree of precision, but it is not an absolute exact number. In contrast, counting objects or using defined mathematical operations can result in exact numbers, as these are not subject to the variability of measurement processes.

Mathematical operations in chemistry are foundational for solving problems and understanding chemical relationships. In the given exercise, the square root of \(144\) is used as an example of an exact number in a mathematical operation. When a value is derived from a precise mathematical operation, such as calculating the square root of a perfect square, the result is an exact number. This differs from measured numbers which can have varying levels of precision due to limitations inherent in measurement techniques.

Chemists not only perform basic mathematical operations but also employ more complex calculations involving logarithms, exponents, and calculus as they model reactions, calculate concentrations, and predict behavior of substances. Mastery over these mathematical tools enables chemists to craft accurate solutions and descriptions for a myriad of chemical phenomena. Understanding when and why some numbers are exact in these operations can help students minimize errors and make more accurate predictions in their scientific endeavors.