Question

Carry out these operations as if they were calculations of experimental results, and express each answer in the correct units and with the correct number of significant figures: (a) \(5.6792 \mathrm{~m}+0.6 \mathrm{~m}+4.33 \mathrm{~m}\) (b) \(3.70 \mathrm{~g}-2.9133 \mathrm{~g}\) (c) \(4.51 \mathrm{~cm} \times 3.6666 \mathrm{~cm}\) (d) \(\left(3 \times 10^{4} \mathrm{~g}+6.827 \mathrm{~g}\right) /\left(0.043 \mathrm{~cm}^{3}-0.021 \mathrm{~cm}^{3}\right)\)

Short answer

Solutions: (a) 10.6 m, (b) 0.79 g, (c) 16.53 cm², (d) 1.6 × 10^6 g/cm³ (to 2 significant figures)

Step-by-step solution

Addition of Measurements

Add the numbers given in exercise (a) which are 5.6792 m, 0.6 m, and 4.33 m. Note that the result must not have more decimal places than the number with the least decimal places, which is 0.6 m (with 1 decimal place).

Subtraction of Measurements

Subtract the numbers provided in exercise (b) which are 3.70 g and 2.9133 g. Note that the result must not have more decimal places than the number with the least decimal places, which is 3.70 g (with 2 decimal places).

Multiplication of Measurements

Multiply the numbers given in exercise (c) which are 4.51 cm and 3.6666 cm. Note that the result must not have more significant figures than the number with the least significant figures, which is 4.51 cm (with 3 significant figures).

Combined Operation of Measurements

Perform the addition of 3 × 10^4 g and 6.827 g, then subtract 0.043 cm³ and 0.021 cm³ as provided in exercise (d). Finally, divide the result of the addition by the result of the subtraction. For the subtraction operation, note that the result must not have more decimal places than the number with the least decimal places, which is 0.043 cm³ (with 2 decimal places). Also, the result of the final division should not have more significant figures than the input with the least number of significant figures.

Key concepts

Experimental Calculations

Experimental calculations are a crucial part of scientific research and experimentation. They involve mathematical operations on measured data. The primary goal here is to ensure that results represent reality as closely as possible by factoring in significant figures. - **Significant Figures**: These are the digits in a measurement that contribute to its accuracy. When performing operations like addition, subtraction, multiplication, or division, it's important to determine how many significant figures the final result should have. - **Precision and Accuracy**: It's important to maintain precision, which is the consistency of repeated measurements and accuracy, which reflects how close a measurement is to the true value. In addition to knowing how calculations are carried out, it's important to report the answers with the correct units and appropriate significant figures. This helps in accurately representing the level of precision available in your measurement process.

Unit Conversion

Unit conversion is often necessary in experimental calculations to ensure consistency and accuracy. Different units are used in different branches of science, and converting them accurately is crucial for a correct interpretation of results. - **Common Units in Science**: Length is often measured in meters, mass in grams, and time in seconds. Sometimes it might be necessary to convert these to centimeters, kilograms, or other units, depending on the dataset you are working with. - **Conversion Factors**: These are used to switch from one unit to another. For example, 1 meter equals 100 centimeters. To maintain the correct number of significant figures throughout the conversion process, remember that when multiplying or dividing measurements by a conversion factor, the number of significant figures should be consistent with the measurement having the smallest number of significant figures.

Scientific Notation

Scientific notation is a way of expressing numbers that are too large or small to be conveniently written in decimal form. This is particularly useful in scientific calculations where extreme values are frequently encountered.- **Structure of Scientific Notation**: A number is written as \( a \times 10^{n} \), where \( a \) is a number greater than or equal to 1 and less than 10, and \( n \) is an integer. For example, 0.00012 can be represented as \( 1.2 \times 10^{-4} \).- **Advantages**: Using scientific notation helps to simplify equations and calculations by reducing the number of zeros or decimal places, which can reduce errors and make computations more manageable. In operations involving scientific notation, significant figures play a crucial role. The number part of scientific notation shows the number of significant figures, making it easier to handle while ensuring accuracy in calculations.