Question

Without actually performing the calculations indicated, tell to how many significant digits the answer to the calculation should be expressed. a. \((10.12+17.381+18.2) /(1.41)\) b. (0.173)\(\left(6.022 \times 10^{23}\right)\) c. \((2.51)(0.08202)(298) /(765.2)\) d. \(\frac{(9.77732)(47.112)}{(273.24)(0.2)}\)

Short answer

The results should have the following significant digits: a. 2 significant digits b. 3 significant digits c. 3 significant digits d. 1 significant digit

Step-by-step solution

Determine decimal places for addition/subtraction

In the numerator, we have three numbers being added: 10.12 (2 decimal places), 17.381 (3 decimal places), and 18.2 (1 decimal place). The minimum number of decimal places is 1.

Determine significant digits for the division

The numbers in the numerator have 1 decimal place, and the number in the denominator (1.41) has 2 decimal places. We have to choose the least number of significant digits, which is 2 (from the denominator 1.41). The result for a. should have 2 significant digits. #b. (0.173)\(\left(6.022 \times 10^{23}\right)\)#

Determine significant digits for the multiplication

The two numbers being multiplied are 0.173 (3 significant digits) and 6.022 x 10^23 (4 significant digits). The minimum number of significant digits is 3. The result for b. should have 3 significant digits. #c. \((2.51)(0.08202)(298) /(765.2)\)#

Determine significant digits for the multiplication

The three numbers being multiplied are 2.51 (3 significant digits), 0.08202 (5 significant digits), and 298 (3 significant digits). The minimum number of significant digits is 3.

Determine significant digits for the division

The result from the multiplication has 3 significant digits, and the number in the denominator (765.2) has 4 significant digits. The least number of significant digits is 3. The result for c. should have 3 significant digits. #d. \(\frac{(9.77732)(47.112)}{(273.24)(0.2)}\)#

Determine significant digits for the multiplication (numerator)

The two numbers being multiplied in the numerator are 9.77732 (6 significant digits) and 47.112 (5 significant digits). The minimum number of significant digits is 5.

Determine significant digits for the multiplication (denominator)

The two numbers being multiplied in the denominator are 273.24 (5 significant digits) and 0.2 (1 significant digit). The minimum number of significant digits is 1.

Determine significant digits for the division

The numbers from the numerator have 5 significant digits, and the numbers from the denominator have 1 significant digit. The least number of significant digits is 1. The result for d. should have 1 significant digit.

Key concepts

Scientific Notation

Scientific notation is a way to express very large or very small numbers in a more concise and manageable form. Instead of writing out long strings of zeros, scientific notation simplifies these numbers using powers of ten. For example, a large number like 6,022,000,000,000,000,000,000,000 becomes more manageable as \(6.022 \times 10^{23}\). This is particularly useful in chemistry and physics, where you're often dealing with extremely large or small values.

Scientific notation is helpful in maintaining precision across mathematical operations. The digits before the multiplication sign (in the example, '6.022') are called the significand or coefficient, and they determine the number's significant digits. The exponent '23' signifies the scale or size. When performing calculations, it's crucial to keep an eye on the significant digits in both the coefficient and the overall result, especially in context like chemistry where precision is paramount.

Decimal Places

Decimal places are the digits that appear to the right of the decimal point in a number. They play an important role in determining the precision of a measurement. In chemistry and mathematics, maintaining precision means paying attention to the number of decimal places used during calculations.

When performing addition or subtraction, the result should be rounded to the least number of decimal places found in any of the numbers. For example, when adding 10.12 (2 decimal places) and 18.2 (1 decimal place), the result should only have 1 decimal place because that's the least among the numbers.

This ensures that the precision of the calculation is consistent with the least precise measurement you started with. It's crucial not to extend the precision of your result beyond the least precise input, as this could misrepresent the accuracy of your data.

Mathematical Operations in Chemistry

Mathematical operations such as addition, subtraction, multiplication, and division are frequent in chemistry calculations. However, the key to maintaining accuracy and precision is how you handle significant figures during these operations.

For multiplication and division, the result should be expressed with the same number of significant figures as the measurement with the smallest number of significant figures. For example, when multiplying 0.173 (3 significant digits) and \(6.022 \times 10^{23}\) (4 significant digits), the result should have 3 significant digits.

In chemical calculations, always consider the significant figures in each of your measurements and calculations. Disregarding them can lead to inaccuracy in your final result. This can be especially critical when determining exact concentrations or reaction yields, where precision is vital.

Precision in Measurements

Precision in measurements refers to how close repeated measurements are to each other. More precise measurements produce data that is consistently similar, which is important in scientific experiments.

When discussing precision in a numerical context, like when handling significant figures, it's about making sure that your calculations correctly reflect the measured data. Using too many decimal places or significant digits implies a false sense of precision in your results. Consequently, when you perform any math with measurements, the precision limits are dictated by your least precise measurement.

A precise measurement is not necessarily an accurate one, so it's crucial to understand what your precision tells you about the quality of your data. Always aim to align the precision of your result with your input data to maintain the integrity of your work in both educational exercises and real-world applications.