Problem 63 - A Complete Resource Book in Chemistry for JEE Main

Question

In the final answer of the expression $$ \frac{(29.2-20.2)\left(1.79 \times 10^{5}\right)}{1.37} $$ The number of significant figures is (a) 2 (b) 4 (c) 6 (d) 7

Step-by-Step Solution

Answer

(a) 2 significant figures.

Step 1: Solve the Operations Inside the Parentheses

Calculate \(29.2 - 20.2\). This gives us \(9.0\). Because we are subtracting, the result should have the same number of decimal places as the quantity with the smallest number of decimal places: \(29.2\) has one decimal place, so the answer \(9.0\) has one decimal place.

Step 2: Evaluate the Exponentiation and Multiplication

Evaluate \(1.79 \times 10^5\) and multiply it by \(9.0\). Calculate the multiplication: \(9.0 \times 1.79 = 16.11\). Since \(9.0\) has two significant figures, \(16.11\) should be rounded to two significant figures: \(16.11 \approx 16\). Continue with \(16 \times 10^5 = 1.6 \times 10^6\).

Step 3: Perform the Final Division

Divide \(1.6 \times 10^6\) by \(1.37\). To maintain the significant figures according to the least significant figures rule, find the division: \(\frac{1.6 \times 10^6}{1.37}\). Both \(1.6 ext{ (2 sig. figs)}\) and \(1.37 ext{ (3 sig. figs)}\) lead the result to have 2 significant figures. The final result keeps 2 significant figures.

Step 4: Determine the Significant Figures in the Final Answer

The calculated result of the expression after performing all operations should have 2 significant figures because the limiting factor is the multiplication and division rule, where the smallest number of significant figures in any number of the calculation determines the number of significant figures in the result. In this case, the smallest was 2 from \(9.0\) and \(1.6\).

Key Concepts

Scientific Notation

Scientific notation is a way of expressing numbers that are too large or too small to conveniently write in standard decimal form. This method is especially useful in scientific calculations. It involves expressing numbers as a product of a base (usually 10) raised to an exponent. The format for scientific notation is:

• A number greater than or equal to 1 but less than 10, followed by
• A multiplication sign, \( \times \), and
• 10 raised to an exponent.

For example, the scientific notation for 179,000 is \(1.79 \times 10^5\). Here, \(1.79\) is the significant portion, and \(10^5\) indicates the number's size or magnitude on the scale of powers of ten.
Using scientific notation simplifies arithmetic operations like multiplication and division, especially when involving very large or very small numbers.

Arithmetic Operations

Arithmetic operations are basic mathematical operations which include addition, subtraction, multiplication, and division. When performing these operations with numbers in scientific notation, it is crucial to follow specific steps to ensure accuracy.

• Multiplication: To multiply numbers in scientific notation, first multiply the significant figures, and then add the exponents together. For instance, multiplying \(2 \times 10^3\) by \(3 \times 10^4\) results in \((2 \times 3) \times 10^{3+4} = 6 \times 10^7\).
• Division: For division, divide the significant figures, and then subtract the exponent of the divisor from the exponent of the dividend. For example, dividing \(8 \times 10^5\) by \(2 \times 10^3\) yields \((8 \div 2) \times 10^{5-3} = 4 \times 10^2\).

Understanding and applying these operations correctly helps in solving expressions systematically, such as the one in the exercise.

Significant Figures Rules

Significant figures are the digits in a number that carry meaning towards its precision. The rules of significant figures help in maintaining uniformity and precision in mathematical calculations and scientific measurements.

• Non-zero digits are always significant: E.g., 123 has three significant figures.
• Any zeros between significant digits are significant: E.g., 1003 has four significant figures.
• Leading zeros are not significant: E.g., 0.0025 has two significant figures.
• Trailing zeros in a decimal number are significant: E.g., 2.50 has three significant figures.

In arithmetic operations:

• For addition/subtraction: The result should have the same number of decimal places as the measurement with the fewest decimal places.
• For multiplication/division: The number of significant figures in the result is determined by the original number with the fewest significant figures.

These rules ensure that the final result of any calculation is represented with the appropriate level of precision. In the given exercise, these rules determine that the final answer should be expressed with two significant figures, consistent with the least precise measurement used in the calculation.