Chapter 1, Problem 28QAP

Perform the indicated calculations. Write your answers with the correct number of significant figures.

(a) $\frac{13.01\text{g}}{1.23}$(b) $17.2\text{lb}+65.18\text{lb}-2.3\text{lb}$

(c) $\left(0.003240\text{cm}\right)\left(1.25\text{cm}\right)\left(1.0\times {10}^2\text{cm}\right)$

(d) $\frac{0.024+\sqrt{{\left(0.024\right)}^2+4\left(6.3\times {10}^{-3}\right)}}{2}$Where 4 and 2 are exact numbers, and all other quantities are measurements.

(e) $\frac{\left(4.63\times {10}^4\right)\left(1.8\times {10}^{-6}\right)}{\left(6.022\times {10}^{23}\right)}$Assume all quantities are measurements.

Interpretation Introduction

(a)

Interpretation:

The following calculation should be done up to correct significant numbers.

$\frac{\text{13}\text{.01 g}}{\text{1}\text{.23 L}}$

Concept introduction:

The significant figures are the numbers which are used to describe a measured quantity. They are calculated with the help of following rules:

1. All of the non-zero digits will be significant.
2. Any zero which is placed between two non-zero digits will be significant.
3. Trailing zeros or if the zero is final, it will be significant only if it is after the decimal.
4. All the zeroes before a non-zero digit will be non-significant.

The exact number is a quantity which is obtained by definition or by counting. The measured number is a quantity which is measured through the measuring device. The measured quantities have some uncertainty and so they can have ambiguity.

Answer to Problem 28QAP

The correct calculation is 10.6 g/L.

Explanation of Solution

Rules for significant figures:

1. For addition and subtraction, the calculation results should have same decimal places as found in the number with least significant digits in the problem.
2. For division and multiplication, the calculation results should have significant numbers as found in the number with least significant digits in the problem.

As per rule 2 of significant figures, the number with least significant number will be used to decide the significant digits of result.

Firstly, looking at all the two numbers, the least decimal place is up to three (1.23 L).

Hence, the calculation result will be made up to correct significant digits as follows:

$\frac{\text{13}\text{.01 g}}{\text{1}\text{.23 L}}\text{=10}\text{.577 g/L }\simeq \text{ 10}\text{.6 g/L}$

The result has 3 significant digits.

The calculation result will be 10.6 g/L.

Interpretation Introduction

(b)

Interpretation:

The following calculation should be done up to correct significant numbers.

$\text{17}\text{.2 lb + 65}\text{.18 lb - 2}\text{.3 lb}$

Concept introduction:

The significant figures are the numbers which are used to describe a measured quantity. They are calculated with the help of following rules:

1. All of the non-zero digits will be significant.
2. Any zero which is placed between two non-zero digits will be significant.
3. Trailing zeros or if the zero is final, it will be significant only if it is after the decimal.
4. All the zeroes before a non-zero digit will be non-significant.

The exact number is a quantity which is obtained by definition or by counting. The measured number is a quantity which is measured through the measuring device. The measured quantities have some uncertainty and so they can have ambiguity.

Answer to Problem 28QAP

The correct calculation is $\text{80}\text{.1 lb}$.

Explanation of Solution

Rules for significant figures:

1. For addition and subtraction, the calculation results should have same decimal places as found in the number with least significant digits in the problem.
2. For division and multiplication, the calculation results should have significant numbers as found in the number with least significant digits in the problem.

As per significant rule 1, the number with least decimal places will be used to decide the significant digits of result.

Firstly, looking at all the three numbers, the least decimal places is 1.

Hence the calculation result will be made up to correct significant digits with 1 decimal place as follows:

$\text{17}\text{.2lb + 65}\text{.18 lb - 2}\text{.3 lb = 80}\text{.08 }\simeq \text{ 80}\text{.1 lb}$

The result has 3 significant digits.

The calculation result will be $\text{80}\text{.1 lb}$.

Interpretation Introduction

(c)

Interpretation:

The following calculation should be done up to correct significant numbers.

$\text{(0}\text{.003240 cm)(1}\text{.25 cm)(1}{\text{.0×10}}^{\text{2}}\text{ cm)}$

Concept introduction:

The significant figures are the numbers which are used to describe a measured quantity. They are calculated with the help of following rules:

1. All of the non-zero digits will be significant.
2. Any zero which is placed between two non-zero digits will be significant.
3. Trailing zeros or if the zero is final, it will be significant only if it is after decimal.
4. All the zeroes before a non-zero digit will be non-significant.

The exact number is a quantity which is obtained by definition or by counting. The measured number is a quantity which is measured through the measuring device. The measured quantities have some uncertainty and so they can have ambiguity.

Answer to Problem 28QAP

The correct calculation is $4.0\times {10}^{-1}c{m}^3$.

Explanation of Solution

Rules for significant figures:

1. For addition and subtraction, the calculation results should have same decimal places as found in the number with least significant digits in the problem.
2. For division and multiplication, the calculation results should have significant numbers as found in the number with least significant digits in the problem.

As per significant rule 2, the number with least significant digit will be used to decide the significant digits of result.

Firstly, looking at all the three numbers, the least significant digit is 2 for (1.0×10²cm).

Hence, the calculation result will be made up to correct significant digits of 5 as follows:

$\text{(0}\text{.003240 cm)(1}\text{.25 cm)(1}{\text{.0×10}}^{\text{2}}\text{ cm)=0}{\text{.00405×10}}^{\text{2}}\text{ cm}$

The result has 3 significant digits. Converting in scientific notation will give the result up to two significant digits as:

$\text{0}{\text{.00405×10}}^{\text{2}}{\text{ cm}}^{\text{3}}\simeq \text{4}{\text{.0×10}}^{\text{-1}}{\text{ cm}}^{\text{3}}$

The calculation result will be $\text{4}{\text{.0×10}}^{\text{-1}}{\text{ cm}}^{\text{3}}$.

Interpretation Introduction

(d)

Interpretation:

The following calculation should be done up to correct significant numbers.

$\frac{0.024+\sqrt{{(0.024)}^2+4(6.3\times {10}^{-3})}}{2}$

Where 4 and 2 are exact numbers and all other quantities are measurements.

Concept introduction:

The significant figures are the numbers which are used to describe a measured quantity. They are calculated with the help of following rules:

1. All of the non-zero digits will be significant.
2. Any zero which is placed between two non-zero digits will be significant.
3. Trailing zeros or if the zero is final, it will be significant only if it is after the decimal.
4. All the zeroes before a non-zero digit will be non-significant.

The exact number is a quantity which is obtained by definition or by counting. The measured number is a quantity which is measured through the measuring device. The measured quantities have some uncertainty and so they can have ambiguity.

Answer to Problem 28QAP

The calculation result will be $0.09$.

Explanation of Solution

Rules for significant figures:

1. For addition and subtraction, the calculation results should have same decimal places as found in the number with least significant digits in the problem.
2. For division and multiplication, the calculation results should have significant numbers as found in the number with least significant digits in the problem.

As per significant rule 1; the number with least decimal place will be used to decide the significant digits of result.

As per significant rule 2; the number with least significant digit will be used to decide the significant digit of result.

First of all, the equation should be solved using BODMAS rule simultaneously with significant rule as follows:

Here, 4 and 2 are exact numbers which can have infinite significant digits so, they will not be counted in least significant digit number.

The under root terms are multiplied and added as per round off rule (2) and (1) as:

$\begin{array}{l}=\frac{0.024+\sqrt{{(0.024)}^2+4(6.3\times {10}^{-3})}}{2}\\ =\frac{0.024+\sqrt{0.00057+0.025}}{2}\\ =\frac{0.024+\sqrt{0.026}}{2}\\ =\frac{0.024+0.16}{2}\\ =\frac{0.18}{2}\\ =0.09\end{array}$

The calculation result will be $0.09$.

Interpretation Introduction

(e)

Interpretation:

The following calculation should be done up to correct significant numbers.

$\frac{(4.63\times {10}^4)(1.8\times {10}^{-6})}{(6.022\times {10}^{23})}$

Concept introduction:

The significant figures are the numbers which are used to describe a measured quantity. They are calculated with the help of following rules:

1. All of the non-zero digits will be significant.
2. Any zero which is placed between two non-zero digits will be significant.
3. Trailing zeros or if the zero is final, it will be significant only if it is after the decimal.
4. All the zeroes before a non-zero digit will be non-significant.

The exact number is a quantity which is obtained by definition or by counting. The measured number is a quantity which is measured through the measuring device. The measured quantities have some uncertainty and so they can have ambiguity.

Answer to Problem 28QAP

The correct calculation is $1.4\times {10}^{-25}$.

Explanation of Solution

Rules for significant figures:

1. For addition and subtraction, the calculation results should have same decimal places as found in the number with least significant digits in the problem.
2. For division and multiplication, the calculation results should have significant numbers as found in the number with least significant digits in the problem.

As per significant rule 2; the number with least significant digits will be used to decide the significant digits of result.

First of all, looking at all the three numbers, the least significant digit is up to two (1.8×10-6).

Hence the calculation result will be made up to correct significant digits as follows:

$\frac{(4.63\times {10}^4)(1.8\times {10}^{-6})}{(6.022\times {10}^{23})}=\frac{8.4\times {10}^{-2}}{6.022\times {10}^{23}}=1.4\times {10}^{-25}$

The result has two significant digits.

Hence, the calculation result will be $1.4\times {10}^{-25}$.