Chapter 1, Problem 1.59P

(a)

Interpretation Introduction

Interpretation:

Average density of lead that is determined by each group has to be calculated and also the percentage error in the average density of each group has to be calculated.

Concept Introduction:

Accuracy can be measured using percentage error.  Percentage error is the difference between the average experimental values and/or true value, divided by true value, multiplied by $100$.  This can be represented in form of equation as shown below.

    $\text{Percentage}\text{error}=\frac{\text{average}\text{value}-\text{true}\text{value}}{\text{true}\text{value}}\times 100$

Explanation of Solution

Average density and percentage error obtained by Group 1:

Three density values of lead is obtained by Group 1 and they are $12.7\text{g}\cdot {\text{cm}}^{-3}$, $11.2\text{g}\cdot {\text{cm}}^{-3}$, and $10.3\text{g}\cdot {\text{cm}}^{-3}$.  Average density can be calculated as shown below.

    $\begin{array}{c}\text{Average}\text{density}=\frac{(12.7+11.2+10.3)\text{g}\cdot {\text{cm}}^{-3}}{3}\\ =\frac{34.2\text{g}\cdot {\text{cm}}^{-3}}{3}\\ =11.4\text{g}\cdot {\text{cm}}^{-3}\end{array}$

Therefore, average density of lead obtained by Group 1 is $11.4\text{g}\cdot {\text{cm}}^{-3}$.

True value of density of lead is $11.3\text{g}\cdot {\text{cm}}^{-3}$.  Average value is $11.4\text{g}\cdot {\text{cm}}^{-3}$.  Percentage error can be calculated as shown below.

    $\begin{array}{c}\text{Percentage}\text{error}=\frac{\text{average}\text{value}-\text{true}\text{value}}{\text{true}\text{value}}\times 100\\ =\frac{11.4\text{g}\cdot {\text{cm}}^{-3}-11.3\text{g}\cdot {\text{cm}}^{-3}}{11.3\text{g}\cdot {\text{cm}}^{-3}}\times 100\\ =\frac{0.1\text{g}\cdot {\text{cm}}^{-3}}{11.3\text{g}\cdot {\text{cm}}^{-3}}\times 100\\ =0.88\%\end{array}$

Percentage error obtained by Group 1 is $0.88\%$.

Average density and percentage error obtained by Group 2:

Three density values of lead is obtained by Group 2 and they are $11.5\text{g}\cdot {\text{cm}}^{-3}$, $11.4\text{g}\cdot {\text{cm}}^{-3}$, and $11.4\text{g}\cdot {\text{cm}}^{-3}$.  Average density can be calculated as shown below.

    $\begin{array}{c}\text{Average}\text{density}=\frac{(11.5+11.4+11.4)\text{g}\cdot {\text{cm}}^{-3}}{3}\\ =\frac{34.3\text{g}\cdot {\text{cm}}^{-3}}{3}\\ =11.4\text{g}\cdot {\text{cm}}^{-3}\end{array}$

Therefore, average density of lead obtained by Group 2 is $11.4\text{g}\cdot {\text{cm}}^{-3}$.

True value of density of lead is $11.3\text{g}\cdot {\text{cm}}^{-3}$.  Average value is $11.4\text{g}\cdot {\text{cm}}^{-3}$.  Percentage error can be calculated as shown below.

    $\begin{array}{c}\text{Percentage}\text{error}=\frac{\text{average}\text{value}-\text{true}\text{value}}{\text{true}\text{value}}\times 100\\ =\frac{11.4\text{g}\cdot {\text{cm}}^{-3}-11.3\text{g}\cdot {\text{cm}}^{-3}}{11.3\text{g}\cdot {\text{cm}}^{-3}}\times 100\\ =\frac{0.1\text{g}\cdot {\text{cm}}^{-3}}{11.3\text{g}\cdot {\text{cm}}^{-3}}\times 100\\ =0.88\%\end{array}$

Percentage error obtained by Group 2 is $0.88\%$.

Average density and percentage error obtained by Group 3:

Three density values of lead is obtained by Group 3 and they are $10.9\text{g}\cdot {\text{cm}}^{-3}$, $11.3\text{g}\cdot {\text{cm}}^{-3}$, and $11.1\text{g}\cdot {\text{cm}}^{-3}$.  Average density can be calculated as shown below.

    $\begin{array}{c}\text{Average}\text{density}=\frac{(10.9+11.3+11.1)\text{g}\cdot {\text{cm}}^{-3}}{3}\\ =\frac{33.3\text{g}\cdot {\text{cm}}^{-3}}{3}\\ =11.1\text{g}\cdot {\text{cm}}^{-3}\end{array}$

Therefore, average density of lead obtained by Group 3 is $11.1\text{g}\cdot {\text{cm}}^{-3}$.

True value of density of lead is $11.3\text{g}\cdot {\text{cm}}^{-3}$.  Average value is $11.1\text{g}\cdot {\text{cm}}^{-3}$.  Percentage error can be calculated as shown below.

    $\begin{array}{c}\text{Percentage}\text{error}=\frac{\text{average}\text{value}-\text{true}\text{value}}{\text{true}\text{value}}\times 100\\ =\frac{11.1\text{g}\cdot {\text{cm}}^{-3}-11.3\text{g}\cdot {\text{cm}}^{-3}}{11.3\text{g}\cdot {\text{cm}}^{-3}}\times 100\\ =\frac{-0.2\text{g}\cdot {\text{cm}}^{-3}}{11.3\text{g}\cdot {\text{cm}}^{-3}}\times 100\\ =-1.77\%\end{array}$

Percentage error obtained by Group 3 is $-1.77\%$.

(b)

Interpretation Introduction

Interpretation:

The most accurate data among the Group’s data has to be identified.

Concept Introduction:

Precision of the quantities that are measured use the significant figures to express the result.  Precision tells about the closeness of values to each other in a measurement.

Accuracy is a state that tells about the result obtained from experiment corresponds to the actual value or true value.

Explanation of Solution

Density of lead is $11.3\text{g}\cdot {\text{cm}}^{-3}$.  Most accurate data can be found by seeing the values that are close to $11.3\text{g}\cdot {\text{cm}}^{-3}$.  Average density of Group 1, Group 2, and Group 3 are $11.4\text{g}\cdot {\text{cm}}^{-3}$, $11.4\text{g}\cdot {\text{cm}}^{-3}$, and $11.1\text{g}\cdot {\text{cm}}^{-3}$ respectively.  Therefore, the most accurate data is found from Group 1 and Group 2.

(c)

Interpretation Introduction

Interpretation:

The most precise data among the Group’s data has to be identified.

Concept Introduction:

Refer part (b).

Explanation of Solution

Density of lead is $11.3\text{g}\cdot {\text{cm}}^{-3}$.  Most precise data can be found by seeing the values that are close to each other in each group.  The values in Group 2 is found to be very close to each other.  Hence, this is the most precise data.