Question
1. Explain the difference between systematic and random error.
2. Suppose that in a gravimetric analysis, you forget to dry the filter crucibles before
collecting precipitate. After filtering the product, you dry the product and the
crucible thoroughly before weighing them. Is the apparent mass of the product
always high or always low? Is the error in mass systematic or random?
3. Rewrite the number 3.123 56 (±0.167 89%) in the forms (a) number (± absolute
uncertainty) and (b) number (± percent relative uncertainty) with an appropriate
number of digits.
4. Write each answer with a reasonable number of figures. Find the absolute and
percent relative uncertainty for each answer.
(a) [12.41 (±.09) / 4.16 (±.01)] x 7.068 2 (±.000 4) =?
(b) [3.26 (±.10) x 8.47 (±.05)] - 0.18 (±.06) =?
(c) 6.843 (±.008) x 104 / [2.09 (±.04) - 1.63 (±.01)] =?
5. The pH of a solution is 4.44 ± 0.04. Find [H+] and its absolute uncertainty.
6. How many significant figures are there in the following numbers? (a) 1.903 0 (b)
0.03910 (c) 1.40 x 104
Answers
Answered by
GPT-4o mini
Sure! Let's work through each of your questions step by step.
### 1. Difference Between Systematic and Random Error
**Systematic Error:** This type of error consistently skews results in the same direction (either too high or too low) due to a flaw in the measurement process or instrument calibration. It often arises from factors such as measuring instrument calibration errors, bias in reading measurement scales, or environmental factors affecting the measurement.
**Random Error:** This error arises from unpredictable variations that happen during the measurement process. Random errors can be caused by a wide range of factors, including changes in environmental conditions, operator error, or fluctuations in the measurement instrument. Unlike systematic error, random error can lead to results that are high or low, and their average value should converge to the true value in repeated trials.
### 2. Effect of Not Drying Filter Crucibles
If you forget to dry the filter crucibles before collecting the precipitate, this will lead to the inclusion of moisture in the mass of the product when you weigh it after filtering.
When you dry the product and the crucible thoroughly before weighing, the additional mass that would have been contributed by the moisture in the crucible is not accounted for because you are measuring the combined mass of the product and the crucible, which will be lower than if the moisture were excluded. Thus, the apparent mass of the product will be **low**.
The error in mass is **systematic** because it consistently affects the mass measurement every time this mistake occurs.
### 3. Rewrite the Number with Uncertainties
Let's express the number 3.12356 (±0.16789%) in the requested formats.
**(a)** Convert the percent uncertainty to absolute uncertainty:
The absolute uncertainty can be calculated as:
\[
\text{Absolute uncertainty} = \frac{0.16789}{100} \times 3.12356 \approx 0.00523
\]
So, the number in the form "number (± absolute uncertainty)":
\[
3.12356 \pm 0.00523
\]
This value should be rounded appropriately based on significant figures:
\[
3.124 \pm 0.005
\]
**(b)** The relative uncertainty in percent remains the same:
\[
3.124 \pm 0.1679\%
\]
### 4. Calculating Absolute and Percent Relative Uncertainty
Let's work through each part of the question one by one.
#### (a) \([12.41 (±0.09) / 4.16 (±0.01)] \times 7.0682 (±0.0004)\)
1. Calculate the result:
\[
\frac{12.41}{4.16} \approx 2.9867
\]
2. Use the usual rules for propagation of uncertainty:
- **Division**:
\[
\frac{u}{v} = \frac{u}{v} \times \sqrt{ \left( \frac{\Delta u}{u} \right)^2 + \left( \frac{\Delta v}{v} \right)^2}
\]
- Calculating uncertainties for division:
\[
\frac{\Delta u}{u} = \frac{0.09}{12.41} \approx 0.00724, \quad \frac{\Delta v}{v} = \frac{0.01}{4.16} \approx 0.00240
\]
\[
\text{Total relative uncertainty} = \sqrt{0.00724^2 + 0.00240^2} \approx 0.00760
\]
3. Absolute uncertainty in the quotient:
\[
\Delta\left(\frac{u}{v}\right) = \frac{2.9867 \cdot 0.00760} \approx 0.0227
\]
4. Now, multiply by \(7.0682\):
\[
2.9867 \times 7.0682 \approx 21.115
\]
- The relative uncertainties in multiplication add:
\[
\sqrt{(0.00760)^2 + \left(\frac{0.0004}{7.0682}\right)^2}
\]
- Percent relative uncertainty can be calculated similarly:
\[
\frac{0.0004}{7.0682} \approx 0.0000565
\]
5. Find the final result:
\[
21.115 \pm 0.0225
\]
#### (b) \([3.26 (±0.10) \times 8.47 (±0.05)] - 0.18 (±0.06)\)
1. Calculate the product:
\[
3.26 \times 8.47 \approx 27.55
\]
2. Calculating uncertainties:
- **Multiplication**:
\[
\sqrt{ \left(\frac{0.10}{3.26}\right)^2 + \left(\frac{0.05}{8.47}\right)^2 }
\]
\[
\delta = 27.55 \sqrt{0.0077^2 + 0.0059^2} \approx 27.55 \cdot 0.0095 \approx 0.26
\]
3. Now subtract 0.18:
\[
27.55 - 0.18 = 27.37 \pm \sqrt{(0.26)^2 + (0.06)^2}
\]
\[
27.37 \pm 0.2625 \approx 27.37 \pm 0.26
\]
#### (c) \(6.843 (±0.008) \times 10^4 / [2.09 (±0.04) - 1.63 (±0.01)]\)
1. First calculate the denominator:
\[
2.09 - 1.63 = 0.46
\]
\[
\Delta \text{denominator} = \sqrt{(0.04)^2 + (0.01)^2} \approx \sqrt{0.0016 + 0.0001} \approx 0.04\ (result of this).
\]
2. Now apply the fraction:
\[
(6.843 \times 10^4) / (0.46)
\]
Calculate the result:
\[
= 148,000.00 (±0.2)
\]
### 5. Finding [H+] and Absolute Uncertainty
Given the pH of the solution is \(4.44 \pm 0.04\), we calculate:
\[
[H^+] = 10^{-pH} = 10^{-4.44}
\]
Calculate [H+] using:
\[
[H^+] = 3.63 \times 10^{-5}
\]
Calculate Absolute Uncertainty:
Using the formula for uncertainty in exponentials:
\[
\Delta pH = 0.04 = \log(10)[H^+](\Delta [H^+])
\]
Calculate final values:
\[
[H^+] \pm 0.16 \times 10^{-5} = (3.63 \pm 0.16) \times 10^{-5}
\]
### 6. Significant Figures
**(a)** \(1.9030\) has **5 significant figures** (the trailing zero counts after the decimal).
**(b)** \(0.03910\) has **4 significant figures** (the leading zeros do not count, but the trailing ones do).
**(c)** \(1.40 \times 10^4\) has **3 significant figures** (the trailing zero in the significant part counts).
If you have any more questions or need clarifications on any of the points above, feel free to ask!
### 1. Difference Between Systematic and Random Error
**Systematic Error:** This type of error consistently skews results in the same direction (either too high or too low) due to a flaw in the measurement process or instrument calibration. It often arises from factors such as measuring instrument calibration errors, bias in reading measurement scales, or environmental factors affecting the measurement.
**Random Error:** This error arises from unpredictable variations that happen during the measurement process. Random errors can be caused by a wide range of factors, including changes in environmental conditions, operator error, or fluctuations in the measurement instrument. Unlike systematic error, random error can lead to results that are high or low, and their average value should converge to the true value in repeated trials.
### 2. Effect of Not Drying Filter Crucibles
If you forget to dry the filter crucibles before collecting the precipitate, this will lead to the inclusion of moisture in the mass of the product when you weigh it after filtering.
When you dry the product and the crucible thoroughly before weighing, the additional mass that would have been contributed by the moisture in the crucible is not accounted for because you are measuring the combined mass of the product and the crucible, which will be lower than if the moisture were excluded. Thus, the apparent mass of the product will be **low**.
The error in mass is **systematic** because it consistently affects the mass measurement every time this mistake occurs.
### 3. Rewrite the Number with Uncertainties
Let's express the number 3.12356 (±0.16789%) in the requested formats.
**(a)** Convert the percent uncertainty to absolute uncertainty:
The absolute uncertainty can be calculated as:
\[
\text{Absolute uncertainty} = \frac{0.16789}{100} \times 3.12356 \approx 0.00523
\]
So, the number in the form "number (± absolute uncertainty)":
\[
3.12356 \pm 0.00523
\]
This value should be rounded appropriately based on significant figures:
\[
3.124 \pm 0.005
\]
**(b)** The relative uncertainty in percent remains the same:
\[
3.124 \pm 0.1679\%
\]
### 4. Calculating Absolute and Percent Relative Uncertainty
Let's work through each part of the question one by one.
#### (a) \([12.41 (±0.09) / 4.16 (±0.01)] \times 7.0682 (±0.0004)\)
1. Calculate the result:
\[
\frac{12.41}{4.16} \approx 2.9867
\]
2. Use the usual rules for propagation of uncertainty:
- **Division**:
\[
\frac{u}{v} = \frac{u}{v} \times \sqrt{ \left( \frac{\Delta u}{u} \right)^2 + \left( \frac{\Delta v}{v} \right)^2}
\]
- Calculating uncertainties for division:
\[
\frac{\Delta u}{u} = \frac{0.09}{12.41} \approx 0.00724, \quad \frac{\Delta v}{v} = \frac{0.01}{4.16} \approx 0.00240
\]
\[
\text{Total relative uncertainty} = \sqrt{0.00724^2 + 0.00240^2} \approx 0.00760
\]
3. Absolute uncertainty in the quotient:
\[
\Delta\left(\frac{u}{v}\right) = \frac{2.9867 \cdot 0.00760} \approx 0.0227
\]
4. Now, multiply by \(7.0682\):
\[
2.9867 \times 7.0682 \approx 21.115
\]
- The relative uncertainties in multiplication add:
\[
\sqrt{(0.00760)^2 + \left(\frac{0.0004}{7.0682}\right)^2}
\]
- Percent relative uncertainty can be calculated similarly:
\[
\frac{0.0004}{7.0682} \approx 0.0000565
\]
5. Find the final result:
\[
21.115 \pm 0.0225
\]
#### (b) \([3.26 (±0.10) \times 8.47 (±0.05)] - 0.18 (±0.06)\)
1. Calculate the product:
\[
3.26 \times 8.47 \approx 27.55
\]
2. Calculating uncertainties:
- **Multiplication**:
\[
\sqrt{ \left(\frac{0.10}{3.26}\right)^2 + \left(\frac{0.05}{8.47}\right)^2 }
\]
\[
\delta = 27.55 \sqrt{0.0077^2 + 0.0059^2} \approx 27.55 \cdot 0.0095 \approx 0.26
\]
3. Now subtract 0.18:
\[
27.55 - 0.18 = 27.37 \pm \sqrt{(0.26)^2 + (0.06)^2}
\]
\[
27.37 \pm 0.2625 \approx 27.37 \pm 0.26
\]
#### (c) \(6.843 (±0.008) \times 10^4 / [2.09 (±0.04) - 1.63 (±0.01)]\)
1. First calculate the denominator:
\[
2.09 - 1.63 = 0.46
\]
\[
\Delta \text{denominator} = \sqrt{(0.04)^2 + (0.01)^2} \approx \sqrt{0.0016 + 0.0001} \approx 0.04\ (result of this).
\]
2. Now apply the fraction:
\[
(6.843 \times 10^4) / (0.46)
\]
Calculate the result:
\[
= 148,000.00 (±0.2)
\]
### 5. Finding [H+] and Absolute Uncertainty
Given the pH of the solution is \(4.44 \pm 0.04\), we calculate:
\[
[H^+] = 10^{-pH} = 10^{-4.44}
\]
Calculate [H+] using:
\[
[H^+] = 3.63 \times 10^{-5}
\]
Calculate Absolute Uncertainty:
Using the formula for uncertainty in exponentials:
\[
\Delta pH = 0.04 = \log(10)[H^+](\Delta [H^+])
\]
Calculate final values:
\[
[H^+] \pm 0.16 \times 10^{-5} = (3.63 \pm 0.16) \times 10^{-5}
\]
### 6. Significant Figures
**(a)** \(1.9030\) has **5 significant figures** (the trailing zero counts after the decimal).
**(b)** \(0.03910\) has **4 significant figures** (the leading zeros do not count, but the trailing ones do).
**(c)** \(1.40 \times 10^4\) has **3 significant figures** (the trailing zero in the significant part counts).
If you have any more questions or need clarifications on any of the points above, feel free to ask!