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Let's work through the questions step-by-step.

### Question 9
#### Step 1: Define Terms
- **IDF (Individual Dilution Factor)**: This is the dilution factor for each individual step in the serial dilution.
- **TDF (Total Dilution Factor)**: This is the cumulative dilution factor from the original solution to each step in the sequence.

#### Step 2: Calculate IDFs and TDFs for Each Step

1. **Dilution from Original to "a"**:
   - **Volume taken**: 1.0 mL from the original solution.
   - **Diluent added**: 99 mL.
   - **Final volume**: \( 1.0 \, \text{mL} + 99 \, \text{mL} = 100 \, \text{mL} \).

   **IDF for "a"**:
   \[
   \text{IDF}_a = \frac{\text{Volume of sample}}{\text{Total volume}} = \frac{1.0}{100} = 10^{-2}
   \]

   **TDF for "a"** (since it's the first dilution, TDF = IDF):
   \[
   \text{TDF}_a = 10^{-2}
   \]

2. **Dilution from "a" to "b"**:
   - **Volume taken**: 10 mL from solution "a".
   - **Diluent added**: 990 mL.
   - **Final volume**: \( 10 \, \text{mL} + 990 \, \text{mL} = 1000 \, \text{mL} \).

   **IDF for "b"**:
   \[
   \text{IDF}_b = \frac{10}{1000} = 10^{-2}
   \]

   **TDF for "b"** (cumulative from original to "b"):
   \[
   \text{TDF}_b = \text{TDF}_a \times \text{IDF}_b = 10^{-2} \times 10^{-2} = 10^{-4}
   \]

3. **Dilution from "b" to "c"**:
   - **Volume taken**: 1.0 mL from solution "b".
   - **Diluent added**: 99 mL.
   - **Final volume**: \( 1.0 \, \text{mL} + 99 \, \text{mL} = 100 \, \text{mL} \).

   **IDF for "c"**:
   \[
   \text{IDF}_c = \frac{1.0}{100} = 10^{-2}
   \]

   **TDF for "c"** (cumulative from original to "c"):
   \[
   \text{TDF}_c = \text{TDF}_b \times \text{IDF}_c = 10^{-4} \times 10^{-2} = 10^{-6}
   \]

#### Summary of Answers for Question 9:
- **IDF\(_a\)** = \( 10^{-2} \)
- **IDF\(_b\)** = \( 10^{-2} \)
- **IDF\(_c\)** = \( 10^{-2} \)
- **TDF\(_a\)** = \( 10^{-2} \)
- **TDF\(_b\)** = \( 10^{-4} \)
- **TDF\(_c\)** = \( 10^{-6} \)

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### Question 10
**Why do we do dilutions in a series rather than making one large dilution to the final concentration?**

Serial dilutions are often performed to achieve very high dilution factors in a more controlled and accurate way. Making a large dilution in a single step can lead to significant measurement errors, especially when working with very small amounts. By performing a series of smaller dilutions, we reduce potential errors and ensure a more accurate and reproducible final concentration.

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Would you like additional explanations for any part of these calculations?

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### Follow-Up Questions
1. What is the formula to calculate an individual dilution factor (IDF)?
2. How does a total dilution factor (TDF) differ from an individual dilution factor?
3. What are some potential sources of error in a serial dilution?
4. How could you verify the accuracy of a final dilution in a laboratory setting?
5. What practical applications make use of serial dilutions?

### Tip
When performing serial dilutions, always label each step carefully to avoid confusion, as concentration decreases quickly with each step.

Calculate the IDF's and the TDF for the following serial dilution. Express IDF's and TDF's in scientific notation.
Why do we do dilutions in a series - why not just make one dilution to the final concentration?

Learn how to calculate the Individual Dilution Factor (IDF) and Total Dilution Factor (TDF) in a serial dilution with scientific notation. Understand why dilutions are performed in a series rather than a single large dilution to achieve accurate and reproducible results. Suitable for high school students.

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