Problem 65 - Chemistry and Chemical Reactivity

Question

The data below were collected in a Millikan oil drop experiment. $$\begin{array}{ll} \hline \text { Oil Drop } & \text { Measured Charge on Drop (C) } \\ \hline 1 & 1.59 \times 10^{-19} \\ 2 & 11.1 \times 10^{-19} \\ 3 & 9.54 \times 10^{-19} \\ 4 & 15.9 \times 10^{-19} \\ 5 & 6.36 \times 10^{-19} \\ \hline \end{array}$$ (a) Use these data to calculate the charge on the electron (in coulombs). (b) How many electrons have accumulated on each oil drop? (c) The accepted value of the electron charge is \(1.60 \times\) \(10^{-19} \mathrm{C} .\) Calculate the percent and error for the value determined by the data in the table.

Step-by-Step Solution

Answer

(a) Charge is approximately \(1.59 \times 10^{-19}\) C. (b) Electrons on drops are 1, 7, 6, 10, 4 respectively. (c) Percent error ~0.625%.

Step 1: Identify the Smallest Charge Unit

Start by observing the smallest charge measured among the given data. The smallest charge detected is for Oil Drop 1: \[\text{Charge} = 1.59 \times 10^{-19} \text{ C}\] This value represents the smallest unit of charge observed, potentially corresponding to the fundamental charge of an electron, denoted as \(e\).

Step 2: Calculate the Number of Electrons for Each Oil Drop

For each oil drop, divide the measured charge by the smallest unit of charge (\(1.59 \times 10^{-19} \text{ C}\)) to find the number of electrons:- **Oil Drop 1**: \[\text{Electrons} = \frac{1.59 \times 10^{-19}}{1.59 \times 10^{-19}} = 1\]- **Oil Drop 2**: \[\text{Electrons} = \frac{11.1 \times 10^{-19}}{1.59 \times 10^{-19}} \approx 7\]- **Oil Drop 3**: \[\text{Electrons} = \frac{9.54 \times 10^{-19}}{1.59 \times 10^{-19}} \approx 6\]- **Oil Drop 4**: \[\text{Electrons} = \frac{15.9 \times 10^{-19}}{1.59 \times 10^{-19}} \approx 10\]- **Oil Drop 5**: \[\text{Electrons} = \frac{6.36 \times 10^{-19}}{1.59 \times 10^{-19}} \approx 4\]

Step 3: Fine-tune the Calculation of Electron Charge

To ensure accuracy in determining the fundamental charge, recalculate \(e\) using the averaged charges from the calculated number of electrons for each oil drop.The average calculated charge can be based on the common divisor of collected data. However, using each count's adjustments: We should find the average charge unit value based on actual common fraction divisions. Regardless, for pedagogical purposes, the calculated effective charge remains quite close to our starting approximation:- **Calculated charge**: \(1.59 \times 10^{-19} \text{ C} \approx e\)

Step 4: Calculate the Percent Error

Use the accepted value of \(e\), which is \(1.60 \times 10^{-19} \text{ C}\), and find the percent error compared with our determined charge \(1.59 \times 10^{-19} \text{ C}\):\[\text{Percent Error} = \left( \frac{1.60 \times 10^{-19} - 1.59 \times 10^{-19}}{1.60 \times 10^{-19}} \right) \times 100\%\]Calculating this gives:\[\text{Percent Error} = \frac{0.01 \times 10^{-19}}{1.60 \times 10^{-19}} \times 100\%\approx 0.625\%\]

Key Concepts

Electron Charge Calculation

In the Millikan oil drop experiment, the primary goal is to calculate the charge on an electron. This is done by observing the charges on oil drops, which are measured in coulombs. By identifying the smallest unit of charge from these measurements, we can infer the charge of a single electron, often denoted by the symbol \( e \). In this specific dataset, the smallest charge recorded is \(1.59 \times 10^{-19}\) C. This value is taken as an approximation of the electron's charge, known as the fundamental charge.

Once we have this base value, we can determine how many electrons are associated with each oil drop. This number is found by dividing the charge of each drop by the smallest observed charge. For instance, if the oil drop has a charge of \(9.54 \times 10^{-19}\) C, you divide this by \(1.59 \times 10^{-19}\) C. The result suggests that approximately 6 electrons are causing this charge.

This process highlights how using the smallest charge unit allows us to understand the quantized nature of electric charge, supporting the idea that charge comes in discrete quantities of the electron charge.

Percent Error Calculation

After estimating the electron charge from the experiment, comparing it to the accepted standard value is essential. Percent error gives us a sense of the deviation of our experimental findings from the recognized value. This standard value is usually \(1.60 \times 10^{-19}\) C for an electron.

To calculate percent error, use the formula:

• Subtract the experimental value from the accepted value.
• Take the absolute difference.
• Divide by the accepted value.
• Multiply the result by 100 to convert to a percentage.

In our case, the experimental value is \(1.59 \times 10^{-19}\) C, leading to a percent error calculation as follows:\[\text{Percent Error} = \left( \frac{1.60 \times 10^{-19} - 1.59 \times 10^{-19}}{1.60 \times 10^{-19}} \right) \times 100\% \approx 0.625\%\]This small percentage indicates that the experiment's approximation of the charge is quite close to the actual accepted value.

Fundamental Charge of Electron

The fundamental charge of an electron, often simply noted as "\( e \)", is a constant that represents the smallest charge an isolated electron can possess. Its value is accepted to be approximately \(1.60 \times 10^{-19}\) C. This charge is significant in physics as it forms the basis of understanding electric charge quantization.

This constant allows scientists and students alike to simplify many calculations in electrostatics and quantum physics. It provides insight into the structure of matter by suggesting that charges are quantized, being naturally observed in integer multiples of this fundamental charge.

The Millikan oil drop experiment was pivotal in proving the quantization of charge. By analyzing the rise and fall of charged oil droplets in an electric field, Millikan could measure the charge to a high degree of accuracy, reinforcing the concept that charge comes in discrete units. This discovery was not only important for physics but also laid groundwork for future studies in atomic theory and quantum mechanics.