Question

A chemist in a galaxy far, far away performed the Millikan oil drop experiment and got the following results for the charges on various drops. Use these data to calculate the charge of the electron in zirkombs. \(2.56 \times 10^{-12}\) zirkombs \(7.68 \times 10^{-12}\) zirkombs \(3.84 \times 10^{-12}\) zirkombs \(6.40 \times 10^{-13}\) zirkombs

Short answer

The charge of the electron in zirkombs is \(6.40 \times 10^{-13}\) zirkombs.

Step-by-step solution

Convert the charges into integers

Since finding the GCD for decimals might be difficult, we can first convert all the charges into integer values. To do this, we can multiply all the charges by \(10^{12}\) so that all decimal places are eliminated:\[2560, 7680, 3840, 640\]

Find the GCD of the charges

Now, we need to find the GCD of these integer values. You can use the Euclidean algorithm, or any other method you're comfortable with. The GCD of the values 2560, 7680, 3840, and 640 is 640.

Convert the GCD back to the original unit

Now, we need to convert the GCD back to its original unit of zirkombs. To do this, we simply divide the integer GCD by \(10^{12}\): \[\frac{640}{10^{12}}\]

Calculate the charge of an electron in zirkombs

Finally, calculate the charge of the electron in zirkombs by dividing the GCD by \(10^{12}\): \[\frac{640}{10^{12}} = 6.40 \times 10^{-13}\] Now we have the charge of the electron in zirkombs: \(6.40 \times 10^{-13}\) zirkombs.

Key concepts

Electron Charge Calculation

Understanding the calculation of an electron's charge via the Millikan oil drop experiment involves analyzing a classic physics procedure. Originally devised by Robert A. Millikan, the experiment measures the charge of electrons in oil droplets that are suspended in an electric field.

The crucial aspect here is to recognize each measurement as possibly being multiple of the fundamental charge of an electron. Imagine you're given multiple pieces of different lengths all being integral multiples of a base unit length. To find out the base unit, you'd look for the smallest length that can be evenly divided into all other lengths. In the context of Millikan's experiment, you are looking for the smallest charge (the smallest piece) which can evenly divide into all the other measured charges.

When a chemist in a galaxy far, far away performs this experiment and records the charges in 'zirkombs', they essentially capture snapshots of this base charge in action. Through calculations—as outlined in the textbook—the chemist can determine the electron's charge by repeatedly identifying the greatest common divisor of all observed charges.

GCD (Greatest Common Divisor) Method

The Greatest Common Divisor (GCD) is a foundational concept in mathematics, particularly in number theory. It offers an efficient way to find the largest number that can divide two or more integers without leaving a remainder. In the context of electron charge calculations based on the Millikan experiment results, the GCD represents the smallest quantum of charge that is a common factor among all observed charges.

Using the GCD method involves first converting all scientific notation measurements into whole numbers, as decimals complicate things. Once the charges are in integer form, algorithms like the Euclidean algorithm come into play. This algorithm involves a series of division steps where you repeatedly subtract the smaller number from the larger one until you reach a common measure.

In simplistic terms, the GCD is like finding the biggest measuring cup that can be used to evenly measure full cups of water from several different containers without spilling a single drop. For our intergalactic chemist, the GCD enables the determination of the fundamental charge unit by analyzing the commonality in the varied measurements.

Scientific Notation

Scientific notation is a method used to express very large or very small numbers in a compact form, which is particularly useful in sciences such as chemistry and physics. It follows the format of a number between 1 and 10 multiplied by a power of 10. This is essential for tasks like quantifying an electron's charge because these values tend to be extremely small.

For example, the value of an electron's charge in 'zirkombs' can be a cumbersome decimal with many zeroes. Scientific notation simplifies these numbers, making them more manageable and less prone to error when calculating. In the results of the Millikan oil drop experiment provided by the chemist, each charge is given in scientific notation to streamline further analysis, like the aforementioned GCD method.

Understanding scientific notation, therefore, is not just about making numbers shorter to write; it's about making the computational process easier and reducing potential errors in calculation, both of which are vital when dealing in the realm of atomic particles where precision is paramount.