Problem 3 - Advanced Problems in Physical Chemistry for Competitive Examinations

Question

An electron at rest is accelerated through a potential difference of \(200 \mathrm{~V}\). If the specific charge of electron is \(1.76 \times 10^{11} \mathrm{C} / \mathrm{kg}\), the speed acquired by the electron is about (a) \(8.4 \times 10^{6} \mathrm{~cm} / \mathrm{s}\) (b) \(8.4 \times 10^{6} \mathrm{~m} / \mathrm{s}\) (c) \(4.2 \times 10^{6} \mathrm{~m} / \mathrm{s}\) (d) \(4.2 \times 10^{6} \mathrm{~cm} / \mathrm{s}\)

Step-by-Step Solution

Answer

The speed acquired by the electron is about (b) 8.4 x 10^6 m/s.

Step 1: Find the Kinetic Energy Gained by the Electron

Calculate the kinetic energy (KE) gained by the electron when it is accelerated through a potential difference (V) using the equation: KE = e * V. Here, e is the charge of an electron which is approximately equal to 1.60 x 10^-19 coulombs.

Step 2: Calculate the Speed of the Electron

Use the relation between kinetic energy and speed for the electron: KE = (1/2) * m * v^2, where m is the mass of the electron and v is its speed. Rearrange the equation to solve for v: v = sqrt(2 * KE / m).

Step 3: Determine the Mass of the Electron

Find the mass of the electron (m) using the specific charge (E/m) given by: m = e / Specific charge.

Step 4: Combine the Results to Calculate the Speed

Insert kinetic energy (KE) from Step 1 and mass (m) from Step 3 into the speed equation from Step 2 to calculate the speed of the electron.

Step 5: Convert Units if Necessary

If the speed is calculated in meters per second (m/s) and the answer choices are in centimeters per second (cm/s), convert the units by multiplying the speed in m/s by 100 to obtain the speed in cm/s.

Key Concepts

Potential Difference

Think of potential difference, often referred to as voltage, as a kind of pushing force for electrons. It's measured in volts (V) and can be visualized like a slope down which the electrons 'slide'. This slope drives the electron's journey from a lower potential energy point to a higher one, just like a ball rolling downhill gains speed thanks to gravity. In our example, an electron is accelerated through a potential difference of 200 V. That's analogous to giving the electron a shove down a hill, turning potential energy into kinetic energy as it rolls down.

• A potential difference causes electrons to accelerate.
• It's measured in volts (V).
• Larger potential differences cause a greater increase in kinetic energy and speed.

Specific Charge of Electron

The specific charge of an electron is a crucial value that tells you how much charge there is per unit mass. For electrons, this is a huge number because they have such a tiny mass and a defined charge. The specific charge of an electron is approximately 1.76 x 10^11 C/kg. This ratio helps us understand how changes in electric fields or potential differences will affect an electron's motion. For instance, it's used to figure out how fast an electron will be traveling after accelerating through a given potential difference.

How Specific Charge Impacts Acceleration

• It dictates the level of acceleration an electron will undergo in an electric field.
• A higher specific charge means that for the same potential difference, the electron's speed will be greater.

Kinetic Energy of Electron

When an electron speeds up due to a potential difference, it gains kinetic energy (KE), which can be calculated using the formula KE = e * V, where 'e' is the charge of the electron, and 'V' is the potential difference it traverses. In simple terms, the kinetic energy is the energy that the electron has due to its motion. This energy can be further related to the speed of the electron via the equation KE = (1/2) * m * v^2. From these relationships, we know that if the potential difference goes up, so does the electron's kinetic energy—and therefore, its speed.

• Kinetic energy is the energy of motion.
• The formula for kinetic energy shows its direct proportionality to speed.

Electron Speed Calculation

To find out the speed of an electron after it's been accelerated, we take the kinetic energy that results from the potential difference and use the electron's mass. You've learned that KE = (1/2) * m * v^2, so rearranging for speed gives us v = sqrt(2 * KE / m). By substituting the kinetic energy we found using the potential difference and the charge of the electron, and the mass of the electron calculated from its specific charge, we can get the speed directly.
After working out the numbers for our specific example, we have to ensure our units match those in the answer choices. If they're in different units, we convert them accordingly. Remember to always check the units of your final answer to make sure it's presented in the same format as the question or the standard scientific conventions.

• Speed is the square root of twice the kinetic energy divided by mass.
• Make sure to match the units to the answer choices provided.