Question

A stationary free electron in a gas is struck by an X-ray with an energy of \(159.98 \mathrm{eV}\). After the collision, the speed of the electron is measured to be \(92.17 \mathrm{~km} / \mathrm{s}\). By how much did the energy of the X-ray decrease?

Short answer

(A): We can find the decrease in the energy of the X-ray after the collision using the conservation of energy principle, with the following steps: 1. Calculate the kinetic energy (KE) of the electron after the collision using the formula KE = (1/2)mv², where m is the mass of the electron (\(9.109 \times 10^{-31} \mathrm{kg}\)), and v is the velocity of the electron after the collision (\(92.17 \mathrm{km/s}\), or \(92.17 * 10^3 \mathrm{m/s}\)). 2. Convert the initial X-ray energy from electron volts (eV) to Joules (J) using the conversion factor \(1 \mathrm{eV} = 1.602 \times 10^{-19} \mathrm{J}\). 3. Find the decrease in the X-ray energy by subtracting the kinetic energy of the electron after the collision from the initial energy of the X-ray. 4. Convert the decrease in the X-ray energy back to electron volts (eV) using the same conversion factor as in Step 2. After performing these calculations, we can find the decrease in the energy of the X-ray after the collision.

Step-by-step solution

Calculate the kinetic energy of the electron after the collision

We will use the formula for the kinetic energy (KE) of an object, given by: KE = (1/2)mv² where m is the mass of the object, and v is its velocity. In this case, the mass of an electron (m) is approximately \(9.109 \times 10^{-31} \mathrm{kg}\), and its velocity (v) after the collision is given as \(92.17 \mathrm{km/s}\). First, we need to convert the speed to m/s: v = 92.17 * 10^3 m/s Now we can calculate the kinetic energy of the electron: KE = (1/2) * (9.109 * 10^{-31} kg) * (92.17 * 10^3 m/s)²

Convert the initial X-ray energy to Joules

The initial energy of the X-ray is given as \(159.98 \mathrm{eV}\). To find the decrease in energy, we need to first convert this value to Joules. The conversion factor is \(1 \mathrm{eV} = 1.602 \times 10^{-19} \mathrm{J}\): Initial X-ray energy (Joules) = 159.98eV * 1.602 * 10^(-19) J/eV

Calculate the decrease in the X-ray energy

Now that we have the initial energy of the X-ray in Joules and the kinetic energy gained by the electron after the collision, we can find the decrease in the X-ray energy by subtracting the kinetic energy of the electron from the initial energy of the X-ray: Decrease in X-ray energy (Joules) = Initial X-ray energy (Joules) - Kinetic energy of electron (Joules)

Convert the decrease in the X-ray energy back to eV

Finally, we need to convert the decrease in the X-ray energy back to electron volts (eV). We will use the same conversion factor as in Step 2: Decrease in X-ray energy (eV) = Decrease in X-ray energy (Joules) / (1.602 * 10^(-19) J/eV) By calculating the above expressions, we will obtain the decrease in the energy of the X-ray after the collision.