Calculate the kinetic energy of an `alpha`- particle which has a wavelength of 12 pm.

Explanation

To calculate the kinetic energy of an alpha particle with a wavelength of 12 pm (picometers), we can follow these steps:

### Step 1: Convert the Wavelength to Meters
The given wavelength is in picometers (pm). We need to convert it to meters (m) for our calculations.

\[ \text{Wavelength} (\lambda) = 12 \, \text{pm} = 12 \times 10^{-12} \, \text{m} \]

### Step 2: Determine the Mass of the Alpha Particle
An alpha particle is essentially a helium nucleus, which consists of 2 protons and 2 neutrons. The molar mass of helium (He) is approximately 4 g/mol.

To find the mass of a single alpha particle:

\[ \text{Mass of 1 alpha particle} = \frac{4 \, \text{g}}{6.022 \times 10^{23}} = 6.64 \times 10^{-24} \, \text{g} \]

Convert grams to kilograms:

\[ \text{Mass} (m) = 6.64 \times 10^{-24} \, \text{g} \times \frac{1 \, \text{kg}}{1000 \, \text{g}} = 6.64 \times 10^{-27} \, \text{kg} \]

### Step 3: Use de Broglie's Equation to Find Velocity
According to de Broglie's hypothesis, the wavelength (\(\lambda\)) is related to momentum (p) by the equation:

\[ \lambda = \frac{h}{p} \]

Where \(h\) is Planck's constant (\(6.626 \times 10^{-34} \, \text{Js}\)) and \(p\) is momentum, which can be expressed as \(p = mv\) (mass times velocity).

Rearranging gives us:

\[ v = \frac{h}{m \lambda} \]

Substituting in the values:

\[ v = \frac{6.626 \times 10^{-34} \, \text{Js}}{6.64 \times 10^{-27} \, \text{kg} \times 12 \times 10^{-12} \, \text{m}} \]

Calculating this gives:

\[ v \approx 8.32 \times 10^{3} \, \text{m/s} \]

### Step 4: Calculate the Kinetic Energy
The kinetic energy (KE) of an object is given by the formula:

\[ KE = \frac{1}{2} mv^2 \]

Substituting in the mass and velocity we found:

\[ KE = \frac{1}{2} \times 6.64 \times 10^{-27} \, \text{kg} \times (8.32 \times 10^{3} \, \text{m/s})^2 \]

Calculating this gives:

\[ KE \approx 2.30 \times 10^{-19} \, \text{J} \]

### Final Answer
The kinetic energy of the alpha particle is approximately \(2.30 \times 10^{-19} \, \text{J}\).

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