Problem 7 - Chemistry

Question

Some radioactive materials emit a type of radiation called alpha particles at high velocity. (a) What is the kinetic energy in joules of an alpha particle with a mass of \(6.6 \times 10^{-27} \mathrm{~kg}\) and a speed of \(1.5 \times 10^{\frac{1}{7}} \mathrm{~m} / \mathrm{s} ?\) Express the number in scientific notation. (b) Express the number and unit using an appropriate prefix.

Step-by-Step Solution

Answer

The kinetic energy is approximately \(7.425 \times 10^{-26} \text{ J}\) or \(74.25 \text{ yJ}\).

Step 1: Understand the kinetic energy formula

The formula for kinetic energy (KE) is given by \(KE = \frac{1}{2}mv^2\), where \(m\) is the mass of the object and \(v\) is the velocity.

Step 2: Plug in the values

We know the mass \(m = 6.6 \times 10^{-27} \text{ kg}\) and the velocity \(v = 1.5 \times 10^{\frac{1}{7}} \text{ m/s}\). Substitute these values into the kinetic energy formula: \(KE = \frac{1}{2} \times 6.6 \times 10^{-27} \times (1.5 \times 10^{\frac{1}{7}})^2\).

Step 3: Simplify the velocity

Calculate the square of the velocity: \((1.5 \times 10^{\frac{1}{7}})^2 = 1.5^2 \times (10^{\frac{1}{7}})^2 = 2.25 \times 10^{\frac{2}{7}}\).

Step 4: Calculate the kinetic energy

Now substitute the square of velocity back into the kinetic energy formula: \(KE = \frac{1}{2} \times 6.6 \times 10^{-27} \times 2.25 \times 10^{\frac{2}{7}}\). This simplifies to \(KE = 7.425 \times 10^{-27 + \frac{2}{7}}\).

Step 5: Simplify to scientific notation

Combine the exponents: \(-27 + \frac{2}{7} = x\), solve for \(x\) (in terms of a standard exponent), giving \(KE \approx 7.425 \times 10^{-26.714} \text{ J}\), rounding leads to \(7.425 \times 10^{-26}\).

Step 6: Convert to appropriate prefix

To express the kinetic energy with a suitable prefix: \(7.425 \times 10^{-26}\) can be expressed as \(74.25 \times 10^{-27}\), or \(74.25 \text{ yJ} \text{ (yoctoJoules)}\).

Key Concepts

Alpha Particles

Alpha particles are a form of radiation that consists of two protons and two neutrons bound together. They are emitted from the nucleus of an unstable atom during radioactive decay. This makes them relatively heavy compared to other forms of radiation, such as beta particles and gamma rays.
Due to their mass and charge, alpha particles have a limited ability to penetrate substances. For instance, they can be stopped by a sheet of paper or even the outer layer of human skin. This means they primarily pose a risk when inhaled or ingested, as they can damage internal tissues.
- High velocity: Despite being relatively large, alpha particles are emitted at high speeds.
- Positive charge: Their charge of +2 affects their interaction with electric and magnetic fields.
Understanding alpha particles is crucial when learning about radiation and its effects.

Radiation

Radiation refers to the emission of energy as electromagnetic waves or as moving subatomic particles. There are several types of radiation, with differing properties and sources.
Alpha radiation, as with alpha particles, is one type. It occurs when unstable nuclei release energy for the atom to reach a more stable state. This release can occur naturally, like in radioactive decay, or artificially, such as in nuclear reactors.
Types of radiation vary in mass, energy, and ability to penetrate materials.

• Alpha Radiation: Safely blocked by paper or skin.
• Beta Radiation: More penetrating than alpha.
• Gamma Radiation: Highly penetrating, requiring dense materials like lead to block.

Learning about different radiation helps us understand its applications and risks.

Scientific Notation

Scientific notation is a method of writing numbers that are too large or too small to be conveniently written in standard decimal form. It is especially useful in scientific fields where measurements can vary greatly in size.
Scientific notation expresses numbers as a product of a number between 1 and 10 and a power of ten. For example, the mass of an alpha particle might be written as:
- Mass: \(6.6 \times 10^{-27}\) kg
Using scientific notation makes calculations easier and more manageable when dealing with extreme values.

• Standard form: A number followed by an exponent of 10.
• Example: The speed of light is \(3 \times 10^8\) m/s.

This notation is efficient and helps avoid errors when performing complex calculations.

Joules

Joules (abbreviated as J) are the unit of measurement for energy in the International System of Units (SI). Energy can be understood as the capacity to do work or produce change, and it comes in various forms, such as kinetic, potential, thermal, and more.
In the context of the problem, when we talk about the kinetic energy of alpha particles, we measure how much energy they possess due to their motion. Kinetic energy is calculated using the equation:
- \(KE = \frac{1}{2}mv^2\) where \(m\) is mass and \(v\) is velocity.
This calculation tells us how energy is transferred as particles move, which is important for understanding phenomena in both physics and chemistry.
Kinetic energy is just one way we use joules to quantify energy, which is critical in everything from moving vehicles to generating electricity.