Problem 3 - College Physics

Question

An isotope of potassium has the same number of neutrons as argon- \(40 .\) Write this potassium isotope in nuclear notation.

Step-by-Step Solution

Answer

The potassium isotope is written as \(^{41}_{19}\text{K}\).

Step 1: Determine Neutrons in Argon-40

The atomic number of argon is 18, so it has 18 protons. Argon-40 has a mass number of 40, so the number of neutrons can be calculated as follows:\[\text{Number of Neutrons in Argon} = 40 - 18 = 22.\]Thus, Argon-40 has 22 neutrons.

Step 2: Find Potassium's Nuclear Composition

Since the potassium isotope must have 22 neutrons, we start with potassium’s atomic number, which is 19 (the number of protons).

Step 3: Calculate Mass Number of Potassium Isotope

To find the mass number of this potassium isotope, we add the number of protons (19) to the number of neutrons (22):\[\text{Mass Number} = 19 + 22 = 41.\]Hence, the potassium isotope’s mass number is 41.

Step 4: Write in Nuclear Notation

Using nuclear notation, we write the potassium isotope as \(^{41}_{19}\text{K}\), where the superscript is the mass number and the subscript is the atomic number.

Key Concepts

Isotopes

Isotopes are different forms of the same element, with each form having the same number of protons but a different number of neutrons. This means that while isotopes share an atomic number, which counts the protons, they have different mass numbers. For example, in our problem, we have discussed potassium and argon isotopes.

• Potassium's atomic number is 19, indicating 19 protons.
• Argon has an atomic number of 18, indicating 18 protons.

Despite sharing elements, isotopes can have varied masses due to the number of neutrons. For instance, Argon-40 and Potassium-41 are examples. The isotope of an element is symbolically represented via nuclear notation, like for Potassium-41 as \(^{41}_{19}\text{K}\), indicating the differences clearly by using a superscript for mass number and a subscript for atomic number. Understanding isotopes is crucial in fields like chemistry and physics, where they are used to study radioactive decay, nuclear reactions, and even drug formulations.

Neutron Calculation

The number of neutrons in an atom can be calculated using the formula: \[\text{Number of Neutrons} = \text{Mass Number} - \text{Atomic Number} \]Knowing how to calculate the number of neutrons is vital because it helps differentiate between various isotopes of the same element. In the case of Argon-40 mentioned in the exercise, the calculation is done by subtracting Argon's atomic number (18) from its mass number (40), which leaves us with 22 neutrons.

• Atomic Number (Z) = Number of Protons
• Mass Number (A) = Total of Protons + Neutrons

For potassium, once we know it needs to have the same number of neutrons (22), knowing its atomic number (19) allows us to determine the mass number of its isotope. By adding the 22 neutrons to its 19 protons, we find the mass number to be 41 for the potassium isotope. This reflects how isotope's neutron count directly affects mass number, aiding in work related to atomic models and nuclear chemistry.

Mass Number

The mass number of an atom is a count of the total particles, protons and neutrons, found in the nucleus of an atom. It's a significant property that allows scientists to differentiate between the isotopes of an element. The mass number helps in identification and forms a part of nuclear notation, which is crucial for atomic representations in physics and chemistry.
In the step-by-step example provided, the mass number is calculated by simply adding the number of protons and neutrons. For Potassium with 19 protons and requiring 22 neutrons, the mass number results in 41.

• Mass numbers do not have units and are typically a simple whole number.
• They do not show the presence of electrons, which are negligible in mass.
• Mass number varies between isotopes due to changes in neutron count while proton count remains constant.

These principles help in solving isotope problems by identifying and writing the nuclear notation, crucial for communication in scientific analysis and study.