Problem 107 - General Chemistry: Principles and Modern Applications

Question

The mass of the isotope \(\frac{84}{36} \mathrm{Xe}\) is 83.9115 u. If the atomic mass scale were redefined so that \(\frac{84}{36} \mathrm{Xe}=84 \mathrm{u},\) exactly, the mass of the \(^{12} \mathrm{C}\) isotope would be (a) \(11.9115 \mathrm{u}\) (b) \(11.9874 \mathrm{u} ;\) (c) \(12 \mathrm{u}\) exactly; \((\mathrm{d}) 120127 \mathrm{u} ;(\mathrm{e}) 12.0885 \mathrm{u}\)

Step-by-Step Solution

Answer

The mass of the carbon-12 isotope on the redefined atomic mass scale is 11.9874 u, so the correct answer is (b) 11.9874 u.

Step 1: Understanding Atomic Mass Scale

Atomic mass is usually measured in atomic mass units (amu) or in this case symbolized as 'u'. A mass scale is often defined with reference to a standard. By our current mass scale, an atom of \(^{12} \mathrm{C}\) mass is exactly 12 atomic mass units.

Step 2: Redefine Atomic Mass Scale

In this problem, the atomic mass scale is redefined such that \(\frac{84}{36} \mathrm{Xe}\) is 84 u exactly. This means one atomic mass unit is now defined as \( \frac{83.9115 \ u}{84 \ u} \).

Step 3: Calculate the New Mass of \(^{12} \mathrm{C}\)

Now, on this redefined scale, the mass of the carbon-12 isotope would be \(12 * \frac{83.9115 \ u}{84 \ u} = 11.9874 \ u\).

Key Concepts

Isotopes

Atoms of the same element can have different numbers of neutrons. These are called isotopes. Despite having the same number of protons, isotopes vary in neutron numbers, leading to different atomic masses. The chemical properties of isotopes of an element are nearly identical, but physical properties, such as stability and mass, can be quite different. This can significantly impact their behavior in scientific experiments and applications.

• The identity of an element is defined by the number of protons in its nucleus, also known as the atomic number.
• The total number of protons and neutrons gives the mass number of the isotope.
• Isotopes are denoted by their mass number, such as \(^{12}\mathrm{C}\) or \(\frac{84}{36}\mathrm{Xe}\).

Understanding isotopes is essential in the context of atomic mass scale exercises, as these exercises often deal with calculating the mass based on different isotopic compositions.

Atomic Mass Unit (amu)

The atomic mass unit (amu) is a crucial concept in chemistry and physics, providing a standard measurement for atomic and molecular masses. Essentially, 1 amu is defined as one-twelfth the mass of a carbon-12 atom. This scale simplifies dealing with atomic masses, as it facilitates comparing elements on a relative scale.

• The amu offers a convenient unit for scientists to express atomic and molecular weights with ease and precision.
• It standardizes measurements, making it easier to understand and predict chemical reactions and processes.
• Note that 1 amu is approximately equal to 1.66053906660 × 10^-27 kilograms.

When the atomic mass scale changes, like in the exercise where the isotope \(\frac{84}{36}\mathrm{Xe}\) is set as 84 u, this affects how 'amu' is perceived, altering the interpretation of masses of other isotopes.

Carbon-12

Carbon-12 (\(^{12}\mathrm{C}\)) is a stable, naturally occurring isotope of carbon and forms the basis for the atomic mass unit (amu) measurement. It has 6 protons and 6 neutrons, which gives it a mass number of 12. This isotope is significant because it serves as the yardstick against which atomic masses of all other isotopes and elements are measured.

• Carbon-12's importance in the atomic mass scale arises from the decision to define the amu in relation to it, making it the international standard.
• The decision to use carbon-12 allows for consistent atomic mass units, which promote uniformity in scientific communications.
• The mass of carbon-12 is exactly 12 amu on the standard scale, but changes to the atomic mass scale can alter its perceived mass, as shown in the exercise where a new reference isotope changes its calculated mass to 11.9874 u.

This reliable reference point helps scientists around the world to compare, contrast, and quantify the elements and their isotopes efficiently.