Problem 91 - Fundamentals of Physics

Question

If the unit for atomic mass were defined so that the mass of \({ }^{1} \mathrm{H}\) were exactly \(1.000000 \mathrm{u},\) what would be the mass of (a) \({ }^{12} \mathrm{C}\) (actual mass \(12.000000 \mathrm{u}\) ) and (b) \({ }^{238} \mathrm{U}\) (actual mass \(238.050785 \mathrm{u}\) )?

Step-by-Step Solution

Answer

New \({ }^{12} \mathrm{C}\) mass: 11.905404 u, New \({ }^{238} \mathrm{U}\) mass: 236.183925 u.

Step 1: Understand the conversion factor

In the current atomic mass unit (amu) scale, the atomic mass of a hydrogen atom \({ }^{1} \mathrm{H}\) is approximately 1.007825 u. If we redefine it to be exactly 1.000000 u, we need to find the conversion factor from the old amu scale to the new one. The conversion factor will be the ratio of the defined mass to the actual mass: \[ \text{Conversion Factor} = \frac{1.000000}{1.007825} \approx 0.992117 \].

Step 2: Calculate the mass of \\({ }^{12} \\mathrm{C}\\) in the new unit

To find the new mass of \({ }^{12} \mathrm{C}\) with the conversion factor, multiply its actual mass by the conversion factor: \[ \text{New Mass of } { }^{12} \mathrm{C} = 12.000000 \, \text{u} \times 0.992117 \approx 11.905404 \, \text{u} \].

Step 3: Calculate the mass of \\({ }^{238} \\mathrm{U}\\) in the new unit

Similarly, to find the new mass of \({ }^{238} \mathrm{U}\), we use the conversion factor: \[ \text{New Mass of } { }^{238} \mathrm{U} = 238.050785 \, \text{u} \times 0.992117 \approx 236.183925 \, \text{u} \].

Step 4: Summarize and present the results

In the newly defined atomic mass unit system, the mass of \({ }^{12} \mathrm{C}\) is approximately 11.905404 u, and the mass of \({ }^{238} \mathrm{U}\) is approximately 236.183925 u.

Key Concepts

Conversion Factor

When we talk about a conversion factor in chemistry, it means a number that helps us switch the measurement of a specific quantity from one unit to another. Imagine it like changing the language of measurement. If you want to change the mass of an atom because the atomic mass unit (amu) scale has been redefined, you use this handy tool.

In the example of hydrogen, we have a situation where the mass of hydrogen, \({ }^{1}\mathrm{H}\), is initially known to be 1.007825 u according to the old scale. To redefine it to exactly 1.000000 u, we need a conversion factor.

• This factor comes from the ratio of the defined new mass divided by the actual old mass.
• Using the formula: \[ \text{Conversion Factor} = \frac{1.000000}{1.007825} \approx 0.992117 \]
• The conversion factor, therefore, is less than 1, indicating a small decrease in the mass values when using the new amu scale compared to the old one.

With this conversion factor in hand, you can now adjust any atomic masses measured with the old scale to align them with the new definitions.

Isotopes

Isotopes are fascinating! They are variants of a particular chemical element that differ in neutron number, though they have the same number of protons. This difference in neutrons causes variations in their atomic masses, which are key to many processes in chemistry and physics.

Often, when we think of an element like carbon, we might just visualize it simply as carbon. But, in reality, elements come in different isotope forms. Carbon has several isotopes, such as \(^{12}\mathrm{C}\) and \(^{14}\mathrm{C}\), with different atomic mass numbers.

• \(^{12}\mathrm{C}\) is the most abundant isotope of carbon and is used as the standard when defining atomic mass units. Its atomic mass is precisely 12.000000 u in the old scale.
• The isotopes have similar chemical properties because they contain the same number of electrons and protons, but the additional neutrons can affect their physical properties and stability.

Understanding isotopes helps scientists determine the age of artifacts, trace environmental changes, and much more. It's one of the reasons why different isotopes are pivotal in fields including geology, archaeology, and medicine.

Mass of Carbon-12

The importance of \(^{12}\mathrm{C}\), or carbon-12, extends throughout chemistry due to its central role in the determination of atomic mass unit (amu).

It's essential to understand why carbon-12 is so special:

• Carbon-12 is composed of 6 protons and 6 neutrons in its nucleus, making it very stable and abundant in the natural world.
• The atomic mass of carbon-12 is set as exactly 12 u. This means that 1 amu is defined as \(\frac{1}{12}\) the mass of a carbon-12 atom.
• Even though the mass of protons and neutrons is not exactly 1 u, this definition provides a convenient average for mass measurements.

Carbon-12's role as a standard allows scientists to precisely measure elements' masses in a relative way, helping them to understand and develop the periodic table more completely. Whenever there are shifts or new definitions in the amu, like changing hydrogen's mass from 1.007825 to 1.000000 u, carbon-12 remains a constant figure for comparison and calibration.