Problem 113 - Chemistry and Chemical Reactivity

Question

Although carbon-12 is now used as the standard for atomic weights, this has not always been the case. Early attempts at classification used hydrogen as the standard, with the weight of hydrogen being set equal to 1.0000. Later attempts defined atomic weights using oxygen (with a weight of 16.0000 ). In each instance, the atomic weights of the other elements were defined relative to these masses. (To answer this question, you need more precise data on current atomic weights: \(\mathrm{H}, 1.00794 ;\) O, \(15.9994 .\)) (a) If \(\mathrm{H}=1.0000 \mathrm{u}\) was used as a standard for atomic weights, what would the atomic weight of oxygen be? What would be the value of Avogadro's number under these circumstances? (b) Assuming the standard is \(\mathrm{O}=16.0000\), determine the value for the atomic weight of hydrogen and the value of Avogadro's number.

Step-by-Step Solution

Answer

(a) Oxygen ≈ 15.8994 u, Avogadro's number ≈ 6.022 x 10^23; (b) Hydrogen ≈ 1.00784 u, Avogadro's number ≈ 6.022 x 10^23.

Step 1: Define the problem

Determine the atomic weight of oxygen assuming hydrogen as the standard and vice versa. Compute Avogadro's number in both situations.

Step 2: Calculate Oxygen's Atomic Weight with Hydrogen Standard

Since hydrogen is set to 1.0000 u, and the current atomic weight of oxygen is 15.9994 u relative to the current standard, we must use the ratio of their atomic weights. Calculate: \[ \text{Oxygen's weight with } \text{H}=1.0000 \text{u} = \frac{15.9994}{1.00794} \times 1.0000 \approx 15.8994 \text{ u} \]

Step 3: Calculate Avogadro's Number with Hydrogen Standard

Traditional Avogadro's number is based on carbon-12. Here, assume the same process:With H = 1.0000 u, Avogadro's Number, \( N_A = \frac{6.022 \times 10^{23}}{1.00794/1.0000} \approx 6.022 \times 10^{23} \). It remains approximately unchanged because the definition in terms of 12 g of carbon-12 is not being altered by this change.

Step 4: Calculate Hydrogen's Atomic Weight with Oxygen Standard

Given oxygen is 16.0000 u, calculate hydrogen's atomic weight:\[ \text{Hydrogen's weight with } \text{O}=16.0000 \text{u} = \frac{1.00794}{15.9994} \times 16.0000 \approx 1.00784 \text{ u} \]

Step 5: Calculate Avogadro's Number with Oxygen Standard

Follow similar logic as Step 3:With O = 16.0000 u, if we redefine using O = 16.0000 u, it does not affect Avogadro's conceptual basis, so the approximate value of \( N_A \) remains unchanged, at \( 6.022 \times 10^{23} \).

Step 6: Summary

Under (a) H = 1.0000 u: Oxygen is approximately 15.8994 u, Avogadro's number nearly unchanged. (b) O = 16.0000 u: Hydrogen is approximately 1.00784 u, Avogadro's number remains approximately 6.022 x 10^23.

Key Concepts

Avogadro's Number

Avogadro's Number is crucial in chemistry, providing the link between the microscopic and macroscopic worlds. It indicates the number of atoms, ions, or molecules in one mole of a substance and is approximately equal to \( 6.022 \times 10^{23} \). This constant is named after the Italian scientist Amedeo Avogadro, who first suggested the principle that would lead to its discovery.

Even when the atomic weight standards change, such as defining atomic weights relative to hydrogen or oxygen instead of carbon-12, Avogadro's number's value remains largely stable. This is because it is derived from the number of atoms in 12 grams of carbon-12. Therefore, whether we set hydrogen at 1.0000 u or oxygen at 16.0000 u, the magnitude of Avogadro's number stays nearly the same, serving as a stable foundation for molecular calculations.

Carbon-12 Standard

The carbon-12 standard is the current benchmark for atomic weights. In this system, the atomic mass unit (amu) is defined such that one mole of carbon-12 atoms weighs exactly 12 grams.

This conversion facilitates an easier and more accurate determination of atomic weights of other elements, as carbon-12 provides a stable and precise reference. It resolved inconsistencies that arose when prior standards, such as hydrogen or oxygen, were used.

The carbon-12 standard allows Avogadro's number to fit seamlessly since the definition of a mole—containing \( 6.022 \times 10^{23} \) entities—is tailored to precisely match 12 grams of carbon-12.

Hydrogen as Standard

In earlier chemical calculations, hydrogen was utilized as the standard for atomic weights, with a weight of exactly 1.0000 u. Given hydrogen's ubiquity and simple atomic structure, it initially seemed a logical reference. However, this standard was not precise enough for the growing complexity of chemical research.

When using hydrogen as the reference, other elements like oxygen had to be calculated relative to it. As mentioned in the exercise, when hydrogen was 1.0000 u, oxygen's atomic weight was recalculated as 15.8994 u using the ratio of their current atomic weights. This adjustment shows how different standards influence the perceived weights of elements.

Oxygen Standard

Oxygen once served as a standard for atomic weights, with an assigned atomic weight of 16.0000. Scientists chose oxygen due to its wide availability and role in various compounds.

When oxygen was used as the standard, the weights of other elements, such as hydrogen, needed conversion. As per the exercise, this yielded hydrogen's atomic weight as approximately 1.00784 u.

Despite oxygen providing a robust reference for its time, its limitations became apparent as measurement technology advanced. Eventually, the carbon-12 standard provided a more accurate and universally applicable system, simplifying atomic weight comparisons across various scientific fields.