91Ó°ÊÓ

One kilogram of hydrogen combines chemically with 8 kilograms of oxygen to form water; about \(10^{8} \mathrm{~J}\) of energy is released. Ten metric tons \(\left(10^{4} \mathrm{~kg}\right)\) of hydrogen combines with oxygen to produce water. (a) Does the resulting water have a greater or less mass than the original hydrogen plus oxygen? (b) What is the numerical magnitude of this difference in mass? (c) A smaller amount of hydrogen and oxygen is weighed, then combined to form water, which is weighed again. A very good chemical balance is able to detect a fractional change in mass of 1 part in \(10^{8}\). By what factor is this sensitivity more than enough -or insufficient - to detect the fractional change in mass in this reaction?

Step by step solution

01

- Determine mass of water produced

Hydrogen and oxygen combine in a ratio of 1 kg of hydrogen to 8 kg of oxygen to produce water. Thus, in 10 metric tons (10000 kg) of hydrogen, they require 80000 kg of oxygen, making the combined mass before the reaction equal to 90000 kg. Therefore, the mass of water produced is approximately equal to the sum of the masses of hydrogen and oxygen, which is 90000 kg.

02

- Calculate the energy released

The reaction releases about 10^8 J of energy for every 1 kg of hydrogen combined. Given 10 metric tons (10000 kg) of hydrogen, the total energy released during the reaction is 10^{8} J/kg × 10000 kg = 10^{12} J.

03

- Apply mass-energy equivalence (Einstein's formula)

Using Einstein’s mass-energy equivalence: E = mc^2, where E is the energy released, m is the mass converted, and c is the speed of light ( c = 3x10^8 m/s). Solve for the mass difference m: m = E / c^2 = 10^{12} J / (3×10^8 m/s)^2 = 10^{12} J / 9×10^{16} m^2/s^2 = 10^{-4} kg = 0.0001 kg.

04

- Compare the mass of water with original

The mass of the water will be less than the combined mass of hydrogen and oxygen before the reaction due to the mass-energy equivalence. The difference in mass is about 0.0001 kg.

05

- Determine the sensitivity of the balance

Given that a very good chemical balance can detect a change in mass of 1 part in 10^8, calculate the fractional change in mass: Fractional change = (0.0001 kg / 90000 kg) ≈ 1.11x10^-9. This is much smaller than 1 part in 10^8.

06

- Evaluate the balance sensitivity

Determine the factor of insufficiency: Sensitivity factor = 10^-8 / 1.11x10^-9 ≈ 9.05. The balance sensitivity is ≈9 times insufficient to detect the change in mass.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Chemical Reaction

A chemical reaction involves the transformation of reactants into products through the breaking and forming of chemical bonds. In our example, hydrogen and oxygen react to form water. This specific reaction requires one kilogram of hydrogen to combine with eight kilograms of oxygen to create water. Understanding the precise ratios of reactants is key, as it impacts the total mass of the products formed.

Mass Difference

After a chemical reaction, there can be a difference in mass between the reactants and the products. In the hydrogen-oxygen reaction to form water, the mass of the resulting water is slightly less than the total mass of hydrogen and oxygen originally combined. This difference is due to the conversion of a small amount of mass into energy during the reaction. The change is usually minimal but significant in scientific measurements and applications.

Energy Release

Chemical reactions often release or absorb energy. For hydrogen and oxygen forming water, the reaction releases a considerable amount of energy, specifically 10^8 J per kilogram of hydrogen. This energy release is a fundamental aspect of the reaction and is closely related to the change in mass according to mass-energy equivalence principles. Understanding the energy dynamics helps appreciate why the mass appears to decrease during the reaction.

Einstein's Formula

Einstein's formula, E=mc^2, explains the relationship between mass and energy. Here, E is the energy, m is the mass, and c is the speed of light (approximately 3x10^8 m/s). In our example, the enormous energy released in the reaction (10^{12} J) corresponds to a tiny loss in mass (~0.0001 kg). The beauty of Einstein’s formula is that it provides a precise method to quantify mass converted into energy, crucial in understanding phenomena at both the macro and micro levels.

Sensitivity of Measurement

Sensitivity of measurement refers to how accurately a device can detect changes. In our problem, a balance that can detect a fractional change of 1 part in 10^8 is considered. The actual fractional change in mass due to the reaction (~1.11x10^-9) is much smaller than what the balance can detect. Thus, this balance is inadequate by a factor of approximately 9.05 times, making it crucial to recognize the limitations of measurement tools in detecting small changes in mass-energy experiments.