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Chapter 36: Problem 88

When two moles of hydrogen molecules \(\left(\mathrm{H}_{2}\right)\) and one mole of oxygen molecules \(\left(\mathrm{O}_{2}\right)\) react to form two moles of water \(\left(\mathrm{H}_{2} \mathrm{O}\right),\) the energy released is 484 \(\mathrm{kJ} .\) How much does the mass decrease in this reaction? What \(\%\) of the total original mass of the system does this mass change represent?

Short Answer

Expert verified
The mass decreases by \(5.38 \times 10^{-12}\) kg, which is approximately \(1.49 \times 10^{-8}\%\) of the total mass.

Step by step solution

01

Understanding the Reaction

The chemical reaction is given by: \ \( 2 \text{H}_2 + \text{O}_2 \rightarrow 2 \text{H}_2\text{O} \). Energy released during this reaction is 484 kJ.
02

Using Einstein's Energy-Mass Equivalence

According to Einstein's equivalence, the energy released (\(E\)) is related to the mass change (\(\Delta m\)) by the equation: \ \[ E = \Delta m \cdot c^2 \] where \( c \) is the speed of light, approximately \( 3 \times 10^8 \text{ m/s} \).
03

Calculate Mass Change

First convert energy to joules: \ \( 484 \text{ kJ} = 484,000 \text{ J} \). Now, calculate \(\Delta m\): \ \[ \Delta m = \frac{E}{c^2} = \frac{484,000 \text{ J}}{(3 \times 10^8 \text{ m/s})^2} \approx 5.38 \times 10^{-12} \text{ kg} \].
04

Calculate Total Initial Mass

Molar masses: \( \text{H}_2 = 2 \cdot 1.008 \text{ g/mol} = 2.016 \text{ g/mol} \); \( \text{O}_2 = 2 \cdot 16.00 \text{ g/mol} = 32.00 \text{ g/mol} \). Total mass of reactants will be: \ \[ \text{Total mass} = (2 \text{ mol} \times 2.016 \text{ g/mol}) + (1 \text{ mol} \times 32.00 \text{ g/mol}) = 4.032 \text{ g} + 32.00 \text{ g} = 36.032 \text{ g} \] Convert to kilograms for consistency with \(\Delta m\): \ \[ 36.032 \text{ g} = 0.036032 \text{ kg} \].
05

Calculate the Percentage of Mass Change

Calculate the percentage change: \ \[ \text{Percentage} = \left( \frac{\Delta m}{\text{Total initial mass}} \right) \times 100 = \left( \frac{5.38 \times 10^{-12} \text{ kg}}{0.036032 \text{ kg}} \right) \times 100 \approx 1.49 \times 10^{-8}\% \].

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Key Concepts

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

Chemical Reactions
Chemical reactions involve the transformation of reactants into products. In the scenario given, hydrogen (\( \text{H}_2 \)) and oxygen (\( \text{O}_2 \)) molecules react to form water (\( \text{H}_2\text{O} \)). Such reactions are fundamental processes in chemistry.
  • Reactants are the starting substances in a chemical reaction, while products are the substances formed as a result of the reaction.
  • The rearrangement of atoms according to the reaction equation \( 2 \text{H}_2 + \text{O}_2 \rightarrow 2 \text{H}_2\text{O} \) leads to the formation of water.
  • This reaction involves breaking and forming of chemical bonds which results in the release or absorption of energy.
Understanding these basic concepts in chemical reactions is crucial, as they form the basis for much of what we observe in the physical world, from the combustion of fuel to metabolic processes in living organisms.
Mass Change
One might find it surprising, but during certain chemical reactions, a small change in mass occurs. This mass change is tied deeply with energy changes in a reaction.
  • According to Einstein鈥檚 Energy-Mass Equivalence principle, energy and mass are interchangeable. This is expressed by the famous equation \( E = mc^2 \), where \( E \) is energy, \( m \) is mass, and \( c \) is the speed of light.
  • In the given reaction, 484 kJ of energy is released, leading to a minuscule mass change. By converting this energy into mass using Einstein鈥檚 principle, we calculate a mass decrease of approximately \( 5.38 \times 10^{-12} \) kg.
  • Even though the change is incredibly small, this concept highlights the interplay between mass and energy, emphasizing the conservation laws in physics.
This mass change is typically negligible in everyday chemical reactions but is quite significant in nuclear reactions where much larger mass-energy conversions occur.
Energy Release
Energy release in chemical reactions is a hallmark of the transformation processes that drive both living systems and mechanical engines. The amount of energy released or absorbed during a chemical reaction is a crucial factor in determining the nature of the reaction.
  • Exothermic reactions, like the formation of water from hydrogen and oxygen, release energy to the surroundings. This is why water formation is sometimes associated with heat and light production, such as in combustion.
  • The reaction described releases 484 kJ of energy, becoming available to do work or to dissipate as heat.
  • Understanding energy release allows chemists to harness these reactions for various practical applications, from powering vehicles with hydrogen fuel cells to understanding the energy flow in biological systems.
Energy transformations are not only pivotal in scientific understanding but also critical in developing technologies for sustainable energy solutions.

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Most popular questions from this chapter

(II) A friend speeds by you in her spacecraft at a speed of \(0.760 c .\) It is measured in your frame to be \(4.80 \mathrm{~m}\) long and \(1.35 \mathrm{~m}\) high. \((a)\) What will be its length and height at rest? (b) How many seconds elapsed on your friend's watch when 20.0 s passed on yours? \((c)\) How fast did you appear to be traveling according to your friend? \((d)\) How many seconds elapsed on your watch when she saw 20.0 s pass on hers?

(II) Reference frame \(\mathrm{S}^{\prime}\) moves at speed \(v=0.92 c\) in the \(+x\) direction with respect to reference frame \(\mathrm{S}\). The origins of \(\mathrm{S}\) and \(\mathrm{S}^{\prime}\) overlap at \(t=t^{\prime}=0 .\) An object is stationary in \(\mathrm{S}^{\prime}\) at position \(x^{\prime}=100 \mathrm{~m} .\) What is the position of the object in \(\mathrm{S}\) when the clock in \(\mathrm{S}\) reads \(1.00 \mu \mathrm{s}\) according to the (a) Galilean and (b) Lorentz transformation equations?

The fictional starship Enterprise obtains its power by combining matter and antimatter, achieving complete conversion of mass into energy. If the mass of the Enterprise is approximately \(6 \times 10^{9} \mathrm{~kg}\), how much mass must be converted into kinetic energy to accelerate it from rest to one-tenth the speed of light?

(II) Two spaceships leave Earth in opposite directions, each with a speed of \(0.60 c\) with respect to Earth. \((a)\) What is the velocity of spaceship 1 relative to spaceship \(2 ?(b)\) What is the velocity of spaceship 2 relative to spaceship \(1 ?\)

(II) Calculate the kinetic energy and momentum of a proton traveling \(2.80 \times 10^{8} \mathrm{~m} / \mathrm{s}\).
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