Chapter 7, Problem 47Q

(a)

Interpretation Introduction

Interpretation:

The mass loss corresponds to $\text{50}\text{.1 KJ}$ of energy released during the combustion of methane has to be calculated using Einstein’s mass-energy equivalence equation.

Concept Introduction:

Einstein’s mass-energy equivalence equation clearly depicts the relation between mass and energy. The mass-energy equivalence equation is as follows,

$\Delta \text{E = }\Delta {\text{m c}}^2$

Here,

    $\Delta \text{E }$ is energy equivalence

  $\Delta \text{m}$ is mass defect

  $\text{c}$ is speed of light = $3.00\times {\text{10}}^8\text{m/s}$

Unit of mass and speed should be taken in SI unit in order to convert energy in terms of joule.

$\text{1J = 1 }\frac{{\text{kg m}}^{\text{2}}}{{\text{s}}^{\text{2}}}$

From the equation it is clear that, anything that has mass will also have energy.

Explanation of Solution

According to Einstein’s equation, the mass change corresponds to the release of $\text{50}\text{.1 kJ}$ energy can be calculated as follows,

$\Delta \text{E = }\Delta {\text{m c}}^2$

Rearranging the equation in terms of mass change is given below.

$\Delta \text{m = }\frac{\Delta \text{E}}{{\text{c}}^2}$

Given,

    $\Delta \text{E = 50}\text{.1 KJ = 50}\text{.1}\times \text{1000 J = 50100 J}$

  $\text{c}$ = speed of light = $3.00\times {\text{10}}^8\text{m/s}$

Substituting these values in the above equation gives mass loss as follows,

$\Delta \text{m = }\frac{\text{50100 J}}{{\left(\text{3}{\text{.00×10}}^{\text{8}}\text{m/s}\right)}^{\text{2}}}=5.566\times {10}^{-13}\text{kg =5}\text{.57 }\times {\text{10}}^{-10}\text{g}$

Therefore, the mass loss corresponds to the release of $\text{50}\text{.1 KJ}$ energy is $\text{5}\text{.57 }\times {\text{10}}^{-10}\text{g}\text{.}$

(b)

Interpretation Introduction

Interpretation:

For the energy $\text{50100 J}$, the ratio of mass of methane burned to the mass loss according to mass-energy equation has to be determined.

Concept Introduction:

During the combustion of methane, methane reacts with oxygen to produce carbon dioxide and water with the release of energy.

The balance chemical equation for the combustion of methane is as follows,

$\frac{\text{802}\text{.3 kJ}}{\cancel{{\text{1 mol CH}}_{\text{4}}}}\text{×}\frac{\cancel{{\text{1 mol CH}}_{\text{4}}}}{\text{16}{\text{.0 g CH}}_{\text{4}}}\text{=50}{\text{.1 kJ/g CH}}_{\text{4}}{\text{ = 50100 J/g CH}}_{\text{4}}$

For $1\text{ mol}$ of methane the energy released is $\text{50100 J}$, so for $\text{1 g}$ of methane the energy released can be calculates as follows,

$\frac{\text{802}\text{.3 kJ}}{\cancel{{\text{1 mol CH}}_{\text{4}}}}\text{×}\frac{\cancel{{\text{1 mol CH}}_{\text{4}}}}{\text{16}{\text{.0 g CH}}_{\text{4}}}\text{=50}{\text{.1 kJ/g CH}}_{\text{4}}{\text{ = 50100 J/g CH}}_{\text{4}}$

Explanation of Solution

For the release of $\text{50100 J}$ energy, the mass of ${\text{CH}}_{\text{4}}$ burned in the reaction is $\text{1 g}\text{.}$

The mass loss for the releases of $\text{50100 J}$ energy according to Einstein’s mass-energy equation is $\text{5}\text{.57 }\times {\text{10}}^{-10}\text{g}\text{.}$

Therefore, the ratio of mass of methane burned by producing $\text{50100 J}$ energy to the mass loss converted to $\text{50100 J}$ energy according to mass-energy equation is given as follows,

$\frac{\text{mass of methane burned}}{\text{mass loss}}=\frac{\text{1g}}{\text{5}{\text{.57 ×10}}^{\text{-10}}\text{g}\text{.}}=1.8\times {10}^9$

That is, the ratio is $1.8\times {10}^9$ to $1.$

(c)

Interpretation Introduction

Interpretation:

The applicability of Einstein’s equation for nuclear reaction than the chemical reaction has to be explained.

.

Concept Introduction:

Einstein’s mass-energy equivalence equation clearly depicts the relation between mass and energy. The mass-energy equivalence equation is as follows,

$\Delta \text{E = }\Delta {\text{m c}}^2$

Here,

    $\Delta \text{E }$ is energy equivalence

  $\Delta \text{m}$ is mass defect

  $\text{c}$ is speed of light = $3.00\times {\text{10}}^8\text{m/s}$

Unit of mass and speed should be taken in SI unit in order to convert energy in terms of joule.

$\text{1J = 1 }\frac{{\text{kg m}}^{\text{2}}}{{\text{s}}^{\text{2}}}$

From the equation it is clear that, anything that has mass will also have energy.

Explanation of Solution

In a chemical reaction, the amount of energy released for each gram of reactant is very much smaller than the energy released in a nuclear reaction for each gram of reactant.

So, in a chemical reaction only very less amount of mass is converted into energy whereas in nuclear reaction the mass converted will be high corresponds to the higher energy produced. Therefore the Einstein’s equation is more applicable to nuclear reaction.