Chapter 5, Problem 5.54P

(a)

Interpretation Introduction

Interpretation: The pressure of three 5 L flasks; A, B and C containing 4 g of each ${\text{H}}_{\text{2}}$ , He and ${\text{CH}}_{\text{4}}$ with pressure gauges and small valves needs to be determined.

Concept Introduction:

The ideal gas equation can be used to calculate P, V, T and number of moles of gases. It is basically the combination of different gas laws such as Boyle’s law, Charles’s law, Gay-Lussac law and Avogadro law. The ideal gas equation can be written as:

  $\begin{array}{l}\text{P}\times \text{V = n}\times \text{R}\times \text{T}\\ \text{P = pressure }\\ \text{V = volume }\\ \text{n = number of moles }\\ \text{R = gas constant }\\ \text{T = temperature }\end{array}$

Answer to Problem 5.54P

The order of pressure is as follows:

  ${\text{CH}}_{\text{4}}$ (C) < He (B) < ${\text{H}}_{\text{2}}$ (A)

Explanation of Solution

From the ideal gas equation:

  $\begin{array}{l}\text{P×V = n×R×T}\\ \text{P×V = }\frac{\text{mass}}{\text{molar mass}}\text{×R×T }\\ \text{molar mass}=\frac{\text{mass}}{\text{P×V}}\text{×R×T }\end{array}$

Hence at constant volume and number of moles, as the molar mass increases, the pressure of gas decreases as pressure is inversely proportional to the molar mass. Since the molar mass of:

  ${\text{H}}_{\text{2}}$ = 2.001 g/mol

  ${\text{CH}}_{\text{4}}$ = 16.0 g/mol

He = 4.00 g/mol

Hence the order of pressure must be:

  ${\text{CH}}_{\text{4}}$ (C) < He (B) < ${\text{H}}_{\text{2}}$ (A)

(b)

Interpretation Introduction

Interpretation: The average molecular kinetic energy of three 5 L flasks; A, B and C containing 4 g of each ${\text{H}}_{\text{2}}$ , He and ${\text{CH}}_{\text{4}}$ with pressure gauges and small valves needs to be determined.

Concept Introduction:

The ideal gas equation can be used to calculate P, V, T and number of moles of gases. It is basically the combination of different gas laws such as Boyle’s law, Charles’s law, Gay-Lussac law and Avogadro law. The ideal gas equation can be written as:

  $\begin{array}{l}\text{P}\times \text{V = n}\times \text{R}\times \text{T}\\ \text{P = pressure }\\ \text{V = volume }\\ \text{n = number of moles }\\ \text{R = gas constant }\\ \text{T = temperature }\end{array}$

Answer to Problem 5.54P

The order of average molecular kinetic energy must be:

  ${\text{CH}}_{\text{4}}$ (C) = He (B) = ${\text{H}}_{\text{2}}$ (A)

Explanation of Solution

According to the kinetic energy formula, the kinetic energy is directly proportional to the velocity and mass.

  $\text{K}\text{.E}{\text{.}}_{\text{av}}\text{ = }\frac{\text{3}}{\text{2}}\text{RT}$

Since all gases are taken 4 g therefore they will have same average molecular kinetic energy.

  ${\text{CH}}_{\text{4}}$ (C) = He (B) = ${\text{H}}_{\text{2}}$ (A)

(c)

Interpretation Introduction

Interpretation: The diffusion rate after the valve is opened of three 5 L flasks; A, B and C containing 4 g of each ${\text{H}}_{\text{2}}$ , He and ${\text{CH}}_{\text{4}}$ with pressure gauges and small valves needs to be determined.

Concept Introduction:

The ideal gas equation can be used to calculate P, V, T and number of moles of gases. It is basically the combination of different gas laws such as Boyle’s law, Charles’s law, Gay-Lussac law and Avogadro law. The ideal gas equation can be written as:

  $\begin{array}{l}\text{P}\times \text{V = n}\times \text{R}\times \text{T}\\ \text{P = pressure }\\ \text{V = volume }\\ \text{n = number of moles }\\ \text{R = gas constant }\\ \text{T = temperature }\end{array}$

Answer to Problem 5.54P

The order of rate of diffusionmust be:

  ${\text{ Rate}}_{{\text{CH}}_{\text{4}}\left(\text{C}\right)}{\text{ < Rate}}_{\text{He }\left(\text{B}\right)}{\text{< Rate}}_{{\text{H}}_{\text{2}}\left(\text{A}\right)}$

Explanation of Solution

According to the Graham's Law, the rate of diffusion of gases inversely proportional to root square of molar mass of gases.

  $\text{Rate of diffusion α}\frac{\text{1}}{\sqrt{\text{molar mass}}}$

Since the molar mass of:

• ${\text{H}}_{\text{2}}$ = 2.001 g/mol
• ${\text{CH}}_{\text{4}}$ = 16.0 g/mol
• He = 4.00 g/mol

Hence the order of rate of diffusion must be:

  ${\text{ Rate}}_{{\text{CH}}_{\text{4}}\left(\text{C}\right)}{\text{ < Rate}}_{\text{He }\left(\text{B}\right)}{\text{< Rate}}_{{\text{H}}_{\text{2}}\left(\text{A}\right)}$

(d)

Interpretation Introduction

Interpretation: The total molecular kinetic energy of three 5 L flasks; A, B and C containing 4 g of each ${\text{H}}_{\text{2}}$ , He and ${\text{CH}}_{\text{4}}$ with pressure gauges and small valves needs to be determined.

Concept Introduction:

The ideal gas equation can be used to calculate P, V, T and number of moles of gases. It is basically the combination of different gas laws such as Boyle’s law, Charles’s law, Gay-Lussac law and Avogadro law. The ideal gas equation can be written as:

  $\begin{array}{l}\text{P}\times \text{V = n}\times \text{R}\times \text{T}\\ \text{P = pressure }\\ \text{V = volume }\\ \text{n = number of moles }\\ \text{R = gas constant }\\ \text{T = temperature }\end{array}$

Answer to Problem 5.54P

The order of total molecular kinetic energy must be:

  ${\text{ E}}_{{\text{CH}}_{\text{4}}\left(\text{C}\right)}{\text{ < E}}_{\text{He }\left(\text{B}\right)}{\text{< E}}_{{\text{H}}_{\text{2}}\left(\text{A}\right)}$

Explanation of Solution

According to the kinetic energy formula, the kinetic energy is directly proportional to the velocity and mass.

  $\text{K}\text{.E = }\frac{\text{1}}{\text{2}}{\text{mv}}^{\text{2}}$

Here velocity is inversely proportional to molar mass, hence gas with less molar mass will have more velocity and more total kinetic energy. The order of total kinetic energy must be:

  ${\text{ E}}_{{\text{CH}}_{\text{4}}\left(\text{C}\right)}{\text{ < E}}_{\text{He }\left(\text{B}\right)}{\text{< E}}_{{\text{H}}_{\text{2}}\left(\text{A}\right)}$

(e)

Interpretation Introduction

Interpretation: The density of gases in three 5 L flasks; A, B and C containing 4 g of each ${\text{H}}_{\text{2}}$ , He and ${\text{CH}}_{\text{4}}$ with pressure gauges and small valves needs to be determined.

Concept Introduction:

Density is calculated from mass and volume as follows:

  $d=\frac{m}{V}$

Here, m is mass and V is volume.

Answer to Problem 5.54P

The order of density must be:

  ${\text{d}}_{{\text{H}}_{\text{2}}}{\text{= d}}_{\text{He }}{\text{= d}}_{{\text{CH}}_{\text{4}}}$

Explanation of Solution

The relation between density, mass and volume is as follows:

  $d=\frac{m}{V}$

Here, m is mass and V is volume.

Since, mass and volume of each gas is 4 g and 5 L respectively thus, density is also equal for each gas.

Therefore,

  ${\text{d}}_{{\text{H}}_{\text{2}}}{\text{= d}}_{\text{He }}{\text{= d}}_{{\text{CH}}_{\text{4}}}$

(f)

Interpretation Introduction

Interpretation: The collision frequency of molecules in three 5 L flasks; A, B and C containing 4 g of each ${\text{H}}_{\text{2}}$ , He and ${\text{CH}}_{\text{4}}$ with pressure gauges and small valves needs to be determined.

Concept Introduction:

The ideal gas equation can be used to calculate P, V, T and number of moles of gases. It is basically the combination of different gas laws such as Boyle’s law, Charles’s law, Gay-Lussac law and Avogadro law. The ideal gas equation can be written as:

  $\begin{array}{l}\text{P}\times \text{V = n}\times \text{R}\times \text{T}\\ \text{P = pressure }\\ \text{V = volume }\\ \text{n = number of moles }\\ \text{R = gas constant }\\ \text{T = temperature }\end{array}$

Answer to Problem 5.54P

The order of collision frequency of molecules must be:

  ${\upsilon }_{{\text{CH}}_{\text{4}}\left(\text{C}\right)}\text{ < }{\upsilon }_{\text{He }\left(\text{B}\right)}\text{< }{\upsilon }_{{\text{H}}_{\text{2}}\left(\text{A}\right)}$

Explanation of Solution

According to the kinetic energy formula, the kinetic energy is directly proportional to the velocity and mass.

  $\text{K}\text{.E = }\frac{\text{1}}{\text{2}}{\text{mv}}^{\text{2}}$

Here, velocity is inversely proportional to molar mass, hence the collision frequency also increases with less molar mass. The order of collision frequency of molecules must be:

  ${\upsilon }_{{\text{CH}}_{\text{4}}\left(\text{C}\right)}\text{ < }{\upsilon }_{\text{He }\left(\text{B}\right)}\text{< }{\upsilon }_{{\text{H}}_{\text{2}}\left(\text{A}\right)}$