Chapter 13, Problem 62A

(a)

Interpretation Introduction

Interpretation:

For the given diagram of helium and neon gases, which gas has greater initial pressure and how much pressure it has, is to be identified.

Concept Introduction :

• The ideal gases are those gases which obey the ideal gas equation and the ideal gas equation is written as

  $\text{PV}\text{= nRT}$

Where, P = pressure of the gas

V = Volume of the gas

R = Gas constant

T = temperature of the gas

• Boyle’s Law gives the inverse relation of volume with pressure at constant temperature and number of moles, such that

  ${\text{P}}_{\text{final}}{\text{V}}_{\text{final}}\text{=}{\text{P}}_{\text{initial}}{\text{V}}_{\text{initial}}$

Answer to Problem 62A

The initial pressure of helium gas is 1.5 times greater than the initial pressure of neon gas.

Explanation of Solution

Given information:

The number of molecules of helium gas = $\text{6}$

The number of molecules of neon gas = 4

Calculation:

The relation between pressure and number of moles of gases at the same temperature and volume is

  $\begin{array}{l}\frac{{\text{P}}_{\text{He}}}{{\text{n}}_{\text{He}}}\text{=}\frac{{\text{P}}_{\text{Ne}}}{{\text{n}}_{\text{Ne}}}\\ \frac{{\text{P}}_{\text{He}}}{{\text{P}}_{\text{Ne}}}\text{=}\frac{{\text{n}}_{\text{He}}}{{\text{n}}_{\text{Ne}}}\\ \frac{{\text{P}}_{\text{He}}}{{\text{P}}_{\text{Ne}}}\text{=}\frac{\text{6}}{\text{4}}\\ \frac{{\text{P}}_{\text{He}}}{{\text{P}}_{\text{Ne}}}\text{=}\frac{\text{3}}{\text{2}}\\ {\text{P}}_{\text{He}}\text{=}\text{1}\text{.5}{\text{P}}_{{\text{P}}_{\text{Ne}}}\end{array}$

So, the initial pressure of helium gas is greater than the initial pressure of neon gas.

(b)

Interpretation Introduction

For the diagram of helium and neon gases, the diagram will looks after opened the stopcock

By neglecting the volume of the connecting tube is to be drawn.

Interpretation:

Concept Introduction :

The ideal gases are those gases which obey the ideal gas equation and the ideal gas equation is written as

  $\text{PV}\text{= nRT}$

Where, P = pressure of the gas

V = Volume of the gas

R = Gas constant

T = temperature of the gas

• The inverse relation of volume with pressure at constant temperature and number of moles, such that

  ${\text{P}}_{\text{final}}{\text{V}}_{\text{final}}\text{=}{\text{P}}_{\text{initial}}{\text{V}}_{\text{initial}}$

Answer to Problem 62A

The diagram after opened the stopcock is

  

Explanation of Solution

When the valve is opened, due to the pressure differences some moles of neon gas are moving from the right side bulb to the left side bulb.

(c)

Interpretation Introduction

Interpretation:

For the given diagram of helium and neon gases, the final pressure in terms of the original pressure of helium and neon gases is to be determined.

Concept Introduction :

• The ideal gases are those gases which obey the ideal gas equation and the ideal gas equation is written as

  $\text{PV}\text{= nRT}$

Where, P = pressure of the gas

V = Volume of the gas

R = Gas constant

T = temperature of the gas

• Boyle’s Law gives the inverse relation of volume with pressure at constant temperature and number of moles, such that

  ${\text{P}}_{\text{final}}{\text{V}}_{\text{final}}\text{=}{\text{P}}_{\text{initial}}{\text{V}}_{\text{initial}}$

Answer to Problem 62A

The final pressure in terms of the original pressures of helium and neon at constant temperature is ${\text{P}}_{\text{total}}\text{=}\frac{\left({\text{P}}_{\text{He}}\text{+}{\text{P}}_{\text{Ne}}\right)}{\text{2}}$ .

Explanation of Solution

Given information:

The number of molecules of helium gas = $\text{6}$

The number of molecules of neon gas = 4

The volume of each bulb = $\text{x}$

Calculation:

Total volume = $\text{x}\text{+}\text{x}\text{=}\text{2x}$

  $\begin{array}{l}{\text{P}}_{\text{total}}{\text{V}}_{\text{total}}\text{=}{\text{P}}_{\text{1}}{\text{V}}_{\text{1}}\text{+}{\text{P}}_{\text{2}}{\text{V}}_{\text{2}}\\ \text{=}{\text{P}}_{\text{He}}\text{X+}{\text{P}}_{\text{Ne}}\text{X}\\ {\text{P}}_{\text{total}}\left(\text{X+X}\right)\text{=}\text{X}\left({\text{P}}_{\text{He}}{\text{+P}}_{\text{Ne}}\right)\\ {\text{P}}_{\text{total}}\left(\text{2X}\right)\text{=}\text{X}\left({\text{P}}_{\text{He}}{\text{+P}}_{\text{Ne}}\right)\\ {\text{P}}_{\text{total}}\text{=}\frac{\left({\text{P}}_{\text{He}}{\text{+P}}_{\text{Ne}}\right)}{\text{2}}\end{array}$

The final pressure in terms of original pressure of neon gas

  $\begin{array}{l}{\text{P}}_{\text{total}}\text{=}\frac{\left({\text{P}}_{\text{He}}\text{+}{\text{P}}_{\text{Ne}}\right)}{\text{2}}\\ \text{but,}{\text{P}}_{\text{He}}\text{=}\text{1}\text{.5}{\text{P}}_{\text{Ne}}\\ \therefore {\text{P}}_{\text{total}}\text{=}\frac{\left(\text{1}\text{.5}{\text{P}}_{\text{Ne}}\text{+}{\text{P}}_{\text{Ne}}\right)}{\text{2}}\\ \text{=}\frac{\text{2}\text{.5}{\text{P}}_{\text{Ne}}}{\text{2}}\\ {\text{P}}_{\text{total}}\text{=}\text{1}{\text{.25P}}_{\text{Ne}}\end{array}$

The final pressure in terms of original pressure of helium gas is

  $\begin{array}{l}{\text{P}}_{\text{total}}\text{=}\frac{\left({\text{P}}_{\text{He}}\text{+}{\text{P}}_{\text{Ne}}\right)}{\text{2}}\\ \text{but,}{\text{P}}_{\text{He}}\text{=}\text{1}\text{.5}{\text{P}}_{\text{Ne}}\text{Þ}{\text{P}}_{\text{Ne}}\text{=}\frac{{\text{P}}_{\text{He}}}{\text{1}\text{.5}}\text{=0}{\text{.66P}}_{\text{He}}\\ \therefore {\text{P}}_{\text{total}}\text{=}\frac{\left({\text{P}}_{\text{He}}\text{+}{\text{P}}_{\text{Ne}}\right)}{\text{2}}\\ {\text{P}}_{\text{total}}\text{=}\frac{\left({\text{P}}_{\text{He}}\text{+}\text{0}{\text{.66P}}_{\text{He}}\right)}{\text{2}}\\ \text{=}\frac{\text{1}{\text{.66P}}_{\text{He}}}{\text{2}}\\ {\text{P}}_{\text{total}}\text{=}\text{0}{\text{.86P}}_{\text{He}}\\ \\ \\ \end{array}$

(d)

Interpretation Introduction

Interpretation:

For the given diagram of helium and neon gases, the final partial pressure of each gas in terms of the final pressure of the system is to be determined.

Concept Introduction :

• The ideal gases are those gases which obey the ideal gas equation and the ideal gas equation is written as

  $\text{PV}\text{= nRT}$

Where, P = pressure of the gas

V = Volume of the gas

R = Gas constant

T = temperature of the gas

• Boyle’s Law gives the inverse relation of volume with pressure at constant temperature and number of moles, such that

  ${\text{P}}_{\text{final}}{\text{V}}_{\text{final}}\text{=}{\text{P}}_{\text{initial}}{\text{V}}_{\text{initial}}$

Answer to Problem 62A

The initial pressure of helium gas is 1.5 times greater than the initial pressure of neon gas.

Explanation of Solution

Given information:

The Initial number of molecules of helium gas = $\text{6}$

The Initial number of molecules of neon gas = 4

Calculation:

The final partial pressure of helium gas in terms of final pressure

  $\begin{array}{l}{\text{P}}_{\text{total}}\text{=}\frac{\left({\text{P}}_{\text{He}}\text{+}{\text{P}}_{\text{Ne}}\right)}{\text{2}}\\ \text{but,}{\text{P}}_{\text{He}}\text{=}\text{1}\text{.5}{\text{P}}_{\text{Ne}}\text{Þ}{\text{P}}_{\text{Ne}}\text{=}\frac{{\text{P}}_{\text{He}}}{\text{1}\text{.5}}\text{=0}{\text{.66P}}_{\text{He}}\\ \therefore {\text{P}}_{\text{total}}\text{=}\frac{\left({\text{P}}_{\text{He}}\text{+}{\text{P}}_{\text{Ne}}\right)}{\text{2}}\\ {\text{P}}_{\text{total}}\text{=}\frac{\left({\text{P}}_{\text{He}}\text{+}\text{0}{\text{.66P}}_{\text{He}}\right)}{\text{2}}\\ \text{=}\frac{\text{1}{\text{.66P}}_{\text{He}}}{\text{2}}\\ {\text{P}}_{\text{total}}\text{=}\text{0}{\text{.86P}}_{\text{He}}\\ \text{By}\text{rearranging}\text{the}\text{above}\text{equation,}\\ {\text{P}}_{\text{He}}\text{=}\frac{\text{5}}{\text{6}}{\text{P}}_{\text{total}}\end{array}$

The final partial pressure of neon gas in terms of final pressure of the system

  $\begin{array}{l}{\text{P}}_{\text{total}}\text{=}\frac{\left({\text{P}}_{\text{He}}\text{+}{\text{P}}_{\text{Ne}}\right)}{\text{2}}\\ \text{but,}{\text{P}}_{\text{He}}\text{=}\text{1}\text{.5}{\text{P}}_{\text{Ne}}\\ \therefore {\text{ P}}_{\text{total}}\text{=}\frac{\left(\text{1}\text{.5}{\text{P}}_{\text{Ne}}\text{+}{\text{P}}_{\text{Ne}}\right)}{\text{2}}\\ \text{=}\frac{\text{2}\text{.5}{\text{P}}_{\text{Ne}}}{\text{2}}\\ {\text{P}}_{\text{total}}\text{=}\text{1}{\text{.25P}}_{\text{Ne}}\\ \text{By}\text{reaaranging}\text{this}\text{arrangment}\\ \Rightarrow {\text{P}}_{\text{Ne}}\text{=}\frac{\text{4}}{\text{5}}{\text{P}}_{\text{total}}\end{array}$