Chapter 13, Problem 76A

(a)

Interpretation Introduction

Interpretation:

The Avogadro law is to be derived.

Concept Introduction :

According to Avogadro law, the volume of a gas is directly related to the number of moles of gas at constant temperature and pressure.

Answer to Problem 76A

According to the ideal gas equation

  $PV=nRT$

By rearranging the above equation

  $\begin{array}{l}\frac{V}{n}=\text{}\frac{RT}{P}\\ \frac{V}{n}\text{=}\text{constant}\end{array}$

When temperature and pressure kept constant, then the value of $\text{}\frac{RT}{P}$ becomes constant

It holds when temperature and pressure are held constant then the ratio of volume of the gas particles to the number of moles of the gas becomes constant.

Explanation of Solution

According to the ideal gas equation,

  $PV=nRT$

Where, P = is the pressure exerted by the gas on the walls of the container

V = is the volume of the gas

n = number of moles

R = gas constant

T = absolute temperature of the gas

By rearranging the equation

  $\begin{array}{l}\frac{V}{n}=\text{}\frac{RT}{P}\\ \frac{V}{n}\text{=}\text{constant}\end{array}$

When temperature and pressure kept constant, then the value of $\text{}\frac{RT}{P}$ becomes constant

Hence, the volume of the gas particles directly varies with the number of moles of gas at constant pressure and temperature.

(b)

Interpretation Introduction

Interpretation:

The Avogadro law based on ideas of the kinetic molecular theory is to be explained.

Concept Introduction :

On the basis of the kinetic molecular theory, the Avogadro law is states that the different gases of equal volumes contain an equal number of molecules under the same temperature and pressure conditions.

Answer to Problem 76A

According to this law based on the kinetic molecular theory, the different gases of equal volumes contain the equal number of molecules under the same condition of temperature and pressure.

Explanation of Solution

The equal volumes of different gases contain an equal number of moles of molecules under constant temperature and pressure.

This relation between the volume and number of moles of the gases is derived from the kinetic molecular theory on the bases of the ideal gas. As the pressure and temperature of a gas are decreased, this is applicable for real gas also.

One mole of each atom contains $\text{6}{\text{.023×10}}^{\text{23}}$ number of atoms. This is the Avogadro law.

(c)

Interpretation Introduction

Interpretation:

The gas with greater volume under the same temperature, and pressure needs to be identified.

Concept Introduction :

Under the same pressure and temperature the volume of the gas is directly depends on the number of moles of the particles.

According to Avogadro Law,

  $\frac{V_{\text{He}}}{n_{\text{He}}}\text{=}\frac{V_{\text{Ar}}}{n_{\text{Ar}}}\text{=}\frac{RT}{P}$

Answer to Problem 76A

The volume of helium gas is greater than as compared to the volume of argon gas under the same temperature and pressure.

Explanation of Solution

Given information:

Mass of helium gas = $\text{10}\text{.0}\text{g}$

Mass of the argon gas= $\text{20g}$

The Molar mass of helium gas = $\text{4}\text{.002}\text{g}$

The Molar mass of argon gas = $\text{18}$

Calculation:

Then, the number of moles of helium gas

  $\begin{array}{c}{n}_{\text{He}}\text{=}\frac{\text{mass}}{\text{molar}\text{mass}}\end{array}$

   $\begin{array}{c}\text{=}\frac{\text{10}}{\text{4}}\text{=}\text{2}\text{.5}\text{mole}\end{array}$

Number of moles of argon gas

  $\begin{array}{c}{n}_{\text{Ar}}\text{=}\frac{\text{mass}}{\text{molar}\text{mass}}\end{array}$

   $\begin{array}{c}\text{=}\frac{\text{20}}{\text{18}}\text{=}\text{2}\text{.5}\text{mole}\end{array}$

Now, according to Avogadro Law,

  $\begin{array}{c}\frac{V_{\text{He}}}{n_{\text{He}}}=\frac{V_{\text{Ar}}}{n_{\text{Ar}}}=\frac{RT}{P}\\ \frac{V_{\text{He}}}{V_{\text{Ar}}}=\frac{n_{\text{He}}}{n_{\text{Ar}}}=\frac{RT}{P}\end{array}$

$\begin{array}{c}or,\frac{V_{\text{He}}}{V_{\text{Ar}}}=\frac{n_{\text{He}}}{n_{\text{Ar}}}\end{array}$

Now,

  $n_{\text{He}}>n_{\text{Ar}}$

Hence,

  $V_{\text{He}}>V_{\text{Ar}}$