Chapter 5, Problem 86E

(a)

Interpretation Introduction

Interpretation:

Whether the kinetic energy, root mean square velocity, frequency of collisions of gas molecules with each other, frequency of collisions of gas molecules with the walls of container and the energy of impact of gas molecules with the container increases, decreases or remain same when the temperature is increased to $100°\text{C}$ .

Concept Introduction:

The energy which is linked with the motion of a body is known as kinetic energy. According to kinetic theory of gases, the average kinetic energy of a gas is calculated by following expression:

  $E_k=\frac{1}{2}m{u}^2$

Where, $E_k$ = average kinetic energy

m = molecular mass and u = average velocity

Another expression of average kinetic energy is:

  $E_K=\frac{3R}{2N_A}T$

Where, R = universal gas constant and N_A = Avogadro number

Average kinetic energy depends upon the temperature.

The expression of root mean square velocity is:

  $u=\sqrt{\frac{3RT}{M}}$

From above expression, the relationship between molar mass and average velocity of gas is:

  $u\propto \sqrt{\frac{1}{M}}$

Average velocity is inversely proportional to the square root of molar mass

The process which refers to the movement of gaseous molecules through a small hole with velocity u is known as effusion. Rate of effusion is defined as the rate at which the gaseous molecules effuse out.

Thus, the expression is given as:

Rate of effusion $\propto \sqrt{\frac{1}{M}}$

The expression for mole fraction:

  $X_A=\frac{n_A}{n_{Total}}$

Where, $X_A$ = Mole fraction of gas A

  $n_A$ = number of moles of gas A

  $n_{Total}$ = Total number of moles

Explanation of Solution

The expression of average kinetic energy is:

  $E_K=\frac{3R}{2N_A}T$

The expression of root mean square velocity is:

  $u=\sqrt{\frac{3RT}{M}}$

When the temperature increases, the average kinetic energy is also increases as the average kinetic energy is directly related with the temperature.

When the temperature increases, the root mean square velocity is also increases as the root mean square velocity is directly related with temperature.

The molecules are moving faster due to which the frequency of collisions of gas molecules with each other increases because the frequency of collisions is directly related with square root of temperature.

Similarly, the molecules are moving faster due to which the frequency of collisions of gas molecules with the walls increases because the frequency of collisions is directly related with square root of temperature in kelvin.

  $Z\alpha \sqrt{T}$

Due to increase in velocity of the molecules, the energy of impact of the gas molecule also increases.

(b)

Interpretation Introduction

Interpretation:

Whether the kinetic energy, root mean square velocity, frequency of collisions of gas molecules with each other, frequency of collisions of gas molecules with the walls of container and the energy of impact of gas molecules with the container increases, decreases or remain same when the temperature is decreased to $-50°\text{C}$ .

Concept Introduction:

The energy which is linked with the motion of a body is known as kinetic energy. According to kinetic theory of gases, the average kinetic energy of a gas is calculated by following expression:

  $E_k=\frac{1}{2}m{u}^2$

Where, $E_k$ = average kinetic energy

m = molecular mass and u = average velocity

Another expression of average kinetic energy is:

  $E_K=\frac{3R}{2N_A}T$

Where, R = universal gas constant and N_A = Avogadro number

Average kinetic energy depends upon the temperature.

The expression of root mean square velocity is:

  $u=\sqrt{\frac{3RT}{M}}$

From above expression, the relationship between molar mass and average velocity of gas is:

  $u\propto \sqrt{\frac{1}{M}}$

Average velocity is inversely proportional to the square root of molar mass

The process which refers to the movement of gaseous molecules through a small hole with velocity u is known as effusion. Rate of effusion is defined as the rate at which the gaseous molecules effuse out.

Thus, the expression is given as:

Rate of effusion $\propto \sqrt{\frac{1}{M}}$

The expression for mole fraction:

  $X_A=\frac{n_A}{n_{Total}}$

Where, $X_A$ = Mole fraction of gas A

  $n_A$ = number of moles of gas A

  $n_{Total}$ = Total number of moles

Explanation of Solution

The expression of average kinetic energy is:

  $E_K=\frac{3R}{2N_A}T$

The expression of root mean square velocity is:

  $u=\sqrt{\frac{3RT}{M}}$

When the temperature decreases, the average kinetic energy is also decreases as the average kinetic energy is directly related with the temperature.

When the temperature decreases, the root mean square velocity is also decreases as the root mean square velocity is directly related with temperature.

The molecules are moving slowly due to which the frequency of collisions of gas molecules with each other decreases because the frequency of collisions is directly related with square root of temperature.

Similarly, the molecules are moving slowly due to which the frequency of collisions of gas molecules with the walls decreases because the frequency of collisions is directly related with square root of temperature in kelvin.

  $Z\alpha \sqrt{T}$

Due to decrease in velocity of the molecules, the energy of impact of the gas molecule also decreases.

(c)

Interpretation Introduction

Interpretation:

Whether the kinetic energy, root mean square velocity, frequency of collisions of gas molecules with each other, frequency of collisions of gas molecules with the walls of container and the energy of impact of gas molecules with the container increases, decreases or remain same when the volume is decreased to $0.5\text{ L}$ .

Concept Introduction:

The energy which is linked with the motion of a body is known as kinetic energy. According to kinetic theory of gases, the average kinetic energy of a gas is calculated by following expression:

  $E_k=\frac{1}{2}m{u}^2$

Where, $E_k$ = average kinetic energy

m = molecular mass and u = average velocity

Another expression of average kinetic energy is:

  $E_K=\frac{3R}{2N_A}T$

Where, R = universal gas constant and N_A = Avogadro number

Average kinetic energy depends upon the temperature.

The expression of root mean square velocity is:

  $u=\sqrt{\frac{3RT}{M}}$

From above expression, the relationship between molar mass and average velocity of gas is:

  $u\propto \sqrt{\frac{1}{M}}$

Average velocity is inversely proportional to the square root of molar mass

The process which refers to the movement of gaseous molecules through a small hole with velocity u is known as effusion. Rate of effusion is defined as the rate at which the gaseous molecules effuse out.

Thus, the expression is given as:

Rate of effusion $\propto \sqrt{\frac{1}{M}}$

The expression for mole fraction:

  $X_A=\frac{n_A}{n_{Total}}$

Where, $X_A$ = Mole fraction of gas A

  $n_A$ = number of moles of gas A

  $n_{Total}$ = Total number of moles

Explanation of Solution

The expression of average kinetic energy is:

  $E_K=\frac{3R}{2N_A}T$

The expression of root mean square velocity is:

  $u=\sqrt{\frac{3RT}{M}}$

Both average kinetic energy and root mean square velocity depends upon temperature not on volume. By considering the temperature constant, there is no change takes place in average kinetic energy and root mean square velocity.

The frequency of collisions of gas molecules with each other increases because the frequency of collisions is inversely related with the volume.

  $Z\alpha \frac{1}{V}$

Similarly, the frequency of collisions of gas molecules with the walls increases because the frequency of collisions is inversely related with the volume.

  $Z\alpha \frac{1}{V}$

The energy of collisions doesn’t change as the molecules with same velocity at constant temperature.

(d)

Interpretation Introduction

Interpretation:

Whether the kinetic energy, root mean square velocity, frequency of collisions of gas molecules with each other, frequency of collisions of gas molecules with the walls of container and the energy of impact of gas molecules with the container increases, decreases or remain same when the number of moles of neon is doubled.

Concept Introduction:

The energy which is linked with the motion of a body is known as kinetic energy. According to kinetic theory of gases, the average kinetic energy of a gas is calculated by following expression:

  $E_k=\frac{1}{2}m{u}^2$

Where, $E_k$ = average kinetic energy

m = molecular mass and u = average velocity

Another expression of average kinetic energy is:

  $E_K=\frac{3R}{2N_A}T$

Where, R = universal gas constant and N_A = Avogadro number

Average kinetic energy depends upon the temperature.

The expression of root mean square velocity is:

  $u=\sqrt{\frac{3RT}{M}}$

From above expression, the relationship between molar mass and average velocity of gas is:

  $u\propto \sqrt{\frac{1}{M}}$

Average velocity is inversely proportional to the square root of molar mass

The process which refers to the movement of gaseous molecules through a small hole with velocity u is known as effusion. Rate of effusion is defined as the rate at which the gaseous molecules effuse out.

Thus, the expression is given as:

Rate of effusion $\propto \sqrt{\frac{1}{M}}$

The expression for mole fraction:

  $X_A=\frac{n_A}{n_{Total}}$

Where, $X_A$ = Mole fraction of gas A

  $n_A$ = number of moles of gas A

  $n_{Total}$ = Total number of moles

Explanation of Solution

The expression of average kinetic energy is:

  $E_K=\frac{3R}{2N_A}T$

The expression of root mean square velocity is:

  $u=\sqrt{\frac{3RT}{M}}$

Both average kinetic energy and root mean square velocity depends upon temperature not on number of moles. By considering the temperature constant, there is no change takes place in average kinetic energy and root mean square velocity.

The frequency of collisions of gas molecules with each other increases because the frequency of collisions is directly related with the number of moles.

  $Z\alpha N$

Similarly, the frequency of collisions of gas molecules with the walls increases because the frequency of collisions is directly related with the number of moles.

  $Z\alpha N$

The energy of collisions doesn’t change as the molecules with same velocity at constant temperature.