Chapter 5, Problem 5.15P

What is the effect of the following on the volume of 1 mol of an ideal gas?

(a) The pressure is reduced by a factor of 4 (at constant T).

(b) The pressure changes from 760 torr to 202 kPa, and the temperature changes from 37°C to 155 K.

(c) The temperature changes from 305 K to 32°C, and the presure changes from 2 atm to 101 kPa.

Interpretation Introduction

Interpretation:

The effect on the volume of 1 mole of an ideal gas should be determined when the pressure is decreased by factor of four at constant temperature.

Concept Introduction:

Boyle's Law gives the relationship between Pressure (P) and Volume (V).

According to Boyle's Law, the volume of gas changes inversely with the pressure of the gas if temperature and amount of a gas are constant.

PV = constant

  $\text{Pα}\frac{\text{1}}{\text{V}}$

The pressure of a gas decreases with increase in volume; volume of a gas decreases with increase in pressure.

Charles’s Law gives the relationship between Volume (V) and Temperature (T)

According to Charles’s Law, the volume of gas has direct relationship with temperature of the gas if pressure and amount of a gas are constant.

  $\frac{\text{V}}{\text{T}}$ = constant

  $\text{VαT}$

If the temperature or volume of a gas changes without any change in amount of a gas and pressure, then the final volume and temperature will give the same $\frac{\text{V}}{\text{T}}$ as the initial volume and temperature. Then, a relationship between initial and final $\frac{\text{V}}{\text{T}}$ can be set as equal to each other.

Charles’s Law can be written as:

  $\frac{{\text{V}}_{\text{1}}}{{\text{T}}_{\text{1}}}\text{=}\frac{{\text{V}}_{\text{2}}}{{\text{T}}_{\text{2}}}$ (Pressure and amount of a gas remain constant)

Where, T₁ and V₁ are the initial temperature and volume.

T₂ and V₂ are the final temperature and volume.

Avogadro's Law:

At same condition of pressure and temperature, equal volume of gases has same number of moles. In other words, at same temperature and pressure; one mole of a gas has the same volume.

According to Avogadro's Law, at STP, 1 mole of a gas consist of $6.02\times {10}^{23}$ occupies 22.4 L volume.

The mathematical expression is given as:

  $\frac{V}{n}=\text{constant}(\text{pressure, temperature fixed)}$

Amonton's Law:

The pressure of a gas is directly related with the absolute temperature at constant number of moles and volume.

The mathematical expression is given as:

  $P\alpha T$

Or,

  $\frac{P}{T}=\text{constant (Volume, number of moles fixed)}$

Answer to Problem 5.15P

At constant temperature, the volume of one mole of a gas is four times the initial volume when the pressure of a gas is reduced by factor four.

Explanation of Solution

Ideal gas law gives the relation between pressure, volume, number of moles and temperature.

The ideal gas law is:

  $\text{PV= nRT}$

Where,

P = Pressure

V = Volume

n = Number of moles

R = Universal gas constant ( $0.0821\text{ atm}\cdot \text{L/mol}\cdot \text{K}$ )

T = Temperature

The new ideal expression is shown below, when the pressure is decreased by factor 4 at constant temperature.

  $\frac{\text{P}}{4}{\text{V}}^{\text{'}}\text{= nRT}$

Now, the new volume is calculated as:

  

Thus, new volume is:

  ${\text{V}}^{\text{'}}\text{= }4\text{V}$

Hence, at constant temperature, the volume of one mole of a gas is four times the initial volume when the pressure of a gas is reduced by factor four.

Interpretation Introduction

Interpretation:

The effect on the volume of 1 mole of an ideal gas should be determined when the pressure changes from 760 torr to 202 kPa and the temperature changes from $37°C$ to 155 K.

Concept Introduction:

Boyle's Law gives the relationship between Pressure (P) and Volume (V).

According to Boyle's Law, the volume of gas changes inversely with the pressure of the gas if temperature and amount of a gas are constant.

PV = constant

  $\text{Pα}\frac{\text{1}}{\text{V}}$

The pressure of a gas decreases with increase in volume; volume of a gas decreases with increase in pressure.

Charles’s Law gives the relationship between Volume (V) and Temperature (T)

According to Charles’s Law, the volume of gas has direct relationship with temperature of the gas if pressure and amount of a gas are constant.

  $\frac{\text{V}}{\text{T}}$ = constant

  $\text{VαT}$

If the temperature or volume of a gas changes without any change in amount of a gas and pressure, then the final volume and temperature will give the same $\frac{\text{V}}{\text{T}}$ as the initial volume and temperature. Then, a relationship between initial and final $\frac{\text{V}}{\text{T}}$ can be set as equal to each other.

Charles’s Law can be written as:

  $\frac{{\text{V}}_{\text{1}}}{{\text{T}}_{\text{1}}}\text{=}\frac{{\text{V}}_{\text{2}}}{{\text{T}}_{\text{2}}}$ (Pressure and amount of a gas remain constant)

Where, T₁ and V₁ are the initial temperature and volume.

T₂ and V₂ are the final temperature and volume.

Avogadro's Law:

At same condition of pressure and temperature, equal volume of gases has same number of moles. In other words, at same temperature and pressure; one mole of a gas has the same volume.

According to Avogadro's Law, at STP, 1 mole of a gas consist of $6.02\times {10}^{23}$ occupies 22.4 L volume.

The mathematical expression is given as:

  $\frac{V}{n}=\text{constant}(\text{pressure, temperature fixed)}$

Amonton's Law:

The pressure of a gas is directly related with the absolute temperature at constant number of moles and volume.

The mathematical expression is given as:

  $P\alpha T$

Or,

  $\frac{P}{T}=\text{constant (Volume, number of moles fixed)}$

Answer to Problem 5.15P

The final volume of a gas is the ¼ times of the initial volume of a gas.

Explanation of Solution

Ideal gas law gives the relation between pressure, volume, number of moles and temperature.

The ideal gas law is:

  $\text{PV= nRT}$

Where,

P = Pressure

V = Volume

n = Number of moles

R = Universal gas constant ( $0.0821\text{ atm}\cdot \text{L/mol}\cdot \text{K}$ )

T = Temperature

The ideal gas law for two given conditions is:

  $\frac{P_1{V}_1}{T_1{n}_1}=\frac{P_2{V}_2}{T_2{n}_2}$

Where, $P_1$ = 760 torr, $P_2$ = 202 kPa, $T_1$ = $37°C$ and $T_2$ = 155 K

  $n_1\text{ and }{n}_2\text{ = 1 mol}$

Convert the value of pressure in torr to kPa.

Since, 1 kPa = 7.50 torr

Thus, Pressure in kPa = $760\text{ torr}\times \frac{1\text{ kPa}}{7.50\text{ torr}}=101\text{ kPa}$

Convert the value of temperature in degree Celsius in Kelvin.

Temperature in K = $37°C+273=310\text{ K}$

Put the values,

  $\frac{101\text{ kPa}\times V_1}{310\text{ K}\times 1\text{ mol}}=\frac{202\text{ kPa}\times V_2}{155\text{ K}\times 1\text{ mol}}$

  $V_2=\frac{1}{4}{V}_1$

Thus, the final volume of a gas is¼ times of the initial volume of a gas.

Interpretation Introduction

Interpretation:

The effect on the volume of 1 mole of an ideal gas should be determined when Temperature changes from 305 K to $32°C$ and Pressure changes from 2 atm to 101 kPa.

Concept Introduction:

Boyle's Law gives the relationship between Pressure (P) and Volume (V).

According to Boyle's Law, the volume of gas changes inversely with the pressure of the gas if temperature and amount of a gas are constant.

PV = constant

  $\text{Pα}\frac{\text{1}}{\text{V}}$

The pressure of a gas decreases with increase in volume; volume of a gas decreases with increase in pressure.

Charles’s Law gives the relationship between Volume (V) and Temperature (T)

According to Charles’s Law, the volume of gas has direct relationship with temperature of the gas if pressure and amount of a gas are constant.

  $\frac{\text{V}}{\text{T}}$ = constant

  $\text{VαT}$

If the temperature or volume of a gas changes without any change in amount of a gas and pressure, then the final volume and temperature will give the same $\frac{\text{V}}{\text{T}}$ as the initial volume and temperature. Then, a relationship between initial and final $\frac{\text{V}}{\text{T}}$ can be set as equal to each other.

Charles’s Law can be written as:

  $\frac{{\text{V}}_{\text{1}}}{{\text{T}}_{\text{1}}}\text{=}\frac{{\text{V}}_{\text{2}}}{{\text{T}}_{\text{2}}}$ (Pressure and amount of a gas remain constant)

Where, T₁ and V₁ are the initial temperature and volume.

T₂ and V₂ are the final temperature and volume.

Avogadro's Law:

At same condition of pressure and temperature, equal volume of gases has same number of moles. In other words, at same temperature and pressure; one mole of a gas has the same volume.

According to Avogadro's Law, at STP, 1 mole of a gas consist of $6.02\times {10}^{23}$ occupies 22.4 L volume.

The mathematical expression is given as:

  $\frac{V}{n}=\text{constant}(\text{pressure, temperature fixed)}$

Amonton's Law:

The pressure of a gas is directly related with the absolute temperature at constant number of moles and volume.

The mathematical expression is given as:

  $P\alpha T$

Or,

  $\frac{P}{T}=\text{constant (Volume, number of moles fixed)}$

Answer to Problem 5.15P

The final volume of a gas is 2 times of the initial volume of a gas.

Explanation of Solution

Ideal gas law gives the relation between pressure, volume, number of moles and temperature.

The ideal gas law is:

  $\text{PV= nRT}$

Where,

P = Pressure

V = Volume

n = Number of moles

R = Universal gas constant ( $0.0821\text{ atm}\cdot \text{L/mol}\cdot \text{K}$ )

T = Temperature

The ideal gas law for two given conditions is:

  $\frac{P_1{V}_1}{T_1{n}_1}=\frac{P_2{V}_2}{T_2{n}_2}$

Where, $P_1$ = 2 atm, $P_2$ = 101 kPa, $T_1$ = 305 K and $T_2$ = $32°C$

  $n_1\text{ and }{n}_2\text{ = 1 mol}$

Convert the value of pressure in kPa to atm.

Since, 1 kPa = 0.00987 atm

Thus, Pressure in atm = $101\text{ kPa}\times \frac{0.00987\text{ atm}}{1\text{ kPa}}=1.0\text{ atm}$

Convert the value of temperature in degree Celsius in Kelvin.

Temperature in K = $32°C+273=305\text{ K}$

Put the values,

  $\frac{2\text{ atm}\times V_1}{305\text{ K}\times 1\text{ mol}}=\frac{1.0\text{ atm}\times V_2}{305\text{ K}\times 1\text{ mol}}$

  $V_2=2V_1$

Thus, the final volume of a gas is 2 times of the initial volume of a gas.