Chapter 5, Problem 5.14P

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

(a) The pressure is tripled (at constant T).

(b) The absolute temperature is increased by a factor of 3.0 (at constant P).

(c) Three more moles of the gas are added (at constant P and T).

Interpretation Introduction

Interpretation:

The effect on the volume of 1 mole of an ideal gas should be determined when the pressure is tripled 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 5.14P

At constant temperature, the volume of one mole of a gas is 1/3 of the initial volume when the pressure of a gas is tripled.

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 tripled.

  ${\text{3PV}}^{\text{'}}\text{= nRT}$

Now, the new volume is calculated as:

  

Thus, new volume is:

  ${\text{V}}^{\text{'}}\text{= }\frac{1}{3}\text{V}$

Hence, at constant temperature, the volume of one mole of a gas is 1/3 of the initial volume when the pressure of a gas is tripled.

Interpretation Introduction

Interpretation:

The effect on the volume of 1 mole of an ideal gas should be determined when the absolute temperature is increased by factor of 3.0 at constant pressure.

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 s $6.02\times {10}^{23}$ 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 5.14P

At constant pressure, the volume of one mole of a gas is triple of the initial volume when the absolute temperature of a gas is increased by factor 3.0.

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 temperature is increased by factor 3.0.

  ${\text{PV}}^{\text{'}}\text{= nR3T}$

Now, the new volume is calculated as:

  

Thus, new volume is:

  ${\text{V}}^{\text{'}}\text{= }3\text{V}$

Hence, at constant pressure, the volume of one mole of a gas is triple of the initial volume when the absolute temperature of a gas is increased by factor 3.0.

Interpretation Introduction

Interpretation:

The effect on the volume of 1 mole of an ideal gas should be determined when three more number of moles of gas is added at constant pressure and 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 5.14P

At constant pressure and temperature, the volume of one mole of a gas is increase by factor 4.0 when three more number of moles of a gas is added.

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 three more number of moles of gas are added.

  ${\text{PV}}^{\text{'}}\text{=(1+3)nRT}$

  ${\text{PV}}^{\text{'}}\text{=4 nRT}$

Now, the new volume is calculated as:

  

Thus, new volume is:

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

Hence, at constant pressure and temperature, the volume of one mole of a gas is increase by factor 4.0 when three more number of moles of a gas is added.