Chapter 5, Problem 5.13P

Each of the following processes caused the gas volume to double, as shown. For each process, tell how the remaining gas variable changed or state that it remained fixed:

1. T doubles at fixed P.
2. T and n are fixed.
3. At fixed T, the reaction is ${\text{CD}}_{\text{2}}(g)\to \text{C}(g)+{\text{D}}_2(g).$ At fixed P, the reaction is ${\text{A}}_{\text{2}}(g)+{\text{B}}_2(g)\to 2\text{A}\text{B}(g).$

Interpretation Introduction

Interpretation:

The change in variable or state of the remaining gas that it remained fixed should be explained when temperature doubles 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 $6.02\times {10}^{23}$ 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 5.13P

At constant pressure and number of moles, the volume of one mole of a gas is doubled when the temperature is doubled.

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 doubled.

  ${\text{PV}}^{\text{'}}\text{= nR2T}$

Now, the new volume is calculated as:

  

Thus, new volume is:

  ${\text{V}}^{\text{'}}\text{= }2\text{V}$

Hence, at constant pressure and number of moles, the volume of one mole of a gas is doubled when the temperature is doubled.

Interpretation Introduction

Interpretation:

The change in variable or state of the remaining gas that it remained fixed should be explained when temperature and number of moles is fixed.

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 5.13P

At constant temperature and number of moles, the pressure of a gas is ½ when the volume is doubled.

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 volume is doubled.

  ${\text{P}}^{\text{'}}\text{2V= nRT}$

Now, the new pressure is calculated as:

  

Thus, new pressure is:

  ${\text{P}}^{\text{'}}\text{= }\frac{P}{2}$

Hence, at constant temperature and number of moles, the pressure of a gas is ½ when the volume is doubled.

Interpretation Introduction

Interpretation:

The change in variable or state of the remaining gas that it remained fixed should be explained when temperature is fixed and the reaction is:

  ${\text{CD}}_{\text{2}}\text{(g)}\to {\text{C(g) + D}}_{\text{2}}\text{(g)}$

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 5.13P

At constant temperature, the pressure of a gas is equal to the initial pressure when the volume and number of moles of a gas is doubled.

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 given reaction is:

  ${\text{CD}}_{\text{2}}\text{(g)}\to {\text{C(g) + D}}_{\text{2}}\text{(g)}$

In above reaction is number of moles of gas increases from 1 to 2.

The new ideal expression is shown below, when temperature is constant and volume, number of gas is doubled.

  ${\text{P}}^{\text{'}}\text{2V=2nRT}$

Now, the new volume is calculated as:

  

Thus, new pressure is:

  ${\text{P}}^{\text{'}}\text{= P}$

Hence, at constant temperature, the pressure of a gas is equal to the initial pressure when the volume and number of moles of a gas is doubled.

Interpretation Introduction

Interpretation:

The change in variable or state of the remaining gas that it remained fixed should be explained when pressure is fixed and the reaction is:

  ${\text{A}}_{\text{2}}{\text{(g)+B}}_{\text{2}}\text{(g)}\to 2\text{AB(g)}$

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 5.13P

At constant pressure and number of moles of a gas, the temperature of a gas is doubled when volume is doubled.

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 given reaction is:

  ${\text{A}}_{\text{2}}{\text{(g)+B}}_{\text{2}}\text{(g)}\to 2\text{AB(g)}$

In above reaction is number of moles of gases are same on both sides that is 2.

The new ideal expression is shown below, when pressure and number of moles is constant and volume is doubled.

  ${\text{P2V=nRT}}^{\text{'}}$

Now, the new temperature is calculated as:

  

Thus, new temperature is:

  ${\text{T}}^{\text{'}}\text{= 2T}$

Hence, at constant pressure and number of moles of a gas, the temperature of a gas is doubled when volume is doubled.