Chapter 5, Problem 5.100P

In A, the picture shows a cylinder with 0.1 mol of a gas that behaves ideally. Choose the cylinder (B, C, or D) that correctly represents the volume of the gas after each of the foflowing changes. If none of the cylinders is correct, specify “none.”

(a) P is doubled at fixed n and T

(b) T is reduced Ironi 400 K to 200 K at fixed n and P.

(c) T is increased from 100°C to 200°C at fixed is and P

(d) 0.1 mol of gas is added at fixed P and T.

(e) 0.1 mol of gas is added and P is doubled at fixed T.

Interpretation Introduction

Interpretation:

The cylinder among B, C and D should be chosen that exactly represents the volume of a gas when the pressure of a gas is doubled at fixed number of moles 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 5.100P

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

Thus, it is represents by Cylinder B.

Explanation of Solution

Given information:

The cylinders are:

  

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 doubled at constant number of moles and temperature.

  $2{\text{PV}}^{\text{'}}\text{= nRT}$

Now, the new volume is calculated as:

  

Thus, new volume is:

  ${\text{V}}^{\text{'}}\text{= }\frac{1}{2}\text{V}$

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

Thus, it is represents by Cylinder B.

Interpretation Introduction

Interpretation:

The cylinder among B, C and D should be chosen that exactly represents the volume of a gas when the temperature of a gas is reduced from 400 K to 200 K at constant number of moles and 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 5.100P

The final volume of a gas is ½ times of the initial volume of a gas and it is described by the Cylinder B.

Explanation of Solution

Given information:

The cylinders are:

  

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, $P_1$ and $P_2$ = P, $T_1$ = 400 K and $T_2$ = 200 K

  $n_1\text{ and }{n}_2\text{ = 0}\text{.1 mol}$

Put the values,

  $\frac{P\times V_1}{400\text{ K}\times 0.1\text{ mol}}=\frac{P\times V_2}{200\text{ K}\times 0.1\text{ mol}}$

  $V_2=\frac{1}{2}{V}_1$

Thus, the final volume of a gas is ½ times of the initial volume of a gas and it is described by the Cylinder B.

Interpretation Introduction

Interpretation:

The cylinder among B, C and D should be chosen that exactly represents the volume of a gas when the temperature of a gas is increased from 100 $°C$ K to 200 $°C$ at constant number of moles and 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 5.100P

The final volume of a gas is 1.26 times of the initial volume of a gas. It is described by “None” cylinder.

Explanation of Solution

Given information:

The cylinders are:

  

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, $P_1$ and $P_2$ = P, $T_1$ = $100°\text{C}$ and $T_2$ = $200°C$

  $n_1\text{ and }{n}_2\text{ = 0}\text{.1 mol}$

Convert the value of temperature in degree Celsius in Kelvin.

Initial temperature in K = $100°C+273=373\text{ K}$

Final temperature in K = $200°C+273=473\text{ K}$

Put the values,

  $\frac{P\times V_1}{373\text{ K}\times 0.1\text{ mol}}=\frac{P\times V_2}{473\text{ K}\times 0.1\text{ mol}}$

  $V_2=1.26V_1$

Thus, the final volume of a gas is 1.26 times of the initial volume of a gas and it is not described any cylinder.

Interpretation Introduction

Interpretation:

The cylinder among B, C and D should be chosen that exactly represents the volume of a gas when 0.1 mole of a gas is added at constant temperature and 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 5.100P

At constant temperature and pressure of a gas, the volume of a gas is 2 times of the initial volume when the 0.1 mole of a gas is added.

Thus, it is represents by Cylinder C.

Explanation of Solution

Given information:

The cylinders are:

  

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 0.1 mole of a gas is added at constant temperature and pressure.

  ${\text{PV}}^{\text{'}}\text{= (0}\text{.1+0}\text{.1) nRT}$

  ${\text{PV}}^{\text{'}}\text{= 0}\text{.2 nRT}$

Now, the new volume is calculated as:

  

Thus, new volume is:

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

Hence, at constant temperature and pressure of a gas, the volume of a gas is 2 times of the initial volume when the 0.1 mole of a gas is added.

Thus, it is represents by Cylinder C.

Interpretation Introduction

Interpretation:

The cylinder among B, C and D should be chosen that exactly represents the volume of a gas when 0.1 mole of a gas is added and Pressure is doubled 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 5.100P

At constant temperature of a gas, the volume of a gas is equal to the initial volume when the 0.1 mole of a gas is added and pressure is doubled.

Thus, it is represents by Cylinder D.

Explanation of Solution

Given information:

The cylinders are:

  

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 0.1 mole of a gas is added and Pressure is doubled at constant temperature

  ${\text{2PV}}^{\text{'}}\text{= (0}\text{.1+0}\text{.1) nRT}$

  ${\text{2PV}}^{\text{'}}\text{= 0}\text{.2 nRT}$

Now, the new volume is calculated as:

  

Thus, new volume is:

  ${\text{V}}^{\text{'}}\text{= V}$

Hence, at constant temperature of a gas, the volume of a gas is equal to the initial volume when the 0.1 mole of a gas is added and pressure is doubled.

Thus, it is represents by Cylinder D.