Equation 2.44 is to be derived from the previous step. Concept introduction: Heat capacity (thermal capacity) is the quantity of heat required to raise the temperature of the system from the lower limit to higher divided by the temperature difference of the system. When the mass of the system is taken as 1gram, the heat capacity is denoted as specific heat capacity. Similarly, when the mass of the system taken as 1 mole, the heat capacity is referred as molar heat capacity. Heat capacity is generally described as the symbol C . Mathematically, the heat capacity of the system between two temperature T 1 and T 2 can be expressed as C (T 2 , T 1 ) = q / (T 2 – T 1 ) Intriguingly, the molar heat capacities of gaseous systems are determined at constant volume and can be expressed as C v = (δU/δT) v

Question

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Answer 1

Question

Chapter 2, Problem 2.59E

Interpretation Introduction

Interpretation:

Equation 2.44 is to be derived from the previous step.

Concept introduction:

Heat capacity (thermal capacity) is the quantity of heat required to raise the temperature of the system from the lower limit to higher divided by the temperature difference of the system. When the mass of the system is taken as 1gram, the heat capacity is denoted as specific heat capacity. Similarly, when the mass of the system taken as 1 mole, the heat capacity is referred as molar heat capacity. Heat capacity is generally described as the symbol C. Mathematically, the heat capacity of the system between two temperature T₁ and T₂ can be expressed as

C (T₂, T₁) = q / (T₂ – T₁)

Intriguingly, the molar heat capacities of gaseous systems are determined at constant volume and can be expressed as

C_v = (δU/δT) v

Expert Solution & Answer

Answer to Problem 2.59E

The derived expression relating molar capacity at constant volume (Cv) of a system with change in volume (Vf and Vi) and change in temperature (Tf and Ti) is given as;

- R ln (Vf/Vi) = Cv ln (Tf/Ti) (or) R ln (Vi/Vf) = Cv ln (Tf/Ti)

Explanation of Solution

In an adiabatic process, the change in work can be expressed in relationship with change in temperature and change in volume.

dw = - Pex dV ...(1)

dw = n Cv dT ...(2)

on comparing equation (1) and (2) we get the following expression,

- Pex dV = n Cv dT ...(3)

Moreover, for an adiabatic process P_ex = P _int and for an ideal gas PV = nRT; the equation (3) can be changed to,

(- nRT/V)dV = n Cv dT ...(4)

On rearrangement of equation (4), we get

(- R/V)dV = (Cv/T) dT ...(5)

Thus, integrating the equation (5), we get;

- R (ln Vf – ln Vi) = Cv (ln Tf – ln Ti)

Thus,

- R ln (Vf/Vi) = Cv ln (Tf/Ti) …(6)

On changing the sign of the equation 6, we get the expression relating heat capacity at constant volume and volume and temperature change as,

R ln (Vi/Vf) = Cv ln (Tf/Ti) ...(7)

or equation 2.44

Conclusion

Thus, equation 2.44 is derived from the previous steps.

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Answer 2

Question

Chapter 2, Problem 2.59E

Interpretation Introduction

Interpretation:

Equation 2.44 is to be derived from the previous step.

Concept introduction:

Heat capacity (thermal capacity) is the quantity of heat required to raise the temperature of the system from the lower limit to higher divided by the temperature difference of the system. When the mass of the system is taken as 1gram, the heat capacity is denoted as specific heat capacity. Similarly, when the mass of the system taken as 1 mole, the heat capacity is referred as molar heat capacity. Heat capacity is generally described as the symbol C. Mathematically, the heat capacity of the system between two temperature T₁ and T₂ can be expressed as

C (T₂, T₁) = q / (T₂ – T₁)

Intriguingly, the molar heat capacities of gaseous systems are determined at constant volume and can be expressed as

C_v = (δU/δT) v

Expert Solution & Answer

Answer to Problem 2.59E

The derived expression relating molar capacity at constant volume (Cv) of a system with change in volume (Vf and Vi) and change in temperature (Tf and Ti) is given as;

- R ln (Vf/Vi) = Cv ln (Tf/Ti) (or) R ln (Vi/Vf) = Cv ln (Tf/Ti)

Explanation of Solution

In an adiabatic process, the change in work can be expressed in relationship with change in temperature and change in volume.

dw = - Pex dV ...(1)

dw = n Cv dT ...(2)

on comparing equation (1) and (2) we get the following expression,

- Pex dV = n Cv dT ...(3)

Moreover, for an adiabatic process P_ex = P _int and for an ideal gas PV = nRT; the equation (3) can be changed to,

(- nRT/V)dV = n Cv dT ...(4)

On rearrangement of equation (4), we get

(- R/V)dV = (Cv/T) dT ...(5)

Thus, integrating the equation (5), we get;

- R (ln Vf – ln Vi) = Cv (ln Tf – ln Ti)

Thus,

- R ln (Vf/Vi) = Cv ln (Tf/Ti) …(6)

On changing the sign of the equation 6, we get the expression relating heat capacity at constant volume and volume and temperature change as,

R ln (Vi/Vf) = Cv ln (Tf/Ti) ...(7)

or equation 2.44

Conclusion

Thus, equation 2.44 is derived from the previous steps.

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Answer 3

Question

Chapter 2, Problem 2.59E

Interpretation Introduction

Interpretation:

Equation 2.44 is to be derived from the previous step.

Concept introduction:

Heat capacity (thermal capacity) is the quantity of heat required to raise the temperature of the system from the lower limit to higher divided by the temperature difference of the system. When the mass of the system is taken as 1gram, the heat capacity is denoted as specific heat capacity. Similarly, when the mass of the system taken as 1 mole, the heat capacity is referred as molar heat capacity. Heat capacity is generally described as the symbol C. Mathematically, the heat capacity of the system between two temperature T₁ and T₂ can be expressed as

C (T₂, T₁) = q / (T₂ – T₁)

Intriguingly, the molar heat capacities of gaseous systems are determined at constant volume and can be expressed as

C_v = (δU/δT) v

Expert Solution & Answer

Answer to Problem 2.59E

The derived expression relating molar capacity at constant volume (Cv) of a system with change in volume (Vf and Vi) and change in temperature (Tf and Ti) is given as;

- R ln (Vf/Vi) = Cv ln (Tf/Ti) (or) R ln (Vi/Vf) = Cv ln (Tf/Ti)

Explanation of Solution

In an adiabatic process, the change in work can be expressed in relationship with change in temperature and change in volume.

dw = - Pex dV ...(1)

dw = n Cv dT ...(2)

on comparing equation (1) and (2) we get the following expression,

- Pex dV = n Cv dT ...(3)

Moreover, for an adiabatic process P_ex = P _int and for an ideal gas PV = nRT; the equation (3) can be changed to,

(- nRT/V)dV = n Cv dT ...(4)

On rearrangement of equation (4), we get

(- R/V)dV = (Cv/T) dT ...(5)

Thus, integrating the equation (5), we get;

- R (ln Vf – ln Vi) = Cv (ln Tf – ln Ti)

Thus,

- R ln (Vf/Vi) = Cv ln (Tf/Ti) …(6)

On changing the sign of the equation 6, we get the expression relating heat capacity at constant volume and volume and temperature change as,

R ln (Vi/Vf) = Cv ln (Tf/Ti) ...(7)

or equation 2.44

Conclusion

Thus, equation 2.44 is derived from the previous steps.

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Answer 4

Question

Chapter 2, Problem 2.59E

Interpretation Introduction

Interpretation:

Equation 2.44 is to be derived from the previous step.

Concept introduction:

Heat capacity (thermal capacity) is the quantity of heat required to raise the temperature of the system from the lower limit to higher divided by the temperature difference of the system. When the mass of the system is taken as 1gram, the heat capacity is denoted as specific heat capacity. Similarly, when the mass of the system taken as 1 mole, the heat capacity is referred as molar heat capacity. Heat capacity is generally described as the symbol C. Mathematically, the heat capacity of the system between two temperature T₁ and T₂ can be expressed as

C (T₂, T₁) = q / (T₂ – T₁)

Intriguingly, the molar heat capacities of gaseous systems are determined at constant volume and can be expressed as

C_v = (δU/δT) v

Expert Solution & Answer

Answer to Problem 2.59E

The derived expression relating molar capacity at constant volume (Cv) of a system with change in volume (Vf and Vi) and change in temperature (Tf and Ti) is given as;

- R ln (Vf/Vi) = Cv ln (Tf/Ti) (or) R ln (Vi/Vf) = Cv ln (Tf/Ti)

Explanation of Solution

In an adiabatic process, the change in work can be expressed in relationship with change in temperature and change in volume.

dw = - Pex dV ...(1)

dw = n Cv dT ...(2)

on comparing equation (1) and (2) we get the following expression,

- Pex dV = n Cv dT ...(3)

Moreover, for an adiabatic process P_ex = P _int and for an ideal gas PV = nRT; the equation (3) can be changed to,

(- nRT/V)dV = n Cv dT ...(4)

On rearrangement of equation (4), we get

(- R/V)dV = (Cv/T) dT ...(5)

Thus, integrating the equation (5), we get;

- R (ln Vf – ln Vi) = Cv (ln Tf – ln Ti)

Thus,

- R ln (Vf/Vi) = Cv ln (Tf/Ti) …(6)

On changing the sign of the equation 6, we get the expression relating heat capacity at constant volume and volume and temperature change as,

R ln (Vi/Vf) = Cv ln (Tf/Ti) ...(7)

or equation 2.44

Conclusion

Thus, equation 2.44 is derived from the previous steps.

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Answer 5

Chapter 2, Problem 2.59E

Interpretation Introduction

Interpretation:

Equation 2.44 is to be derived from the previous step.

Concept introduction:

Heat capacity (thermal capacity) is the quantity of heat required to raise the temperature of the system from the lower limit to higher divided by the temperature difference of the system. When the mass of the system is taken as 1gram, the heat capacity is denoted as specific heat capacity. Similarly, when the mass of the system taken as 1 mole, the heat capacity is referred as molar heat capacity. Heat capacity is generally described as the symbol C. Mathematically, the heat capacity of the system between two temperature T₁ and T₂ can be expressed as

C (T₂, T₁) = q / (T₂ – T₁)

Intriguingly, the molar heat capacities of gaseous systems are determined at constant volume and can be expressed as

C_v = (δU/δT) v

Expert Solution & Answer

Answer to Problem 2.59E

The derived expression relating molar capacity at constant volume (Cv) of a system with change in volume (Vf and Vi) and change in temperature (Tf and Ti) is given as;

- R ln (Vf/Vi) = Cv ln (Tf/Ti) (or) R ln (Vi/Vf) = Cv ln (Tf/Ti)

Explanation of Solution

In an adiabatic process, the change in work can be expressed in relationship with change in temperature and change in volume.

dw = - Pex dV ...(1)

dw = n Cv dT ...(2)

on comparing equation (1) and (2) we get the following expression,

- Pex dV = n Cv dT ...(3)

Moreover, for an adiabatic process P_ex = P _int and for an ideal gas PV = nRT; the equation (3) can be changed to,

(- nRT/V)dV = n Cv dT ...(4)

On rearrangement of equation (4), we get

(- R/V)dV = (Cv/T) dT ...(5)

Thus, integrating the equation (5), we get;

- R (ln Vf – ln Vi) = Cv (ln Tf – ln Ti)

Thus,

- R ln (Vf/Vi) = Cv ln (Tf/Ti) …(6)

On changing the sign of the equation 6, we get the expression relating heat capacity at constant volume and volume and temperature change as,

R ln (Vi/Vf) = Cv ln (Tf/Ti) ...(7)

or equation 2.44

Conclusion

Thus, equation 2.44 is derived from the previous steps.

Answer 6

Chapter 2, Problem 2.59E

Interpretation Introduction

Interpretation:

Equation 2.44 is to be derived from the previous step.

Concept introduction:

Heat capacity (thermal capacity) is the quantity of heat required to raise the temperature of the system from the lower limit to higher divided by the temperature difference of the system. When the mass of the system is taken as 1gram, the heat capacity is denoted as specific heat capacity. Similarly, when the mass of the system taken as 1 mole, the heat capacity is referred as molar heat capacity. Heat capacity is generally described as the symbol C. Mathematically, the heat capacity of the system between two temperature T₁ and T₂ can be expressed as

C (T₂, T₁) = q / (T₂ – T₁)

Intriguingly, the molar heat capacities of gaseous systems are determined at constant volume and can be expressed as

C_v = (δU/δT) v

Expert Solution & Answer

Answer to Problem 2.59E

The derived expression relating molar capacity at constant volume (Cv) of a system with change in volume (Vf and Vi) and change in temperature (Tf and Ti) is given as;

- R ln (Vf/Vi) = Cv ln (Tf/Ti) (or) R ln (Vi/Vf) = Cv ln (Tf/Ti)

Explanation of Solution

In an adiabatic process, the change in work can be expressed in relationship with change in temperature and change in volume.

dw = - Pex dV ...(1)

dw = n Cv dT ...(2)

on comparing equation (1) and (2) we get the following expression,

- Pex dV = n Cv dT ...(3)

Moreover, for an adiabatic process P_ex = P _int and for an ideal gas PV = nRT; the equation (3) can be changed to,

(- nRT/V)dV = n Cv dT ...(4)

On rearrangement of equation (4), we get

(- R/V)dV = (Cv/T) dT ...(5)

Thus, integrating the equation (5), we get;

- R (ln Vf – ln Vi) = Cv (ln Tf – ln Ti)

Thus,

- R ln (Vf/Vi) = Cv ln (Tf/Ti) …(6)

On changing the sign of the equation 6, we get the expression relating heat capacity at constant volume and volume and temperature change as,

R ln (Vi/Vf) = Cv ln (Tf/Ti) ...(7)

or equation 2.44

Conclusion

Thus, equation 2.44 is derived from the previous steps.

Answer 7

Chapter 2, Problem 2.59E

Interpretation Introduction

Interpretation:

Equation 2.44 is to be derived from the previous step.

Concept introduction:

Heat capacity (thermal capacity) is the quantity of heat required to raise the temperature of the system from the lower limit to higher divided by the temperature difference of the system. When the mass of the system is taken as 1gram, the heat capacity is denoted as specific heat capacity. Similarly, when the mass of the system taken as 1 mole, the heat capacity is referred as molar heat capacity. Heat capacity is generally described as the symbol C. Mathematically, the heat capacity of the system between two temperature T₁ and T₂ can be expressed as

C (T₂, T₁) = q / (T₂ – T₁)

Intriguingly, the molar heat capacities of gaseous systems are determined at constant volume and can be expressed as

C_v = (δU/δT) v

Answer 8

Expert Solution & Answer

Answer to Problem 2.59E

The derived expression relating molar capacity at constant volume (Cv) of a system with change in volume (Vf and Vi) and change in temperature (Tf and Ti) is given as;

- R ln (Vf/Vi) = Cv ln (Tf/Ti) (or) R ln (Vi/Vf) = Cv ln (Tf/Ti)