Thermodynamic derivation of heat capacity

Maxwell relations that can be used to solve it, but the application is not clear.In summary, the conversation is discussing the equation cp=cv+TV?^2/?, which involves finding the specific heat at constant pressure (cp) for a substance using the Maxwell relations. The application of these relations is not clear and further help is needed.
  • #1
tarletontexan
30
0

Homework Statement



cp=cv+TV?^2/?

Homework Equations




cp=T/N([itex]\partialS[/itex]/[itex]\partialT[/itex])p

The Attempt at a Solution


I have the equation, just not sure how to apply it? Any help would be appreciated
 
Physics news on Phys.org
  • #2
tarletontexan said:

Homework Statement



cp=cv+TV?^2/?

Homework Equations




cp=T/N([itex]\partialS[/itex]/[itex]\partialT[/itex])p

The Attempt at a Solution


I have the equation, just not sure how to apply it? Any help would be appreciated

I am not sure what the question is. Are we dealing with an ideal gas?

AM
 
  • #3
yes, I know that there are several maxwell relations to get to the solution I just don't know how to apply them.
 
  • #4
tarletontexan said:
yes, I know that there are several maxwell relations to get to the solution I just don't know how to apply them.
Start with:

TdS = dU + PdV

CP = (∂Q/∂T)P = T(∂S/∂T)P = (∂U/∂T)P + P(∂V/∂T)P

AM
 
  • #5
.

To understand the thermodynamic derivation of heat capacity, we first need to understand the concepts of specific heat and heat capacity. Specific heat is the amount of heat required to raise the temperature of a unit mass of a substance by one degree. Heat capacity, on the other hand, is the amount of heat required to raise the temperature of a substance by one degree, regardless of its mass.

The equation cp=cv+TV?^2/? is derived from the first law of thermodynamics, which states that the change in internal energy of a system is equal to the heat added to the system minus the work done by the system. In mathematical terms, this can be written as dU = dQ - dW.

For a system at constant pressure (p), the work done by the system is equal to the pressure (p) times the change in volume (dV). This can be written as dW = pdV.

Substituting this into the first law of thermodynamics equation, we get dU = dQ - pdV.

Using the definition of enthalpy (H), which is equal to the internal energy (U) plus the product of pressure (p) and volume (V), we can rewrite the equation as dH = dQ + Vdp.

At constant pressure, the change in enthalpy (dH) is equal to the heat added to the system (dQ). This gives us the equation dH = cpdT, where cp is the heat capacity at constant pressure.

We can also define the specific heat at constant volume (cv) as the change in internal energy (dU) divided by the change in temperature (dT) at constant volume. This gives us the equation cv = dU/dT.

Using the above equations, we can derive the equation cp = cv + TV?^2/?. This equation relates the heat capacity at constant pressure (cp) to the heat capacity at constant volume (cv) and the temperature (T) and volume (?).

In summary, the equation cp = cv + TV?^2/? is derived from the first law of thermodynamics and relates the heat capacity at constant pressure to the heat capacity at constant volume and the temperature and volume of a system. It is an important equation in thermodynamics and is used to understand the behavior of substances when heat is added or removed from a system.
 

1. What is the thermodynamic derivation of heat capacity?

The thermodynamic derivation of heat capacity is a method used to calculate the amount of heat required to increase the temperature of a substance by a certain amount. It is based on the principles of thermodynamics, specifically the first law which states that energy cannot be created or destroyed, only transferred.

2. How is heat capacity related to the thermodynamic derivation?

Heat capacity is the amount of heat required to raise the temperature of a substance by one degree. In the thermodynamic derivation, heat capacity is calculated by dividing the change in internal energy of the substance by the change in temperature.

3. What is the difference between specific heat and heat capacity?

Specific heat is the amount of heat required to raise the temperature of a unit mass of a substance by one degree, while heat capacity is the amount of heat required to raise the temperature of the entire substance by one degree. Specific heat is an intensive property, while heat capacity is an extensive property.

4. What are the units of heat capacity?

The units of heat capacity depend on the system of measurement being used. In the SI system, the units are joules per kelvin (J/K). In the CGS system, the units are calories per gram per degree Celsius (cal/g·°C). In the imperial system, the units are British thermal units per pound per degree Fahrenheit (BTU/lb·°F).

5. How is the thermodynamic derivation of heat capacity used in real-world applications?

The thermodynamic derivation of heat capacity is used in various industries, including engineering, chemistry, and materials science. It is used to design and optimize systems that involve heat transfer, such as power plants, refrigeration systems, and chemical reactors. It is also used in the development of new materials with specific heat capacities for various applications.

Similar threads

  • Advanced Physics Homework Help
Replies
1
Views
938
  • Advanced Physics Homework Help
Replies
2
Views
831
Replies
2
Views
390
  • Advanced Physics Homework Help
Replies
1
Views
871
  • Advanced Physics Homework Help
Replies
4
Views
1K
Replies
32
Views
2K
  • Advanced Physics Homework Help
Replies
2
Views
1K
Replies
2
Views
4K
Replies
22
Views
1K
  • Thermodynamics
Replies
28
Views
1K
Back
Top