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Constant-pressure specific heat, denoted by \(C_p\), is an important thermodynamic property. It represents the amount of heat required to raise the temperature of a substance by one degree while maintaining constant pressure. This is crucial in processes where the volume changes, such as in an open system with fluid flow.
Mathematically, \(C_p = \frac{dH}{dT}\), where \(H\) is the enthalpy and \(T\) is the temperature. To estimate \(C_p\), we can use the finite-difference method as follows:
\( C_p \approx \frac{\triangle h}{\triangle T} \),
where \(\triangle h\) is the change in specific enthalpy and \(\triangle T\) is the change in temperature. In the exercise, specific enthalpies are given at 80°F and 120°F, allowing us to calculate \(\triangle h\).
Given the enthalpies:
\( h_{80°F} = 117.63 \text{ Btu/lbm} \)
\( h_{120°F} = 126.39 \text{ Btu/lbm} \)
we find:
\( \triangle h = h_{120°F} - h_{80°F} = 126.39 - 117.63 = 8.76 \text{ Btu/lbm} \).
Using a temperature difference \( \triangle T = 40°F \), we get:
\( C_p \approx \frac{8.76 \text{ Btu/lbm}}{40 \text{ °F}} = 0.219 \text{ Btu/lbm-°F} \).

Constant-volume specific heat, denoted by \(C_v\), refers to the amount of heat needed to raise the temperature of a substance by one degree while the volume is kept constant. This property is particularly relevant in closed systems.
Mathematically, \(C_v = \frac{dU}{dT}\), where \(U\) is the internal energy and \(T\) is the temperature. Using the finite-difference method, we can approximate \(C_v\) as follows:
\( C_v \approx \frac{\triangle u}{\triangle T} \),
where \(\triangle u\) is the change in specific internal energy. For the given exercise, specific internal energies at 80°F and 120°F facilitate this calculation.
Given the internal energies:
\( u_{80°F} = 107.59 \text{ Btu/lbm} \)
\( u_{120°F} = 115.47 \text{ Btu/lbm} \),
we find:
\( \triangle u = u_{120°F} - u_{80°F} = 115.47 - 107.59 = 7.88 \text{ Btu/lbm} \).
Using a temperature difference \( \triangle T = 40°F \), we get:
\( C_v \approx \frac{7.88 \text{ Btu/lbm}}{40 \text{ °F}} = 0.197 \text{ Btu/lbm-°F} \).

The finite-difference method is a numerical approach used to approximate derivatives. It's helpful in estimating changes in thermodynamic properties when exact analytical solutions are difficult to obtain.
In the specific heat calculations, the finite-difference method aids in determining \(C_p\) and \(C_v\) by using known values at discrete points. Instead of dealing with complex analytical derivatives, the method uses:
\( \frac{dy}{dx} \approx \frac{\triangle y}{\triangle x} \),
where \(\triangle y\) and \(\triangle x\) are the changes in the respective quantities.
For \(C_p\) and \(C_v\), values of enthalpy and internal energy at different temperatures are used to estimate:
\( C_p \approx \frac{\triangle h}{\triangle T} \) and
\( C_v \approx \frac{\triangle u}{\triangle T} \).
The accuracy of the finite-difference method improves with smaller intervals for \(\triangle T\). However, in many practical cases, limited data points are sufficient for a reasonable estimate.

Thermodynamic properties are characteristics of a system that describe its physical state and behavior under varying conditions. Key properties include:

• Temperature (T)
• Pressure (P)
• Specific volume (v)
• Enthalpy (H)
• Internal energy (U)
• Entropy (S)

These properties are interrelated through various thermodynamic laws and equations.
For instance, specific enthalpy \(H\) (or simply enthalpy) combines internal energy \(U\) and the product of pressure and volume, as in:
\( H = U + PV \).
This linkage explains why both \(C_p\) and \(C_v\) are necessary. They describe heat capacity depending on whether the pressure or volume remains constant during heating.
Understanding and calculating these properties accurately are essential for designing efficient thermal systems and predicting system behaviors.

Enthalpy \(H\) and internal energy \(U\) are fundamental thermodynamic concepts.
Internal energy is the total energy contained within a system due to the kinetic and potential energies of its molecules. It's a state function, meaning it depends only on the state of the system and not on the path taken to reach that state.
Enthalpy, on the other hand, is defined as:
\( H = U + PV \),
where \(P\) is pressure, and \(V\) is volume. It represents the total heat content of a system.
For constant-pressure processes, changes in enthalpy (\translate_h) are directly proportional to the heat added or removed. This is why \(C_p\), related to \( \translate_h \), is such a critical measure.
Similarly, for constant-volume processes, changes in internal energy (\translate_u) reflect the heat exchange, making \(C_v\) relevant.
When dealing with specific heats, we analyze energy changes concerning temperature changes. Thus, these properties provide insight into system energy dynamics and help understand heat transfer processes effectively.