Which of the following is correct? [h = specific enthalpy of substance, u = specific internal energy of substance, P = pressure of substance, ρ = density of substance.]
In thermodynamics, enthalpy is a fundamental property that helps us understand the total energy content of a system. When we talk about specific enthalpy, we are referring to the enthalpy per unit mass of a substance. This concept is crucial for analyzing energy changes in various processes.
Enthalpy Definition and Components
The total enthalpy (H) of a substance is defined as the sum of its internal energy (U) and the product of its pressure (P) and volume (V). Mathematically, this is expressed as:
\[H = U + PV\]
For convenience in many thermodynamic analyses, we often work with intensive properties, which are properties per unit mass. Thus, we define specific enthalpy (h), specific internal energy (u), and specific volume (v). Dividing the extensive enthalpy equation by mass (m), we get:
\[\frac{H}{m} = \frac{U}{m} + P\frac{V}{m}\]
This leads to the definition of specific enthalpy (h):
\[h = u + Pv\]
Where:
- h is the specific enthalpy of substance (energy per unit mass).
- u is the specific internal energy of substance (internal energy per unit mass).
- P is the pressure of substance.
- v is the specific volume of substance (volume per unit mass).
Specific Volume and Density Relationship
The question provides density of substance (\(\rho\)) instead of specific volume (v). We know that density is defined as mass per unit volume:
\[\rho = \frac{m}{V}\]
Conversely, specific volume is defined as volume per unit mass:
\[v = \frac{V}{m}\]
From these definitions, it is clear that specific volume is the reciprocal of density:
\[v = \frac{1}{\rho}\]
Deriving Enthalpy from Pressure and Density
Now, we can substitute the relationship between specific volume and density into our equation for specific enthalpy. Starting with:
\[h = u + Pv\]
Substitute \(v = \frac{1}{\rho}\):
\[h = u + P \left(\frac{1}{\rho}\right)\]
Simplifying this expression gives us the final relationship:
\[h = u + \frac{P}{\rho}\]
This equation correctly relates specific enthalpy (h) to specific internal energy (u), pressure (P), and density (\(\rho\)) of a substance.
Comparing Options for Specific Enthalpy
Let's evaluate each given option against the derived correct formula \(h = u + \frac{P}{\rho}\):
| Option | Expression | Correctness |
|---|---|---|
| 1 | \(h = u - \left(\frac{P}{\rho}\right)\) | Incorrect. It uses subtraction instead of addition. |
| 2 | \(h = u + P \rho\) | Incorrect. Pressure should be divided by density, not multiplied. |
| 3 | \(h = u + \left(\frac{P}{\rho}\right)\) | Correct. This matches the derived thermodynamic definition. |
| 4 | \(h = u - P\rho\) | Incorrect. It uses subtraction and multiplies pressure by density. |
Conclusion on Enthalpy Relation
Based on the fundamental definitions in thermodynamics, the correct relationship connecting the specific enthalpy of substance (h), specific internal energy of substance (u), pressure of substance (P), and density of substance (\(\rho\)) is \(h = u + \frac{P}{\rho}\). This relationship is vital for understanding energy transformations in various systems.
Important Questions from Thermodynamic Relations
Helmholtz function is expressed as:
- The Joule -Thompson coefficient for an ideal gas is _______.
The property relation for enthalpy change, dh is:
________ is known as the inversion curve to pass through the isenthalpes'.
If the temperature remains constant, then enthalpy
