Question

Employing the ideal gas model, determine the change in specific entropy between the indicated states, in \(\mathrm{kJ} / \mathrm{kg} \cdot \mathrm{K}\). Solve three ways: Use the appropriate ideal gas table, \(I T\), and a constant specific heat value from Table A-20. (a) air, \(p_{1}=100 \mathrm{kPa}, T_{1}=20^{\circ} \mathrm{C}, p_{2}=100 \mathrm{kPa}, T_{2}=\) \(100^{\circ} \mathrm{C} .\) (b) air, \(p_{1}=1\) bar, \(T_{1}=27^{\circ} \mathrm{C}, p_{2}=3\) bar, \(T_{2}=377^{\circ} \mathrm{C}\). (c) carbon dioxide, \(p_{1}=150 \mathrm{kPa}, T_{1}=30^{\circ} \mathrm{C}, p_{2}=300 \mathrm{kPa}\), \(T_{2}=300^{\circ} \mathrm{C}\) (d) carbon monoxide, \(T_{1}=300 \mathrm{~K}, v_{1}=1.1 \mathrm{~m}^{3} / \mathrm{kg}, T_{2}=500 \mathrm{~K}\), \(v_{2}=0.75 \mathrm{~m}^{3} / \mathrm{kg}\) (e) nitrogen, \(p_{1}=2 \mathrm{MPa}, T_{1}=800 \mathrm{~K}, p_{2}=1 \mathrm{MPa}\), \(T_{2}=300 \mathrm{~K}\)

Short answer

Use ideal gas tables, integrate specific heat capacity, and assume constant cp to find the change in specific entropy. For air: calculate for given temperatures and pressures.

Step-by-step solution

- Identify given data

Start by noting the provided data for each part of the problem. For example, part (a) states: air at initial conditions \ \( p_1 = 100 \ \mathrm{kPa}, T_1 = 20^{\circ} \mathrm{C} \ \) and final conditions \ \( p_2 = 100 \ \mathrm{kPa}, T_2 = 100^{\circ} \mathrm{C} \ \).

- Determine specific entropy change

To calculate the specific entropy change \( \ \Delta s \), use the ideal gas relation: \ \( \ \Delta s = s_2 - s_1 \ \). There are three methods we need to use: ideal gas table, integrating the specific heat capacity \( c_p \), and assuming a constant specific heat.

- Use ideal gas table

From the ideal gas table, locate the specific entropy \( s \) values at the given temperatures and pressures. For instance, for part (a), find \( s_{20^{\circ}C} \) and \( s_{100^{\circ}C} \) but with constant pressure condition. Subtract the initial specific entropy from the final: \ \( \ \Delta s = s_{100^{\circ}C} - s_{20^{\circ}C} \ \)

- Use integration with temperature-dependent specific heat

For air, the specific heat at constant pressure \( \ c_p \) is temperature-dependent. Integrate the function provided in Table A-20: \ \( \ \ \Delta s = \int_{T_1}^{T_2} \frac{c_p(T)}{T} dT \ \) . Perform the integral to find \( \ \Delta s \ \).

- Assume constant specific heat

Using the average value for \( c_p \), the specific entropy change can be approximated as: \ \( \ \Delta s \approx c_p^{ave} \ln \left(\frac{T_2}{T_1}\right) \ \), where \( \ c_p^{ave}\ \) is the averaged value of specific heat between \( \ T_1 \ \) and \( \ T_2 \ \). Calculate \( \ \Delta s \) using this approximation.

- Repeat for other cases

Apply steps 1 through 5 to each of the given cases (b), (c), (d), and (e), making sure to use the properties and values specific to the substances and conditions given in each problem.

Key concepts

ideal gas model

The ideal gas model is a crucial concept in thermodynamics. It simplifies the behavior of gases by assuming that the gas molecules have elastic collisions and no intermolecular forces. This allows us to use the ideal gas equation: \text{ideal gas} = PV = nRT where:

• \(P\) is the pressure
• \(V\) is the volume
• \(n\) is the number of moles
• \(R\) is the universal gas constant
• \(T\) is the temperature in Kelvin

When dealing with specific entropy change, the ideal gas law helps in determining the states and properties at different conditions. Since entropy is a state function, changes in entropy only depend on the initial and final states of the system, not on the path taken. The ideal gas model becomes particularly useful in calculating entropy changes under constant volume, pressure, or temperature conditions. For example, you can calculate the specific entropy change by using the equation: \[ \frac{c_p(T)}{T} dT \] Integrating this expression between the given initial and final temperatures provides the answer.

specific heat capacity

Specific heat capacity, often denoted as \(c_p\) for constant pressure or \(c_v\) for constant volume, measures how much heat energy is required to change the temperature of a unit mass of a substance by one degree Celsius or Kelvin. The units of specific heat capacity are usually \( \text{J/kg.K} \) or \( \text{kJ/kg.K} \).

In our exercise, we often use \(c_p\) because many problems assume constant pressure conditions. The specific heat capacity changes with temperature, so using an average value or integrating gives more accurate results. Here are the steps to calculate the specific entropy change using specific heat capacity:

• Identify the specific heat capacity function: For air and other ideal gases, tables or functions for \(c_p(T)\) are often provided in thermodynamic charts.
• Perform Integration: Integrate the specific heat capacity over the temperature range. For a temperature-dependent function, the equation is: \[ \Delta s = \int_{T_1}^{T_2} \frac{c_p(T)}{T} dT \]
• Use Average Value: If assuming constant specific heat capacity, use an average value: \( c_p^{ave} \approx \frac{c_p(T1) + c_p(T2)}{2} \).
• Calculate Specific Entropy Change: Substitute the average value into the equation: \[ \Delta s \approx c_p^{ave} \ln \left(\frac{T_2}{T_1} \right) \]

Understanding how to use specific heat capacity ensures more accurate and reliable results when determining specific entropy changes.

thermodynamic properties

In thermodynamics, properties like pressure, volume, temperature, and entropy are essential to describe the state of a system. Each property provides insight into the energy interactions and efficiencies of processes. Here are some key points to understand:

• Pressure (\(P\)): The force exerted by gas particles per unit area. It can change during processes, affecting other properties like volume and temperature.
• Volume (\(V\)): The space occupied by the gas. Volume changes under constant pressure or temperature conditions can impact entropy.
• Temperature (\(T\)): A measure of the average kinetic energy of gas molecules. Entropy changes largely depend on the temperature difference between initial and final states.
• Specific Entropy (\(s\)): A measure of disorder or randomness. For ideal gases, changes in specific entropy can be calculated using tables, integrations, and specific heat assumptions.

By understanding and correctly applying these thermodynamic properties, students can determine changes in specific entropy accurately for various substances under different conditions. For instance, consider the exercise where we have different initial and final states of air, carbon dioxide, carbon monoxide, and nitrogen. Calculating the specific entropy change involves:

• Using ideal gas tables to find specific entropy values directly
• Integrating \(c_p(T)\) over temperature range
• Assuming constant \(c_p\) for simplified calculations

Mastering these concepts and their interrelations is critical for solving real-world problems involving energy transfer and efficiency in thermodynamic systems.