Question

Nitrogen \(\left(\mathrm{N}_{2}\right)\) initially occupying \(0.5 \mathrm{~m}^{3}\) at \(1.0\) bar, \(20^{\circ} \mathrm{C}\) undergoes an internally reversible compression during which \(p V^{1.30}=\) constant to a final state where the temperature is \(200^{\circ} \mathrm{C}\). Determine assuming the ideal gas model (a) the pressure at the final state, in bar. (b) the work and heat transfer, each in \(\mathrm{kJ}\). (c) the entropy change, in \(\mathrm{kJ} / \mathrm{K}\).

Short answer

Final pressure \( p_2 \) is calculated using the polytropic relation, work and heat transfer are found using the first law of thermodynamics, and entropy change is evaluated using the entropy formula for ideal gases.

Step-by-step solution

Use the Ideal Gas Law to Find Initial Moles of N2

First, determine the number of moles of nitrogen using the initial conditions and the ideal gas law. The ideal gas law is given by \[ pV = nRT \] where: \( p \) = 1.0 bar = 100 kPa, \( V \) = 0.5 m\textsuperscript{3}, \( R \) = 8.314 J/(mol·K), and \( T \) = 293 K. Calculate the number of moles (\( n\)).

Determine Final Pressure Using Polytropic Relation

Since the process follows a polytropic process where \( pV^{1.30} = \text{constant} \), use the relation \[ p_1V_1^{1.30} = p_2V_2^{1.30} \] to find the final pressure \( p_2 \). Use the ideal gas law again to find \( V_2 \) at the final temperature \( T_2 \) = 473 K.

Calculate Work Done During Compression

The work done during a polytropic process is given by \[ W = \frac{p_2V_2 - p_1V_1}{1 - n} \] where \( n \) = 1.30. Plug in known values to calculate the work \( W \).

Calculate Heat Transfer Using First Law of Thermodynamics

Using the first law of thermodynamics \[ Q = \Delta U + W \] where \( \ \Delta U \) is the change in internal energy. For ideal gases, \( \ \Delta U = n \cdot C_v \cdot \ \Delta T \), where \( C_v \) is the specific heat at constant volume. Calculate \( \Delta U \), and use it to find \( Q \) using the value of \( W \) from the previous step.

Determine Entropy Change

Entropy change of ideal gas during a polytropic process can be calculated using \[ \ \Delta S = n \cdot C_v \, ln \( \frac{T_2}{T_1} \) + n \cdot R \, ln \( \frac{V_2}{V_1} \) \]. Calculate \( \ \Delta S \) with the known values.

Key concepts

Ideal Gas Law

The ideal gas law is a fundamental equation in thermodynamics represented as \( pV = nRT \). Here, \( p \) stands for pressure, \( V \) is volume, \( n \) is the number of moles of gas, \( R \) is the ideal gas constant, and \( T \) is temperature. You can use this law to find unknown quantities if you know the other values. For example, given the initial conditions (\( p = 1.0 \) bar, \( V = 0.5 \) m\( ^3 \), and \( T = 293 \) K), you can calculate the number of moles, \( n \), for nitrogen gas by rearranging and solving the equation: \( n = \frac{{ p \, V }}{{ R \, T }} \).

Polytropic Process

A polytropic process is one where the pressure and volume of a gas are related by the equation \( pV^n = \text{constant} \). Here, \( n \) is the polytropic index. In this exercise, \( n \) is given as 1.30. For polytropic processes, the relation can be used to find unknown pressures or volumes at different states if one pair (initial or final) is known. Here, you used the equation \( p_1V_1^{1.30} = p_2V_2^{1.30} \) to solve for the final pressure after finding the final volume using the ideal gas law.

Entropy Change

Entropy change \( (\Delta S) \) in a thermodynamic process gives a measure of the disorder or randomness. For an ideal gas undergoing a polytropic process, the entropy change can be calculated using the formula: \[ \Delta S = n \, C_v \, ln \left( \frac{T_2}{ T_1} \right) + n \, R \, ln \left( \frac{V_2}{V_1} \right) \]. Here,\( T_1 \) and\( T_2 \) are initial and final temperatures, and\( V_1 \) and\( V_2 \) are initial and final volumes. This exercise applies these formulas to find the change in entropy for nitrogen gas as it undergoes compression.

First Law of Thermodynamics

The first law of thermodynamics is a statement of energy conservation and can be expressed as \( Q = \Delta U + W \). Here, \( Q \) is the heat added to the system, \( \Delta U \) is the change in internal energy, and \( W \) is the work done by or on the system. For an ideal gas, the change in internal energy \( \Delta U \) can be found using \( \Delta U = n \, C_v \, \Delta T \), where \( C_v \) is the specific heat capacity at constant volume and \( \Delta T \) is the change in temperature. Once \( \Delta U \) and\( W \) are known, \( Q \) can be calculated.

Work and Heat Transfer

Work and heat transfer are central concepts in thermodynamics. In any thermodynamic process, the work done \( W \) and the heat transferred \( Q \) are key quantities to determine. For a polytropic process, the work done is given by \[ W = \frac{p_2V_2 - p_1V_1}{1 - n} \], where \( n \) is the polytropic index. After determining the work, the heat transfer can be found using the first law of thermodynamics. Heat transfer is then calculated using \( Q = \Delta U + W \). This exercise showcases how to apply these formulas to find the work done and heat transfer for the nitrogen gas compression process.