(II) The P V diagram in Fig. 15-23 shows two possible states...

(II) The $P V$ diagram in Fig. $15-23$ shows two possible states of a system containing 1.35 moles of a monatomic ideal gas. $\left(P_{1}=P_{2}=455 \mathrm{N} / \mathrm{m}^{2}, \quad V_{1}=2.00 \mathrm{m}^{3}, V_{2}=\right.$ $8.00 \mathrm{m}^{3} . )(a)$ Draw the process which depicts an isobaric expansion from state 1 to state $2,$ and label this process A. (b) Find the work done by the gas and the change in internal energy of the gas in process A. $(c)$ Draw the two-step process which depicts an isothermal expansion from state 1 to the volume $V_{2},$ followed by an isovolumetric increase in temperature to state $2,$ and label this process $\mathrm{B}$ . ( $d$ ) Find the change in internal energy of the gas for the two-step process $\mathrm{B}$ .

(II) The $P V$ diagram in Fig. $15-23$ shows two possible states of a system containing 1.35 moles of a monatomic ideal gas. $\left(P_{1}=P_{2}=455 \mathrm{N} / \mathrm{m}^{2}, \quad V_{1}=2.00 \mathrm{m}^{3}, V_{2}=\right.$ $8.00 \mathrm{m}^{3} . )(a)$ Draw the process which depicts an isobaric
expansion from state 1 to state $2,$ and label this process A. (b) Find the work done by the gas and the change in internal energy of the gas in process A. $(c)$ Draw the two-step process which depicts an isothermal expansion from state 1 to the volume $V_{2},$ followed by an isovolumetric increase in temperature to state $2,$ and label this process $\mathrm{B}$ . ( $d$ ) Find the change in internal energy of the gas for the two-step process $\mathrm{B}$ .

Key Concepts

P-V Diagram Interpretation

P-V diagrams are graphical representations of the relationship between pressure and volume in a system. They are essential for visualizing thermodynamic processes and for calculating work done during a process, as the area under the curve represents the work exchanged. They help in understanding complex processes and transitions between different states of a system.

First Law of Thermodynamics

The first law of thermodynamics, which states that ?U = Q - W, is a statement of energy conservation for thermodynamic systems. It relates the changes in internal energy to the net heat added to the system and the work done by the system, serving as a guideline for analyzing energy transfer during various thermodynamic processes.

Change in Internal Energy

The change in internal energy of a system is linked to the change in temperature and is particularly straightforward for ideal gases. For a monatomic ideal gas, the internal energy is proportional to its temperature, and any process that changes the temperature, such as isobaric or isochoric, will change the internal energy in accordance with ?U = (degrees of freedom/2) nR?T.

Work in Thermodynamic Processes

Work in thermodynamic processes is defined as the energy transferred when a system undergoes a change in volume under pressure. On a P-V diagram, work is represented by the area under the process curve. In different processes, such as isobaric, isothermal, or isochoric, the work calculation varies depending on the specific constraints imposed.

Isobaric Process

An isobaric process is one in which the pressure remains constant. In such a process, the work done by or on the gas is determined by the change in volume, and the heat added or removed can be related through the first law of thermodynamics. It often involves using PV = nRT to find temperature changes as the volume changes.

Isothermal Process

An isothermal process is characterized by a constant temperature throughout the process. For an ideal gas undergoing an isothermal process, the internal energy change is zero, and any work done by or on the system is exactly balanced by heat flow. This process is key to understanding reversible work in thermodynamic cycles.

Ideal Gas Law

The ideal gas law, expressed as PV = nRT, links pressure, volume, and temperature for an ideal gas. This equation is fundamental for predicting the behavior of gases during various thermodynamic processes, providing a basis for calculating changes in state variables under different conditions.

Isochoric Process

An isochoric process, also known as a constant volume process, occurs when the volume of a system remains fixed. In this process, no work is done since there is no volume change, but the internal energy can change with temperature, as dictated by the first law of thermodynamics.