(II) The P V diagram in Fig. 19-31 shows two possible states...

(II) The $P V$ diagram in Fig. $19-31$ shows two possible states of a system containing 1.55 moles of a monatomic ideal gas. $\left(P_{1}=P_{2}=455 \mathrm{~N} / \mathrm{m}^{2}, \quad V_{1}=2.00 \mathrm{~m}^{3}, \quad V_{2}=8.00 \mathrm{~m}^{3} .\right)$ (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 B. ( $d$ ) Find the change in internal energy of the gas for the two-step process $B$.

(II) The $P V$ diagram in Fig. $19-31$ shows two possible states of a system containing 1.55 moles of a monatomic ideal gas. $\left(P_{1}=P_{2}=455 \mathrm{~N} / \mathrm{m}^{2}, \quad V_{1}=2.00 \mathrm{~m}^{3}, \quad V_{2}=8.00 \mathrm{~m}^{3} .\right)$
(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
B. ( $d$ ) Find the change in internal energy of the gas for the two-step process $B$.

Key Concepts

Ideal Gas Law

The ideal gas law, expressed as PV = nRT, relates the pressure, volume, temperature, and amount of an ideal gas. It provides the fundamental connection between the state variables of a gas and is critical for analyzing the behavior of gases under various process conditions.

First Law of Thermodynamics

The first law of thermodynamics states that the change in internal energy of a system is equal to the heat added to the system minus the work done by the system. This law is essential for understanding energy conservation and accounting for energy exchanges in thermodynamic processes.

Isobaric Process

An isobaric process is one in which the pressure remains constant while the volume and temperature may change. In such processes, calculating the work done by the gas is straightforward—work is given by the product of constant pressure and the change in volume.

Isothermal Process

An isothermal process occurs at a constant temperature throughout, meaning that for an ideal gas, the internal energy does not change despite the change in volume. This process follows specific relationships between pressure and volume, often involving logarithmic expressions when calculating work.

Isovolumetric (Isochoric) Process

An isovolumetric process is one in which the volume remains constant. Because work is defined as the change in volume multiplied by pressure, no work is done during an isovolumetric process, and any heat exchanged directly changes the internal energy of the system.

Internal Energy of a Monatomic Ideal Gas

For a monatomic ideal gas, the internal energy is solely a function of temperature and is given by U = (3/2) nRT. This relationship implies that any change in the internal energy during a process is directly related to the change in temperature, independent of the type of process.

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