Figure 20-6 is the P-V diagram for 25.0 g of an enclosed...

Figure 20-6 is the $P-V$ diagram for $25.0$ g of an enclosed ideal gas. At $A$ the gas temperature is $200^{\circ} \mathrm{C}$. The value of $c_{v}$ for the gas is $0.150 \mathrm{cal} / \mathrm{g} \cdot{ }^{\circ} \mathrm{C} .(a)$ What is the temperature of the gas at $B$ ? (b) Find $\Delta U$ for the portion of the cycle from $A$ to $B$. (c) Find $\Delta W$ for this same portion. $(d)$ Find $\Delta Q$ for this same portion.

Key Concepts

P-V Diagram Analysis

A Pressure-Volume (P-V) diagram is a graphical representation of the changes in pressure and volume of a system during a thermodynamic process. Such diagrams help in visualizing the work done by or on the system (as the area under the process curve) and in analyzing various cycles. They simplify the understanding of complex thermodynamic paths and processes by providing a visual tool to assess changes in state variables.

Internal Energy

In ideal gases, the internal energy is a state function that depends solely on the temperature of the gas. This means that any change in internal energy (?U) can be calculated from the temperature change, often using the specific heat at constant volume. This concept simplifies energy calculations because it establishes that internal energy changes are independent of the process path taken between two thermodynamic states.

Thermodynamic Work

Thermodynamic work in processes involving gases is generally defined as the integral of pressure with respect to volume change (W = ?PdV). Work done by or on the system during a process is path-dependent, which makes understanding the nature of the process, as represented on a P-V diagram, crucial for determining how energy is transferred via work.

First Law of Thermodynamics

The First Law of Thermodynamics states that the change in the internal energy of a system is equal to the heat added to the system minus the work done by the system (?U = Q - W). This principle is crucial in energy analysis during thermodynamic processes, as it links work, heat, and internal energy changes regardless of the specific path taken during the process.

Ideal Gas Law

The ideal gas law (PV = nRT) relates the pressure, volume, and temperature of an ideal gas and is fundamental in understanding how changes in one state variable affect the others. This relationship provides the basis for converting among different thermodynamic states in cyclic processes and supports the calculations of temperature, pressure, and volume when one or more variables are known.

Specific Heat at Constant Volume

The specific heat at constant volume (c_v) is defined as the amount of heat required to raise the temperature of a unit mass of a substance by one degree Celsius, while keeping the volume constant. In the context of ideal gases, c_v is directly used to calculate the change in internal energy when the temperature changes, emphasizing its role in energy transfer during processes where the volume might be constrained.

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