0.020 mol of a diatomic gas, with initial temperature 20^∘...

0.020 mol of a diatomic gas, with initial temperature $20^{\circ} \mathrm{C},$ are compressed from $1500 \mathrm{cm}^{3}$ to $500 \mathrm{cm}^{3}$ in a process in which $p V^{2}=$ constant a. What is the final temperature (in $^{\circ} \mathrm{C}$ ) $?$ b. How much heat is added during this process? c. Draw the $p V$ diagram, include proper scales on both axes.

0.020 mol of a diatomic gas, with initial temperature $20^{\circ} \mathrm{C},$ are compressed from $1500 \mathrm{cm}^{3}$ to $500 \mathrm{cm}^{3}$ in a process in which $p V^{2}=$ constant
a. What is the final temperature (in $^{\circ} \mathrm{C}$ ) $?$
b. How much heat is added during this process?
c. Draw the $p V$ diagram, include proper scales on both axes.

Key Concepts

Pressure-Volume Diagram

A p-V diagram graphically represents the relationship between pressure and volume during a thermodynamic process. It helps visualize the process path, the work done (represented by the area under the curve), and the nature of the process, including scaling and ensuring the correct interpretation of state changes in experimental and theoretical analyses.

First Law of Thermodynamics

This law 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, expressed as ?U = Q - W. It is crucial for analyzing energy transfers during any thermodynamic process and for computing either the work performed or the heat exchanged when the other quantities are known.

Ideal Gas Law

This is a fundamental relation in thermodynamics that links the pressure, volume, and temperature of an ideal gas through the equation pV = nRT. It serves as the basis for analyzing the state of the gas and is used to connect changes in one variable to changes in the others during different thermodynamic processes.

Polytropic Process

A polytropic process is one in which the pressure and volume of a gas are related by the equation pV^n = constant, where n is the polytropic index. This concept generalizes various specific processes such as isothermal (n = 1) and adiabatic (n = ?). In the given context, using a process with a specific exponent (n = 2 in this case) shows how the state variables change while obeying a defined constraint.

Diatomic Gas Heat Capacities

Diatomic gases have particular molar heat capacities (C_V and C_P) that depend on their degrees of freedom, including translational and rotational motions. These heat capacities are essential for determining the change in internal energy and the amount of heat exchanged in processes, as they directly relate to temperature changes through ?U = nC_V?T.

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