A 1.00 -mol sample of an ideal monatomic gas is at an...

A 1.00 -mol sample of an ideal monatomic gas is at an initial temperature of 300 $\mathrm{K}$ . The gas undergoes an isovolumetric process acquiring 500 $\mathrm{J}$ of energy by heat. It then undergoes an isobaric process losing this same amount of energy by heat. Determine (a) the new temperature of the gas and (b) the work done on the gas.

Key Concepts

Isobaric Process

In an isobaric (constant pressure) process, the pressure remains constant while the volume may change. The work done during an isobaric process is given by W = P?V, and because the ideal gas law links volume and temperature (?V = nR?T/P), the work can also be expressed in terms of temperature change as W = nR?T.

First Law of Thermodynamics

The first law of thermodynamics, which states that ?U = Q - W, provides a framework for analyzing energy changes in a system. Here, ?U is the change in internal energy, Q is the heat added to the system, and W is the work done by the system. This principle is used to balance energy during different thermodynamic processes and determine state variable changes.

Ideal Gas Behavior

Ideal gases obey the ideal gas law, PV = nRT, which relates pressure, volume, and temperature. This relationship is fundamental when analyzing processes such as constant volume (isovolumetric) and constant pressure (isobaric) processes, as it allows one to relate changes in temperature to changes in other state variables.

Isovolumetric Process

An isovolumetric (or constant volume) process is one in which the volume of the gas remains constant. Since no work is performed (W = 0) during such processes, any heat added or removed changes the internal energy of the gas directly, according to ?U = Q, with the temperature change being determined by the relation ?U = nCv?T.

Specific Heat Capacities for a Monatomic Ideal Gas

For a monatomic ideal gas, the molar specific heat capacity at constant volume is Cv = (3/2)R and at constant pressure is Cp = (5/2)R. These values allow one to calculate the temperature changes resulting from heat transfer in constant volume and constant pressure processes, respectively.

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