Let 20.9 J of heat be added to a particular ideal gas. As a...

Let $20.9 \mathrm{~J}$ of heat be added to a particular ideal gas. As a result, its volume changes from $63.0$ to $113 \mathrm{~cm}^{3}$ while the pressure remains constant at $1.00 \mathrm{~atm}$. (a) By how much does the internal energy of the gas change? $(b)$ If the quantity of gas present is $2.00 \times 10^{-3} \mathrm{~mol}$, find the molar heat capacity at constant pressure. (c) Find the molar heat capacity at constant volume.

Key Concepts

Heat Capacities at Constant Pressure and Constant Volume

For an ideal gas, the molar heat capacity at constant pressure (Cp) includes both the heat needed to raise the internal energy and the heat required to do work due to volume change, while the molar heat capacity at constant volume (Cv) involves only the change in internal energy. Their difference is fundamental and relates to the gas constant, reflecting the extra energy needed at constant pressure conditions.

Internal Energy

The internal energy of a system is a measure of the total energy contained within it, primarily in the form of the kinetic energy of molecules for an ideal gas. Understanding how this energy changes with heat addition or work done on the system is essential when applying thermodynamic principles, especially for ideal gases where internal energy is directly related to temperature.

Molar Heat Capacity

Molar heat capacity is defined as the amount of heat required to raise the temperature of one mole of a substance by one degree. It comes in two forms: at constant pressure (Cp) and at constant volume (Cv). These quantities help characterize how a substance absorbs heat and are connected via the work done during expansion or compression in thermodynamic processes.

First Law of Thermodynamics

This principle states that the change in the internal energy of a system is equal to the heat supplied to the system minus the work done by the system on its surroundings. It serves as a statement of energy conservation and is crucial for analyzing energy transfers in thermodynamic processes, such as changes in a gas when heat is added or removed.

Work in a Constant Pressure Process

In any process at constant pressure, the work done by the gas can be calculated by multiplying the constant pressure by the change in volume of the gas. This concept is fundamental when applying the first law of thermodynamics to find the change in internal energy, as part of the heat added will be used to perform work as the gas expands.

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