# Heat Capacities of an Ideal Gas I

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Physik
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Heat Capacities of an Ideal Gas I

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Consider two identical cylinders A and B, filled with an ideal gas. Cylinder A is sealed, while B has a movable piston.

Let some amount of heat be applied to both cylinders to increase the gas temperature equally.

The volume of gas in the enclosed cylinder remains unchanged, so no work is done. In cylinder B, the gas is free to expand, resulting in some work being done under constant external pressure.

From the First law of thermodynamics, the heat applied increases the system's internal energy in cylinder A.  In cylinder B, a portion of heat is used to do work.

As the temperature is the same in both the cylinders, their internal energy change is also the same.

Since more heat should be added to cylinder B than A to raise their temperature equally, two distinct molar heat capacities are defined.

CV and Cp are the amounts of heat needed to raise the temperature of one mole of gas by 1 unit under constant volume and pressure, respectively.

## Heat Capacities of an Ideal Gas I

Heat capacity is the ratio of heat absorbed by the substance corresponding to its temperature change. It is also called thermal capacity and the SI unit of heat capacity is J/K. Whereas, specific heat capacity is defined as the amount of heat necessary to change the temperature of 1 kg of a substance by 1 K and is also called massic heat capacity. Its SI unit is J/kg⋅K.

Molar heat capacity quantifies the ratio of the amount of heat added (or removed) to increase (or decrease) the temperature of 1 mole of a substance by 1 K, either measured under a constant volume or constant pressure. Molar heat capacity is one of the characteristics of a substance and is also called molar specific heat. Its SI unit is J/mol·K. Measuring the molar heat capacity at constant volume is the easiest. The system acquires infinite number of possible heat capacities if neither the pressure nor the volume is constant. In most cases, the molar heat capacity at constant pressure (CP) is higher than the molar heat capacity at constant volume (CV). For example, the air has 40% greater CP than CV.