20.11
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 un…
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.
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Q1: Why do cylinders with sealed and movable pistons require different amounts of heat to reach the same temperature?
A sealed cylinder's gas cannot expand, so all applied heat increases internal energy. A cylinder with a movable piston allows gas expansion, meaning some heat performs work against external pressure. Since internal energy change depends only on temperature, more total heat must be added to the expandable cylinder to achieve the same temperature rise as the sealed one.
Q2: What is the difference between CV and Cp for an ideal gas?
CV is the molar heat capacity at constant volume, measuring heat needed to raise one mole's temperature by 1 K without volume change. Cp is the molar heat capacity at constant pressure, measuring heat needed under constant external pressure. Cp exceeds CV because additional heat performs expansion work. For example, air's Cp is approximately 40% greater than its CV.
Q3: How does the first law of thermodynamics explain why heat capacity differs between constant volume and constant pressure conditions?
The first law states that heat added equals internal energy change plus work done. At constant volume, no work occurs, so all heat increases internal energy. At constant pressure, heat is divided between increasing internal energy and performing expansion work. Therefore, reaching the same temperature requires more heat at constant pressure, defining two distinct molar heat capacities.
Q4: What is molar heat capacity and how is it measured?
Molar heat capacity quantifies the heat required to change one mole of a substance's temperature by 1 K, expressed in J/mol·K. It can be measured at constant volume or constant pressure. Measuring at constant volume is easiest because no work is performed. If neither pressure nor volume remains constant, the system acquires infinite possible heat capacities.
Q5: How does work affect the relationship between heat and internal energy in an expanding gas?
When gas expands against external pressure, it performs work, converting some applied heat into mechanical energy. According to the first law, heat input is partitioned between increasing internal energy and doing work. In a sealed container, all heat increases internal energy; in an expandable system, only the portion not used for work contributes to internal energy change.
Q6: Why is the molar heat capacity at constant pressure always greater than at constant volume?
At constant pressure, applied heat must both increase internal energy and perform expansion work against external pressure. At constant volume, all heat increases internal energy without work. Since the same temperature increase requires identical internal energy change in both cases, constant pressure conditions demand additional heat to account for work performed during expansion.
Q7: What role does internal energy play in determining heat capacity differences between the two cylinders?
Internal energy depends only on temperature, so both cylinders experience identical internal energy changes when heated to the same final temperature. However, the sealed cylinder converts all applied heat into internal energy change, while the expandable cylinder diverts some heat to perform work. This difference in heat distribution, not internal energy change, defines the two distinct molar heat capacities.