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The Ideal Gas Law is a fundamental equation in thermodynamics that describes how gases behave under various conditions of temperature (T), pressure (P), and volume (V). The law is expressed as \( PV = nRT \), where \( P \) is pressure, \( V \) is volume, \( n \) is the amount of gas in moles, \( R \) is the ideal gas constant (8.314 J/mol·K), and \( T \) is temperature in Kelvin.

This relationship helps determine how gases expand or contract when the temperature changes, or when they are compressed or allowed to expand. In the context of the exercise, the Ideal Gas Law was used to predict how the nitrogen gas behaves when it is first heated at constant volume and then allowed to expand at constant pressure. Understanding the interplay of these variables is key to predicting the behavior of an ideal gas in various scenarios, such as calculating the final temperature or the amount of work done by the gas.

When heat is transferred to a gas, it typically results in a change in temperature, depending on the conditions of the process. The heat transferred either increases the internal energy of the gas or does work by expanding against an external pressure.

In the specified exercise, the heat added during constant volume heating is calculated using the formula \( q = nC_v \Delta T \), where \( C_v \) is the specific heat at constant volume, \( n \) is the number of moles, and \( \Delta T \) is the change in temperature. This calculation provides insight into how the gas's temperature changes due to heat input.

Later, when the gas is allowed to expand at constant pressure, heat transfer is described by another formula: \( q = nC_p \Delta T \), where \( C_p \) is the specific heat at constant pressure. This part of the process shows how additional heat is used not just to increase internal energy but also to do work as the gas expands.

Work done by an ideal gas is an important concept when the gas expands or contracts. The work \( W \) done by the gas during an expansion or compression at constant pressure is calculated using the relationship: \( W = P \Delta V \), or more specifically, in terms of temperature change: \( W = nR(T_3 - T_2) \).

In the given problem, part of the work done results from the gas doubling its volume during expansion. This process requires energy, which is provided by the added heat. The calculation of work allows us to see how much energy is used in expanding the gas against a constant pressure.

• This process is essentially an energy conversion where heat energy is turned into mechanical work.
• Knowing the work done is crucial in thermodynamics to analyze energy efficiency and changes within systems.

The internal energy change \( \Delta U \) of a system is a central concept in thermodynamics that reflects the net change in the energy contained within the system. This change can be understood as the balance between heat added to the system and the work done by it.

Mathematically, it is expressed by the equation: \( \Delta U = q_{cv} + q_{cp} - W \), where \( q_{cv} \) is the heat at constant volume, \( q_{cp} \) is the heat added at constant pressure, and \( W \) is the work done by the system. In the context of the exercise, the total change in internal energy combines the effects of heating and expansion.

• This balance highlights the first law of thermodynamics, which states that energy cannot be created or destroyed.
• Understanding \( \Delta U \) is essential for analyzing how energy flows into and out of systems during various processes.