Question

Consider a process in which an ideal gas changes from state 1 to state 2 in such a way that its temperature changes from \(300 \mathrm{~K}\) to \(200 \mathrm{~K}\). (a) Describe how this change might be carried out while keeping the volume of the gas constant. (b) Describe how it might be carried out while keeping the pressure of the gas constant. (c) Does the change in \(\Delta E\) depend on the particular pathway taken to carry out this change of state? Explain.

Short answer

(a) In an isochoric process, the volume is constant, and the temperature decrease from 300 K to 200 K can be achieved by removing heat from the system. The change in internal energy, ΔE, is given by ΔE = mcΔT. (b) In an isobaric process, the pressure is constant, and the temperature decrease can be achieved by allowing the gas to expand and do work. According to the First Law of Thermodynamics, ΔE = Q - W. (c) The change in internal energy, ΔE, only depends on the initial and final temperatures, not on the particular pathway taken during the change of state. Therefore, ΔE will be the same for both isochoric and isobaric processes.

Step-by-step solution

(a) Isochoric process

In an isochoric process, the volume of the gas remains constant. To decrease the temperature of the gas from 300 K to 200 K at constant volume, we can remove heat from the system. Since the volume is constant, no work is done by the gas, and the heat exchange (Q) is equal to the change in internal energy, ΔE = mcΔT, where m is the mass of the gas, c is the specific heat capacity at constant volume, and ΔT is the change in temperature.

(b) Isobaric process

In an isobaric process, the pressure of the gas remains constant while the temperature changes. To achieve a decrease in temperature from 300 K to 200 K at constant pressure, we can allow the gas to expand against an external force, doing work in the process. As the gas loses energy to perform work, its temperature will decrease. So, in this case, both work (W) and heat exchange (Q) are involved. According to the First Law of Thermodynamics, ΔE = Q - W, where Q is the heat added to the system, and W is the work done by the system.

(c) Dependence of ΔE on the pathway

The change in internal energy, ΔE, only depends on the initial and final temperatures, and not on the particular pathway taken during the change of state. This is because for an ideal gas, internal energy only depends on the temperature. It doesn't matter if the process is isochoric or isobaric; the change in internal energy will be the same for the same temperature change.

Key concepts

Ideal Gas Law

The Ideal Gas Law is an equation of state for a hypothetical ideal gas. It is one of the fundamental equations in thermodynamics for gases and provides an approximation of the behavior of many gases under a variety of conditions. The equation is given by:\[ PV = nRT \]Here,

• \( P \) is the pressure of the gas,
• \( V \) is the volume,
• \( n \) is the amount of substance in moles,
• \( R \) is the ideal gas constant (approximately 8.314 J/(mol·K)),
• \( T \) is the absolute temperature in Kelvin.

The Ideal Gas Law combines several simpler gas laws and elucidates how pressure, volume, temperature, and moles of gas are interrelated. It is particularly useful when calculating any property if the other three are known.
This law assumes no interactions between the gas particles and that the particles occupy no volume themselves, which is only approximate, but often very useful in practical calculations.

Isochoric Process

An isochoric process, also known as an isovolumetric or constant-volume process, is a thermodynamic process in which the volume remains unchanged. This means that no work is done by or on the system since work (\( W \)) is defined as the product of pressure and change in volume.
In an isochoric process where the temperature of an ideal gas changes, the heat transfer (\( Q \)) can be directly related to the change in internal energy (\( \Delta E \)). This relationship is described by the formula:\[ \Delta E = Q = mc\Delta T \]where

• \( c \) is the specific heat capacity at constant volume,
• \( \Delta T \) is the change in temperature,
• \( m \) is the mass of the gas.

During an isochoric process, despite the constant volume, a temperature change implies there is heat exchange with the surroundings, affecting the internal energy of the gas directly.

Isobaric Process

An isobaric process is characterized by a constant pressure throughout the process. In this scenario, the volume of the gas may change as the temperature shifts.
If an ideal gas undergoes an isobaric process, work is done by the gas as it expands or contracts. The relationship among the heat exchange (\( Q \)), work done (\( W \)), and change in internal energy (\( \Delta E \)) can be explained using the first law of thermodynamics:\[ \Delta E = Q - W \]For an isobaric process, the work done by the gas is calculated using:\[ W = P \Delta V \]where

• \( P \) is the constant pressure,
• \( \Delta V \) is the change in volume.

The heat added to the gas results in both a change in internal energy and work done by the gas to change its volume, offering a slightly more complex energy exchange than the isochoric process.

Internal Energy

Internal energy is a crucial concept in thermodynamics, representing the total energy contained within a system. For an ideal gas, its internal energy primarily depends on its temperature, which is a reflection of the kinetic energy of the gas particles.
The change in internal energy (\( \Delta E \)) during a process is given by:\[ \Delta E = mc\Delta T \]For an ideal gas, this change is independent of the path taken and only depends on the initial and final temperatures.

• Any work done or heat exchanged during transitions between states is accounted for by changes in kinetic energy related to temperature shifts.

Whether undergoing an isochoric or isobaric transition, the internal energy of an ideal gas modifies solely based on temperature changes, without regard to the process taken. This path-independence underlines the importance of temperature in defining internal energy for ideal gases.