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Chapter 3: Problem 17

There is no change in the internal energy of an ideal gas undergoing an isothermal process since the internal energy depends only on the temperature. Is it therefore correct to say that an isothermal process is the same as an adiabatic process for an ideal gas? Explain your answer.

Short Answer

Expert verified
An isothermal process is not the same as an adiabatic process for an ideal gas. Although there is no change in internal energy during an isothermal process due to constant temperature, an adiabatic process has no heat exchange with the surroundings, potentially leading to temperature changes. The internal energy changes in both processes are accounted for differently, with work done playing a key role.

Step by step solution

01

Define an isothermal process and its relation to internal energy

An isothermal process is a thermodynamic process in which the temperature remains constant throughout the entire process. Since the internal energy of an ideal gas depends only on temperature, then during the isothermal process, there is no change in the internal energy of the ideal gas.
02

Define an adiabatic process and its relation to internal energy

An adiabatic process is a thermodynamic process where the system doesn't exchange any heat with its surroundings. In an adiabatic process, the internal energy change of the ideal gas is given by the work done, \(Q=0\) (where \(Q\) represents heat exchange). The temperature of the gas, in general, doesn't remain constant in an adiabatic process.
03

Compare isothermal and adiabatic processes

While both isothermal and adiabatic processes can involve a change in the internal energy of an ideal gas, they are not the same. The isothermal process involves a constant temperature, while the adiabatic process involves no heat exchange between the system and surroundings. The internal energy change of an ideal gas undergoing an isothermal process is accounted for completely by work done on/by the gas since the temperature doesn't change. In an adiabatic process, the internal energy change depends on the work done without heat exchange. Thus, the result that internal energy remains constant in an isothermal process doesn't mean that the two processes are the same.
04

Conclusion

An isothermal process is not the same as an adiabatic process for an ideal gas. While it is true that there is no change in internal energy for an ideal gas undergoing an isothermal process (due to constant temperature), the adiabatic process differs as it involves no heat exchange between the system and surroundings, potentially leading to temperature changes.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Isothermal Process
In an isothermal process, the temperature of a system remains constant. This is significant because, for an ideal gas, the internal energy is solely determined by its temperature. Since there's no change in temperature during an isothermal process, there is also no change in the internal energy. In practical terms:
  • This kind of process might occur in a perfectly insulated environment with perfect heat exchange with the surroundings鈥攚hich is hard to achieve in reality.
  • Energy entering or leaving the system in the form of work is perfectly balanced by energy entering or leaving the system as heat, in order to maintain constant temperature.
While the temperature doesn鈥檛 change, the system does essential work. It can expand or compress, and this work is compensated by an identical amount of heat exchange to maintain thermal equilibrium.
Adiabatic Process
An adiabatic process skips exchanging heat with the environment. This means that all energy changes within the system happen due to work done on or by the system, affecting its internal energy. Key characteristics include:
  • The total heat exchange, symbolized as \(Q\), is zero \(Q=0\).
  • Temperature alterations are commonplace as a result of work done on or by the gas.
For example:
  • Compressing a gas adiabatically usually increases its temperature.
  • Expanding it lowers the temperature.
Despite involving changes in internal energy (due to work), adiabatic processes can differ vastly in how they physically affect the system's temperature compared to other processes.
Internal Energy
Internal energy refers to the complete energy within an ideal gas, being primarily reliant on its temperature. For different processes:
  • Isothermal: Internal energy remains stable since temperature stays constant.
  • Adiabatic: Internal energy can change since the absence of heat exchange means changes are due to work effects, which often result in a temperature shift.
Remember:
  • Internal energy transitions directly relate to temperature changes in an ideal gas. If temperature fluctuates, so does internal energy.
  • Since there's no heat input or output in an adiabatic process, work done directly shifts internal energy levels, manifesting usually as temperature variation.
Clearly distinguishing these processes and their impact on internal energy highlights the rich variety of thermodynamic behaviors exhibited by ideal gases.

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Most popular questions from this chapter

A hand-driven tire pump has a piston with a 2.50-cm diameter and a maximum stroke of \(30.0 \mathrm{cm}\). (a) How much work do you do in one stroke if the average gauge pressure is \(2.4 \times 10^{5} \mathrm{N} / \mathrm{m}^{2}\) (about \(35 \mathrm{psi}\) )? (b) What average force do you exert on the piston, neglecting friction and gravitational force?

An ideal gas expands isothermally along AB and does 700 J of work (see below). (a) How much heat does the gas exchange along AB? (b) The gas then expands adiabatically along BC and does 400 J of work. When the gas returns to A along CA, it exhausts 100 J of heat to its surroundings. How much work is done on the gas along this path?

Consider a transformation from point \(A\) to \(B\) in a two-step process. First, the pressure is lowered from 3 MPa at point \(A\) to a pressure of \(1 \mathrm{MPa}\), while keeping the volume at 2 L by cooling the system. The state reached is labeled C. Then the system is heated at a constant pressure to reach a volume of \(6 \mathrm{L}\) in the state \(B\). (a) Find the amount of work done on the \(A C B\) path. (b) Find the amount of heat exchanged by the system when it goes from \(A\) to \(B\) on the \(A C B\) path. (c) Compare the change in the internal energy when the \(A B\) process occurs adiabatically with the AB change through the two-step process on the \(A C B\) path.

On an adiabatic process of an ideal gas pressure, volume and temperature change such that \(p V^{\gamma}\) is constant with \(\gamma=5 / 3\) for monatomic gas such as helium and \(\gamma=7 / 5\) for diatomic gas such as hydrogen at room temperature. Use numerical values to plot two isotherms of 1 mol of helium gas using ideal gas law and two adiabatic processes mediating between them. Use T_{1}=500 \mathrm{K}, V_{1}=1 \mathrm{L}, \text { and } T_{2}=300 \mathrm{K} \text { for your plot. }

A great deal of effort, time, and money has been spent in the quest for a so- called perpetual-motion machine, which is defined as a hypothetical machine that operates or produces useful work indefinitely and/or a hypothetical machine that produces more work or energy than it consumes. Explain, in terms of the first law of thermodynamics, why or why not such a machine is likely to be constructed.
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