A quantity of an ideal gas is compressed to half" its initial volume. The process may be adiabatic, isothermal or isobaric. Rank those three processes in order of the work required of an external agent, least to greatest.
doesn't adiabatic mean no work, so it would be the least, correct? and what about the other two?
Ah, the wonderful world of thermodynamics! You're correct that an adiabatic process means no heat exchange takes place. Therefore, it doesn't require any heat transfer work, making it the least in terms of work required.
Now, let's talk about the other two processes. In an isothermal process, the temperature remains constant throughout the compression. This means that heat is constantly transferred to maintain the temperature, resulting in the highest work required among the three processes.
Lastly, we have the isobaric process, where the pressure remains constant during compression. In this case, some heat transfer work is required, but not as much as in the isothermal process. So, we can rank them in terms of work required from least to greatest as: adiabatic, isobaric, and isothermal.
Hope that clarifies things without making you "compress" your brain too much!
You are correct that in an adiabatic process, no heat is exchanged between the system and its surroundings. However, this does not imply that no work is done. In fact, in an adiabatic process, work is done on or by the system to compress or expand the gas.
Now, let's consider the three processes:
1. Isothermal Process: In an isothermal process, the temperature remains constant. This means that the gas will expand or contract while transferring heat to or from its surroundings to maintain a constant temperature. The work done by an external agent in an isothermal process is given by the equation W = nRT ln(Vf/Vi), where n is the number of moles of gas, R is the gas constant, T is the temperature, Vf is the final volume, and Vi is the initial volume. Since the volume is halved in this case, the work done will be negative, indicating work done by the system. The negative sign indicates that an external agent has to do positive work on the gas, making the work required by an external agent the greatest in an isothermal process.
2. Adiabatic Process: As mentioned earlier, in an adiabatic process, no heat is exchanged between the system and its surroundings. The work done in an adiabatic process is given by the equation W = (PfVf - PiVi)/(1 - γ), where Pf and Pi are the final and initial pressures, Vf and Vi are the final and initial volumes, and γ is the heat capacity ratio or adiabatic index of the gas. In compression, the volume decreases, so Vf < Vi, and the work done by an external agent will be positive. Since no heat is lost or gained during the process, the work required by an external agent in an adiabatic process is greater than zero but less than in an isothermal process.
3. Isobaric Process: In an isobaric process, the pressure remains constant. This means that the gas can expand or contract freely, and only the volume changes. As the volume halves, the work done is given by W = PΔV, where P is the constant pressure, and ΔV is the change in volume. In this case, the work done is negative, indicating work done by the system. The negative sign implies that an external agent needs to do positive work on the gas in order to compress it. Therefore, the work required by an external agent in an isobaric process is less than in an isothermal process but greater than in an adiabatic process.
So, the order of the work required of an external agent, from least to greatest, would be:
1. Adiabatic Process
2. Isobaric Process
3. Isothermal Process
To rank the three processes (adiabatic, isothermal, and isobaric) in terms of the work required of an external agent, you need to consider the specific characteristics of each process.
1. Adiabatic process: An adiabatic process is one in which no heat is exchanged with the surroundings. Since there is no heat transfer in an adiabatic process, the work done on or by the gas is solely due to the change in its internal energy. For an adiabatic compression process, work is done on the gas to reduce its volume. Therefore, compared to the other two processes, the adiabatic process requires the most work from an external agent.
2. Isothermal process: An isothermal process is performed at a constant temperature. In this process, the gas exchanges heat with the surroundings to ensure that the temperature remains constant. When compressing an ideal gas isothermally, the gas cools down due to the energy transfer, and the work required to compress the gas is less compared to the adiabatic process. So, an isothermal process requires less work from an external agent than an adiabatic process.
3. Isobaric process: An isobaric process occurs at a constant pressure. In this process, the gas can exchange heat with the surroundings to maintain constant pressure. When considering compression, the work done in an isobaric process is greater than in an isothermal process. This is because, in an isobaric process, the gas can expand or contract against a constant external pressure while exchanging heat, which requires more work than an isothermal process. Therefore, an isobaric process requires the most work from an external agent among the three processes.
So, to answer your question, the ranking in terms of the work required from an external agent, from least to greatest, would be: isothermal < isobaric < adiabatic.