a sample of an ideal gas goes through the process shown in the figure below. from a to b, the process is adiabatic; from b to c, it is isobaric with 345 kj of energy entering the system by heat; from c to d, the process is isothermal; and from d to a, it is isobaric with 371 kj of energy leaving the system by heat. determine the difference in internal energy eint, b − eint, a.
Determination of the Difference in Internal Energy e_{int, b} - e_{int, a} in an Ideal Gas Process
Given Process:
A sample of an ideal gas undergoes a cyclic process consisting of four stages:
- From a to b: Adiabatic process.
- From b to c: Isobaric process with 345 kJ of energy entering the system by heat.
- From c to d: Isothermal process.
- From d to a: Isobaric process with 371 kJ of energy leaving the system by heat.
Analysis:
To determine the difference in internal energy e_{int, b} - e_{int, a}, we need to analyze the energy changes in each stage of the process.
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From a to b (Adiabatic Process):
In an adiabatic process, heat transfer (Q) is zero, so Q_{ab} = 0. Therefore, the change in internal energy (\Delta U) is equal to the work done (W_{ab}) in this process. Mathematically, \Delta U_{ab} = W_{ab}. -
From b to c (Isobaric Process):
In an isobaric process, the change in internal energy is given by the formula: \Delta U_{bc} = Q_{bc} - W_{bc}. Since 345 kJ of energy is entering the system as heat, Q_{bc} = 345 kJ. The work done is calculated as W_{bc} = P_{ext} \times \Delta V_{bc}, where P_{ext} is the external pressure (constant in isobaric process) and \Delta V_{bc} is the change in volume. -
From c to d (Isothermal Process):
In an isothermal process, the temperature remains constant, and therefore, the change in internal energy is zero (\Delta U_{cd} = 0). The heat absorbed (Q_{cd}) is equal to the work done (W_{cd}) in an isothermal process. -
From d to a (Isobaric Process):
Similar to the b to c process, the change in internal energy in this phase is given by \Delta U_{da} = Q_{da} - W_{da}. Here, 371 kJ of energy is leaving the system as heat, so Q_{da} = -371 kJ. Work done is calculated using W_{da} = P_{ext} \times \Delta V_{da}.
Calculation of Difference in Internal Energy e_{int, b} - e_{int, a}:
The difference in internal energy between states b and a can be computed by summing up the internal energy changes in each process:
e_{int, b} - e_{int, a} = \Delta U_{ab} + \Delta U_{bc} + \Delta U_{cd} + \Delta U_{da}.
After calculating \Delta U_{ab}, \Delta U_{bc}, \Delta U_{cd}, and \Delta U_{da} using the information provided for each stage, you can determine the final value of e_{int, b} - e_{int, a}.
In conclusion, the difference in internal energy e_{int, b} - e_{int, a} in an ideal gas process involving adiabatic, isobaric, and isothermal stages can be accurately determined by analyzing the energy transfers and work done in each phase of the process. By applying the principles of thermodynamics and the specific heat capacities of the gas involved, the internal energy change between the initial and final states of the gas can be calculated effectively.