Constant volume process or isochoric process - When the gas is heated at a constant volume, its temperature and pressure will increase. Since there is no change in its volume, no external work is done by the gas. All the heat supplied is stored in the body of the gas in the form of internal energy. It may be noted that this process is governed by Gay-Lussac law.
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)
Now consider m kg of a certain gas being heated at a constant volume from an initial temperature T1 to final temperature T2. This process is shown on the p-v (i.e. pressure-volume)* diagram in Fig. 5.1
*It may be noted that the area below the p—v diagram of a thermodynamic process represents the work done during the process to some scale.
We know that
![](data:image/png;base64,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)
We also know that the internal energy,
![](data:image/png;base64,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)
Now consider m kg of a certain gas being heated at a constant volume from an initial temperature T1 to final temperature T2. This process is shown on the p-v (i.e. pressure-volume)* diagram in Fig. 5.1
*It may be noted that the area below the p—v diagram of a thermodynamic process represents the work done during the process to some scale.
We know that
We also know that the internal energy,
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