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# Explain the microscopic mechanisms for the relation of macroscopic variables for an isothermal process, an isobaric process, and an isochoric process occurring in an ideal gas.

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For an isothermal process, we have the relation

$pV=const,$

which on a microscopic level means that the temperature remains the same, and with it the average kinetic energy of particles. If we increase the volume, the particles with the same energy will now occupy a larger space and have contact with larger walls of a container. This means that on average fewer particles will collide with the walls of the container, which equals to a smaller pressure, just as the previous relation suggests. Conversely, if the volume decreases, more particles will collide with the available walls, which means an increased pressure.

For an isobaric process, we have the relation

$\frac{V}{T}=const.$

On a microscopic level if we increase the volume of a container, the gas will occupy a larger space and fewer particles will collide with the walls. This means that we have to increase the kinetic energy of the particles by increasing the temperature so that even though fewer particles collide with the walls, they will collide more often due to their larger velocities. If we decrease the volume, more particles will collide with the walls, which means that we have to reduce their energy by reducing the temperature.

For an isochoric process, we have the relation

$\frac{p}{T}=const.$

If we increase the temperature while keeping the volume constant, the particles will collide with the walls more often, which means the pressure increases. If the temperature decreases, the particles collide with the walls less frequently, which means that the pressure decreases.

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