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To understand why pressure does not change during an isobaric process, we need to first understand what "pressure" is. Pressure is directly proportional to the frequency at which the gas molecules collide with the piston and their speed. In order to conserve the pressure throughout this process while changing the volume of the system, heat must be transferred into and out of the system to compensate for the constant pressure. For example, during an isobaric compression, the volume of the system would decrease, which would mean that there would be a greater frequency at which the gas molecules collide with the piston and pressure of the system would normally increase. However, in order for the pressure to stay constant, heat must be transferred out of the system, thus decreasing the frequency at which the molecules collide with the piston, and keeping pressure the same as before the system was compressed. This concept is a little hard to grasp, but I hope I did not confuse you even more. Hope this helps.
Whenever you change the volume of a gas, the temperature must change if you want to keep the pressure the same. An expansion of gas would cause pressure and temperature to decrease and compression of gas would cause the pressure and temperature to increase. To counteract these and keep the pressure constant, you need to add heat to an expansion of gas and remove heat from a compression of gas
The pressure of a system is not completely defined by the number of molecules in a system, but also their average kinetic energy, or temperature. Pressure can be conceptualized by molecules hitting a surface, and "pushing on it," in a way. The more collisions that there are, and the higher the energy those collisions are, the harder they will push. If we decrease the volume of a system, we are decreasing the volume of the system, and thus increasing the number of collisions. If we, though, decrease the temperature of the system, we can balance the increase in pressure from the number of collisions. By doing this, we can keep the pressure constant.
If you're compressing a gas and the process is isobaric, then most likely the temperature is decreasing since the equation PV=nRT holds that if volume is going down (compression) and pressure is staying the same (isobaric), then either n, R, or T is going to change. R is a constant so we can rule that out and generally unless the problem explicitly says so the number of moles is always going to remain constant. Therefore we can assume that if the left half of the equation is decreasing, then the temperature is decreasing as well since it is the only variable parameter.
It helps me to use the equations to remember how these operate. Using PV=nRT, when you decrease the volume through the compression, some value on the right side of the equation should also decrease. Since we're keeping the number of moles consistent and R never changes, it's safe to assume that the temperature would decrease in order to keep the pressure the same. The other replies explained the conceptual side to this perfectly.
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