DATA You place a quantity of gas into a metal cylinder that...

DATA You place a quantity of gas into a metal cylinder that has a movable piston at one end. No gas leaks out of the cylinder as the piston moves. The external force applied to the piston can be varied to change the gas pressure as you move the piston to change the volume of the gas. A pressure gauge attached to the interior wall of the cylinder measures the gas pressure, and you can calculate the volume of the gas from a measurement of the piston's position in the cylinder. You start with a pressure of 1.0 atm and a gas volume of 3.0 L. Holding the pressure constant, you increase the volume to $5.0 \mathrm{~L}$ Then, keeping the volume constant at $5.0 \mathrm{~L},$ you increase the pressure to 3.0 atm. Next you decrease the pressure linearly as a function of volume until the volume is $3.0 \mathrm{~L}$ and the pressure is 2.0 atm. Finally, you keep the volume constant at $3.0 \mathrm{~L}$ and decrease the pressure to 1.0 atm, returning the gas to its initial pressure and volume. The walls of the cylinder are good conductors of heat, and you provide the required heat sources and heat sinks so that the necessary heat flows can occur. At these relatively high pressures, you suspect that the ideal-gas equation will not apply with much accuracy. You don't know what gas is in the cylinder or whether it is monatomic, diatomic, or polyatomic. (a) Plot the cycle in the $p V$ -plane. (b) What is the net heat flow for the gas during this cycle? Is there net heat flow into or out of the gas?

Key Concepts

Thermodynamic Cycle

A thermodynamic cycle is a series of processes that return a system to its initial state, meaning that all state variables (such as pressure, volume, and temperature) are the same at the beginning and the end of the cycle. Cycles are fundamental in understanding how work is extracted or input into a system and how heat is transferred during the various processes.

pV Diagram

A pressure-volume (pV) diagram graphically represents the states of a system and the processes it undergoes. Each point on the diagram corresponds to a particular state of pressure and volume, while the paths connecting these points illustrate different processes. The area under or enclosed by the curves can indicate work done by or on the system during these processes.

First Law of Thermodynamics

The First Law of Thermodynamics, which is essentially a statement of energy conservation, relates the change in a system's internal energy to the net heat flow into the system and the work done by the system. In cyclic processes, because the system returns to its original state, the change in internal energy over one complete cycle is zero, and therefore the net heat added is equal to the net work done by the gas.

Process Types (Isobaric and Isochoric Processes)

Various thermodynamic processes are characterized by the conditions maintained during the process. In an isobaric process, the pressure remains constant while volume and temperature may change, and in an isochoric process, the volume remains constant while pressure changes. Recognizing these processes is crucial for analyzing how heat and work are transferred during each segment of a cycle.

Non-Ideal Gas Behavior

Real gases, especially at relatively high pressures or low temperatures, often deviate from the ideal gas law due to intermolecular forces and finite molecular volumes. Understanding non-ideal gas behavior is important when analyzing systems where ideal-gas approximations become inaccurate, impacting calculations of state variables, work, and heat flow.

Heat Flow in Cyclic Processes

Heat flow in a thermodynamic cycle involves energy entering or leaving the system during various processes. In a cyclic process, while the internal energy returns to its original value over the complete cycle, the net heat flow is balanced by the net work done by the system. Analyzing the direction and magnitude of heat transfer in each process is key to understanding energy conversion and efficiency.

Transcript

00:01 In this problem, we're going to talk about the first law of thermodynamics.

00:03 So what we need to remember is that the first law of thermodynamics tells us that delta u, the variation in internal energy of a gas, is equal to the heat provided to the gas, minus the work done by the gas.

00:21 And it's important to remember this convention because if the heat were provided from the gas to the system, then the heat would be negative.

00:28 And the work if it was done on the gas then it would be negative as well okay so what we have in a problem are several processes that a gas undergoes initially the gas has a pressure of one atm and a volume of three liters and it expands at constant pressure to five liters then its pressure increases at constant volume to 3 atm, then its pressure decreases linearly with the volume to 2 atm and 3 liters.

01:24 And finally, the gas loses pressure at constant volume to 1 atm and 3 liters.

01:36 Okay? okay, and our going question a is to draw a pv diagram of our gas, of our process actually.

01:50 So here for p we have 1 atm, 2 atm, 3 atm, and the volume is 3 liters, actually, yeah, 3 liters and 5 liters.

02:11 So initially, the initial point is this one here, okay, this point.

02:23 Then the gas expands at constant pressure to this point here.

02:36 Then it keeps the volume constant and expands to three atm, expense, actually not expends it, increases pressure to three atm.

02:50 Then the pressure decreases linearly with the volume to 2 atm and the volume to 3 liters and finally the pressure decreases to the original value of pressure and volume at constant volume so this is our graph and in question b we have to calculate how much heat flowed into the system or out of the system during this process.

03:35 We know the delta u that is a variation in internal energy is zero.

03:40 And why do we know that? the internal energy of a system depends only on its point in the pvt space, meaning that even that p, v and t are the same, then u is going to be the same...

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