A cylinder contains a perfect gas in thermodynamic...

A cylinder contains a perfect gas in thermodynamic equilibrium at $p, V, T, U$ (internal energy) and $S$ (entropy). The cylinder is surrounded by a very large heat reservoir at the same temperature $T$. The cylinder walls and piston can be either perfect thermal conductors or perfect thermal insulators. The piston is moved to produce a small volume change $\pm \Delta V$. "Slow" or "fast" means that during the volume change the speed of the piston is very much less than, or very much greater than, molecular speeds at temperature $T$. For each of the five processes below show (on your answer sheet) whether the changes (after the reestablishment of equilibrium) in the other quantities have been positive, negative, or zero. FIGURE CAN'T COPY. 1. $(+\Delta V)$ (slow) (conduct) 2. $(+\Delta V)$ (slow) (insulate) 3. $(+\Delta V)$ (fast) (insulate) 4. $(+\Delta V)$ (fast) (conduct) 5. $(-\Delta V)$ (fast) (conduct) TABLE CAN'T COPY.

A cylinder contains a perfect gas in thermodynamic equilibrium at $p, V, T, U$ (internal energy) and $S$ (entropy). The cylinder is surrounded by a very large heat reservoir at the same temperature $T$. The cylinder walls and piston can be either perfect thermal conductors or perfect thermal insulators. The piston is moved to produce a small volume change $\pm \Delta V$. "Slow" or "fast" means that during the volume change the speed of the piston is very much less than, or very much greater than, molecular speeds at temperature $T$. For each of the five processes below show (on your answer sheet) whether the changes (after the reestablishment of equilibrium) in the other quantities have been positive, negative, or zero.
FIGURE CAN'T COPY.
1. $(+\Delta V)$ (slow) (conduct)
2. $(+\Delta V)$ (slow) (insulate)
3. $(+\Delta V)$ (fast) (insulate)
4. $(+\Delta V)$ (fast) (conduct)
5. $(-\Delta V)$ (fast) (conduct)
TABLE CAN'T COPY.

Key Concepts

Thermal Conduction and Insulation

The effectiveness of heat exchange between a system and its surroundings depends on the thermal properties of the boundaries. Perfect thermal conductors allow rapid equilibration with a heat reservoir, leading to isothermal conditions if the process is slow, while perfect insulators block heat exchange, making the system behave adiabatically even if in contact with a reservoir. This distinction is key to predicting the evolution of thermodynamic state variables during a process.

Reversible and Irreversible Processes

This concept distinguishes between processes where the system remains infinitesimally close to equilibrium (reversible, often slow, quasi?static changes) and those where the system is driven far from equilibrium (irreversible, often fast changes). In reversible processes, state functions change in a controlled, predictable manner, whereas in irreversible processes, additional entropy production and non-equilibrium effects occur, altering the final values of thermodynamic quantities.

Isothermal Processes

An isothermal process occurs at constant temperature, usually maintained by a large heat reservoir that can absorb or supply heat as needed. In thermodynamic analysis, if a system is in contact with such a reservoir and the process is slow enough for heat exchange to occur, the temperature remains fixed, simplifying the relationship between work done and heat transfer in accordance with the ideal gas law.

Adiabatic Processes

An adiabatic process is one in which no heat is exchanged with the surroundings, typically due to insulation or because the process occurs too quickly for thermal interaction. In such processes the internal energy change is solely determined by the work done, leading to a change in temperature that depends on the nature of the work (compression or expansion), and this affects other state functions such as pressure.

First Law of Thermodynamics

This fundamental principle, stating that the change in internal energy of a system equals the heat added to the system minus the work done by the system, is crucial in analyzing energy exchanges during any process. It provides the basis for determining how internal energy, temperature, and work interrelate, especially when distinguishing between processes with and without heat exchange.

Entropy

Entropy is a measure of the disorder or randomness in a system, and in thermodynamic processes, it reflects the degree of irreversibility. In reversible processes, the entropy change is determined solely by the heat exchange divided by temperature, while irreversible processes produce additional entropy even if the initial and final temperatures are the same. Understanding entropy is essential to predicting the spontaneity and outcome of a process.

Perfect Gas Behavior in Thermodynamics

Ideal or perfect gases follow the relation PV = nRT and have internal energy that depends solely on temperature. This simplifies the analysis of processes, since changes in temperature directly correspond to changes in internal energy. The behavior of perfect gases under various processes (isothermal, adiabatic) provides a model framework for understanding more complex thermodynamic systems.

Transcript

00:01 For this problem on the topic of entropy, we had to suppose that a mole of monotomic ideal gas is taken from an initial pressure p1 and volume v1 through two steps.

00:08 Firstly, an isothermal expansion to volume 2 v1, and then a pressure increased to 2 p1 and constant volume.

00:14 We want to find various quantities related to this two -step process.

00:19 The gas is then returned to its initial state and then again taken to the same final state but through two different steps.

00:26 Firstly, in isothermal compression to pressure 2 p1.

00:29 And then a volume increased to 2v1 at constant pressure.

00:32 And we want to find the same quantities for this new process.

00:36 Now, we note the connection between molar heat capacity and degrees of degrees of freedom gives us the molar heat capacity at constant volume.

00:43 Cv to be three times the gas constant r over 2.

00:48 And that for constant pressure, cp is 5r over 2.

00:57 And the constant gamma is equal to 5 over 3.

01:03 Now for part a, since this is an ideal gas, equation 19 .45 holes, which implies that the change in internal energy delta e is equal to zero for this process.

01:14 And also equation 19 .14 applies.

01:18 So using the first law of thermodynamics, we get q equal to 0 plus w, which becomes nr, t1, times the natural log of v2 over v1, which is simply p1 v1 times the natural log of 2.

01:39 This gives us the ratio that we require, q divided by p1 v1 to be the natural log of 2, which is 0 .693.

01:57 Next for part b, the gas law in ratio form implies that the pressure decreased by a factor of 2 during the isothermal expansion process to v2 equal to 2 v1 so it needs to increase by a factor of 4 in the step in order to reach a final pressure of p2 is equal to 2 p1 that same ratio form now applied to this constant volume process yields 4 is equal to t2 t1 which is used in the following equation the heat energy q is equal to the number of moles n times cv times delta t and we can write this as n times 3 over 2 r into t 2 minus t 1, which is 3 over 2 nr t 1 into t 2 over t 1 minus 1.

02:55 And so this becomes 3 over 2 p1 v1 into 4 minus 1, which is 9.

03:06 9 over 2 p1 v1 which then implies that q over p1 v1 is equal to 9 over 2 which is 4 .5 that's the ratio that we require now for part c the work done during the isothermal expansion process may be obtained by equation 1914 and this worked on w is nr t1 times the natural log of v2 by v1 which is simply p1 v1 times the natural log of 2 which means that w over p1 v1 is the natural log of 2 which is simply 0 .690 3.

04:14 For part d, in step 2, where the volume is kept constant, the work done is equal to 0, since dv is equal to 0...

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