Proof: S (or entropy) is a valid state variable | Thermodynamics | Physics | Khan Academy

Khan Academy

17 Sept 200915:38

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TLDRThis video explains the concept of state variables in thermodynamics, focusing on internal energy (U) and the limitations of using heat (Q) as a state variable. The instructor uses a Carnot cycle to illustrate how internal energy remains constant regardless of the process path. The video then introduces entropy (S) as a new state variable, defined by the heat added to the system divided by the temperature. Through mathematical derivation, it's shown that entropy remains constant over a complete cycle, validating it as a legitimate state variable.

Takeaways

🔄 State variables like internal energy (U) should have the same value when a system returns to its initial state, regardless of the path taken.
🔍 Heat is not a state variable because its value can change depending on the process, unlike internal energy.
📉 The concept of 'heat content' as a state variable is flawed because it accumulates based on the path taken, not just the state.
♻️ The Carnot cycle is used to illustrate the properties of state variables, showing that internal energy remains constant.
🔄 The net heat added to a system in a cycle equals the work done if the internal energy does not change.
🔑 The area inside a cycle on a PV diagram represents the work done by or on the system.
🌡️ A new state variable 'S' is proposed, defined as the change in entropy, which is the heat added divided by the temperature at which it was added.
🔄 For 'S' to be a valid state variable, its change (delta S) around the Carnot cycle must be zero, indicating no change in entropy.
📚 The integral of pressure times volume change (PdV) is used to calculate the work done during isothermal processes in a Carnot cycle.
∫ The integral of nRT/V from one volume to another on an isotherm simplifies to nRT times the natural logarithm of the volume ratio.
🔍 The change in entropy (delta S) for a Carnot cycle is shown to be zero, confirming entropy as a valid state variable, although its deeper meaning is yet to be explored.

Q & A

What is a state variable?
-A state variable is a property of a system that depends only on the current state of the system, not on the path taken to reach that state. In the script, internal energy (U) is given as an example of a state variable.
Why can't heat be considered a state variable?
-Heat cannot be considered a state variable because its value depends on the path taken by the system. For example, the change in heat content can vary depending on the process, even if the system returns to the same state.
What is the Carnot cycle and why is it significant in this context?
-The Carnot cycle is an idealized thermodynamic cycle that consists of two isothermal processes and two adiabatic processes. It is significant because it is used to illustrate the concept of state variables and to derive the relationship between heat transfer and work in an idealized engine.
What is the relationship between the net heat added to a system and the work done during a Carnot cycle?
-In a Carnot cycle, the net heat added to the system is equal to the work done by the system, since the internal energy does not change during the cycle.
How is work related to heat transfer in the context of the Carnot cycle?
-Work in the Carnot cycle can be represented as the area under the curve on a PV diagram during an isothermal process. The heat added to the system is equal to the work done by the system in the isothermal expansion from A to B.
What is the integral calculation for heat transfer (Q1) during an isothermal process in a Carnot cycle?
-The integral calculation for heat transfer (Q1) during an isothermal process in a Carnot cycle is the integral from the initial volume (VA) to the final volume (VB) of nRT/V dV, where n is the number of moles, R is the gas constant, T is the temperature, and V is the volume.
What is the significance of the state variable S introduced in the script?
-The state variable S, which is defined as the change in entropy, is significant because it is a legitimate state variable that remains constant during a Carnot cycle, unlike heat content which depends on the path.
How is the change in entropy (ΔS) calculated for a Carnot cycle?
-The change in entropy (ΔS) for a Carnot cycle is calculated as the sum of the heat added to the system (Q1) divided by the temperature at which it was added (T1), plus the heat removed from the system (Q2) divided by the temperature at which it was removed (T2).
What is the final result of the change in entropy (ΔS) for a complete Carnot cycle?
-The final result of the change in entropy (ΔS) for a complete Carnot cycle is zero, indicating that entropy is indeed a state variable and remains constant regardless of the path taken during the cycle.
What does the natural log of the ratio of volumes (VB/VA and VD/VC) equal to 1 imply in the context of the Carnot cycle?
-In the context of the Carnot cycle, the natural log of the ratio of volumes (VB/VA and VD/VC) equal to 1 implies that the change in entropy (ΔS) is zero, confirming that entropy is a state variable that does not change during a complete cycle.

Outlines

00:00

🔍 Understanding State Variables and the Carnot Cycle

The paragraph introduces the concept of state variables, using internal energy (U) as an example. It explains that state variables, like U, should remain constant for a system returning to its initial state after a cycle, regardless of the path taken. The script contrasts this with heat, which is not a state variable due to its dependency on the process. The Carnot cycle is used to illustrate this, showing that the net heat added to a system during a cycle equals the work done, assuming no change in internal energy. The area inside the cycle on a PV diagram represents the work done, which is also the net heat added. The paragraph concludes by suggesting that a new state variable, related to heat, might be defined that remains constant over a cycle.

05:03

🔧 Calculating Heat and Work in the Carnot Cycle

This paragraph delves into the specifics of calculating heat transfer (Q1 and Q2) and work done during the Carnot cycle. It explains that the heat added to the system (Q1) can be equated to the work done when there is no change in internal energy. The work is expressed as the integral of pressure times the differential change in volume over the cycle's path on a PV diagram. The script then discusses how to evaluate these integrals by using the ideal gas law and the fact that the processes occur at constant temperatures, leading to the conclusion that Q1 and Q2 can be calculated as the product of the number of moles, gas constant, temperature, and the natural logarithm of the volume ratio between initial and final states.

10:11

🌡️ Defining a New State Variable Related to Heat Transfer

The speaker hypothesizes a new state variable, S, which changes proportionally to the heat added divided by the temperature at which it was added. The paragraph explores the validity of S as a state variable by examining its behavior over the Carnot cycle. It concludes that the change in S (delta S) over a complete cycle is zero, making it a legitimate state variable. The calculation involves summing the heat transfer divided by temperature for each isothermal process in the cycle and simplifies to the natural logarithm of the volume ratio, which has been proven to be consistent across all processes in the cycle, thus validating S.

15:13

🔄 The Implications of the State Variable S in Cyclic Processes

The final paragraph reflects on the implications of the newly defined state variable S, which remains unchanged after a complete cycle, indicating its legitimacy as a state variable. It emphasizes that although the exact physical meaning of S is not yet clear, its mathematical behavior is consistent with the properties of a state variable. The speaker suggests that further exploration in future videos will be needed to develop a deeper understanding of S at a more intuitive level.

Mindmap

Keywords

💡State Variable

A state variable is a property of a system that is solely dependent on the current state of the system, not on the path taken to reach that state. In the video, internal energy (U) is given as an example of a state variable, where its value at a point in a PV diagram should remain constant regardless of the process that led to that state.

💡Carnot Cycle

The Carnot cycle is a theoretical thermodynamic cycle that is the most efficient cycle possible for a heat engine. It is composed of two isothermal processes and two adiabatic processes. In the script, the Carnot cycle is used to illustrate the concept of state variables and to explore the properties of heat and work within this idealized cycle.

💡Heat Content

Heat content, as mentioned in the video, is a hypothetical state variable that the speaker defines to explore the nature of heat as a state variable. It is an illustrative concept used to show that the amount of heat added to a system during a cycle can vary depending on the path taken, thus violating the definition of a state variable.

💡Adiabatic Process

An adiabatic process is one in which no heat is exchanged with the surroundings. In the context of the Carnot cycle discussed in the video, the transitions from B to C and D to A are adiabatic, meaning no heat is added or removed during these stages of the cycle.

💡Isothermal Process

An isothermal process is one in which the temperature of the system remains constant. In the Carnot cycle, the transitions from A to B and C to D are isothermal, with heat exchange occurring at constant temperatures T1 and T2, respectively.

💡Internal Energy

Internal energy is the total energy contained within a system, which includes kinetic and potential energies of its molecules. It is a state variable, and the video emphasizes that it should not change during a complete Carnot cycle, as it is only dependent on the system's state.

💡Work

In thermodynamics, work is the transfer of energy that occurs when a force is applied over a distance. The script explains that during an isothermal process in the Carnot cycle, the heat added to the system is equal to the work done by the system.

💡Enthalpy

Enthalpy is a thermodynamic potential that is often used in place of internal energy when the system allows heat transfer. While not explicitly mentioned in the script, the concept of enthalpy is related to the discussion of heat and work in the Carnot cycle.

💡Entropy

Entropy is a thermodynamic property that measures the degree of disorder or randomness in a system. The script introduces a new state variable, S, which is later understood to represent entropy. The change in entropy (ΔS) is defined as the heat added to the system divided by the temperature at which it was added.

💡Ideal Gas Law

The ideal gas law is given by PV = nRT, where P is pressure, V is volume, n is the number of moles, R is the gas constant, and T is temperature. The script uses the ideal gas law to derive expressions for the heat exchanged during isothermal processes in the Carnot cycle.

💡Natural Logarithm

The natural logarithm is the logarithm to the base e and is used in various mathematical and scientific contexts, including thermodynamics. In the script, the natural logarithm is used to integrate the reciprocal of volume with respect to volume, which is part of the calculation for the heat exchanged in isothermal processes.

Highlights

The state variable U (internal energy) should remain constant at a given point in the PV diagram, regardless of the path taken to return to that point.

Heat cannot be used as a state variable because it depends on the path taken in the PV diagram.

Introducing a hypothetical 'heat content' variable demonstrates why heat is not a state variable due to its dependence on the process path.

In a Carnot cycle, the net heat added to the system equals the net work done by the system, as the internal energy change is zero.

A legitimate state variable is independent of the process path and depends only on the state of the system.

Defining a new state variable S as the change in heat added to the system divided by the temperature at which it was added.

To validate S as a state variable, the change in S (ΔS) around the Carnot cycle should be zero.

During the Carnot cycle, the calculation of ΔS involves Q1/T1 and Q2/T2, where Q1 is the heat added and Q2 is the heat removed.

Q1 can be calculated using the area under the PV curve during the isothermal expansion from A to B.

Q2 can be calculated similarly during the isothermal compression from C to D.

By integrating pressure as a function of volume for isothermal processes, Q1 and Q2 are determined using the ideal gas law.

The final expression for ΔS around the Carnot cycle involves the natural logarithms of volume ratios.

The natural log relationship shows that the volumes at different stages of the cycle are proportional, leading to the conclusion that ΔS is zero.

Proving that ΔS is zero confirms that S is a legitimate state variable.

S represents a state variable related to heat, though its micro-level interpretation requires further exploration.

Transcripts

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