Thermodynamic State Variables And Equation Of State

An equilibrium state of a thermodynamic system is described by its thermodynamic state variables, which are macroscopic properties that have definite values for a system in equilibrium.

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Examples of state variables: Pressure ($P$), Volume ($V$), Temperature ($T$), Mass ($m$), composition.

Not all states are equilibrium states (e.g., a gas undergoing rapid expansion or an explosive reaction). Non-equilibrium states don't have uniform values for variables like P or T throughout the system.

Thermodynamic state variables describe only equilibrium states.

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The state variables of a system are not all independent. The relation connecting the state variables is called the equation of state.

Example: For an ideal gas, the equation of state is $PV = \mu RT$. This means for a fixed amount of gas ($\mu$), knowing two state variables (like P and V) determines the third (T).

A curve showing the relation between two state variables (like P and V) while another is kept constant (like T) is useful. A P-V curve at constant temperature is called an isotherm.

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State variables can be classified into two types:

• Extensive variables: Depend on the size or extent of the system (e.g., Volume V, Mass M, Internal Energy U). If the system is divided into two equal parts, the value of an extensive variable for each part is half the value for the whole system.
• Intensive variables: Do not depend on the size of the system (e.g., Pressure P, Temperature T, Density $\rho$). If the system is divided into two equal parts, the value of an intensive variable for each part remains the same as for the whole system.

Heat ($\Delta Q$) and work ($\Delta W$) are not state variables; they are quantities of energy transfer that depend on the process path.

Thermodynamic Processes

A thermodynamic process is a change in the state of a thermodynamic system.

Processes can take a system from an initial equilibrium state to a final equilibrium state. However, the system may pass through non-equilibrium states during the process if the change is not carried out carefully.

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Quasi-Static Process

A quasi-static process is an idealized thermodynamic process that proceeds so slowly that the system remains in thermal and mechanical equilibrium with its surroundings at every infinitesimal step.

• The difference between the system's pressure and the external pressure is infinitesimally small.
• The difference between the system's temperature and the temperature of the surrounding reservoir is infinitesimally small.

A quasi-static process is a hypothetical construct, infinitely slow. Real processes can approximate quasi-static processes if they are carried out sufficiently slowly.

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Common types of quasi-static thermodynamic processes:

• Isothermal Process: Temperature ($T$) is kept constant throughout the process. For an ideal gas, $PV = \text{constant}$ (Boyle's Law). $\Delta U = 0$ for an ideal gas. $W = \mu RT \ln(V_f/V_i)$.
• Adiabatic Process: No heat exchange occurs between the system and its surroundings ($\Delta Q = 0$). The system is insulated. For an ideal gas, $PV^\gamma = \text{constant}$, where $\gamma = C_P/C_V$. $W = -\Delta U$.
• Isochoric Process: Volume ($V$) is kept constant throughout the process. No work is done ($\Delta W = 0$). Heat exchanged changes internal energy: $\Delta Q = \Delta U$.
• Isobaric Process: Pressure ($P$) is kept constant throughout the process. Work done by the system is $W = P \Delta V$. $\Delta Q = \Delta U + P \Delta V$.
• Cyclic Process: The system returns to its initial state after a series of processes. Since $U$ is a state variable, $\Delta U = 0$ over a complete cycle. The First Law gives $\Delta Q_{cycle} = \Delta W_{cycle}$.

These processes are often represented on a P-V diagram (pressure versus volume), where the area under the curve represents the work done in a quasi-static process.

The curve for an adiabatic process ($PV^\gamma=$ constant, $\gamma > 1$) is steeper than the curve for an isothermal process ($PV=$ constant) at any given point on a P-V diagram passing through that point.

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Isothermal Process

In a quasi-static isothermal process, the temperature $T$ of the system remains constant. For an ideal gas, $U$ depends only on $T$, so $\Delta U = 0$. The First Law becomes $\Delta Q = \Delta W$.

Work done by an ideal gas during an isothermal expansion from $V_1$ to $V_2$ is calculated by integrating $W = \int_{V_1}^{V_2} P \, dV$. Using $P = \mu RT/V$:

$$ W = \int_{V_1}^{V_2} \frac{\mu RT}{V} \, dV = \mu RT \int_{V_1}^{V_2} \frac{1}{V} \, dV = \mu RT [\ln V]_{V_1}^{V_2} $$ $$ W = \mu RT \ln\left(\frac{V_2}{V_1}\right) $$

The heat absorbed during isothermal expansion equals this work done: $Q = W$. In isothermal compression ($V_2 < V_1$), $W < 0$ and $Q < 0$, meaning work is done *on* the system, and heat is *released* by the system.

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Adiabatic Process

In a quasi-static adiabatic process, no heat is transferred between the system and surroundings ($\Delta Q = 0$). The First Law becomes $0 = \Delta U + \Delta W$, or $\Delta W = -\Delta U$. Work done by the system decreases its internal energy, and work done on the system increases its internal energy.

For an ideal gas undergoing a quasi-static adiabatic process, the pressure and volume are related by:

$$ PV^\gamma = \text{constant} $$

where $\gamma = C_P/C_V$ is the ratio of specific heats ($\gamma > 1$ for all gases). The work done during an adiabatic change from $(P_1, V_1, T_1)$ to $(P_2, V_2, T_2)$ is $W = \int_{V_1}^{V_2} P \, dV$. Using $P V^\gamma = K$ (constant), $P = K V^{-\gamma}$.

$$ W = \int_{V_1}^{V_2} K V^{-\gamma} \, dV = K \left[\frac{V^{-\gamma+1}}{-\gamma+1}\right]_{V_1}^{V_2} = \frac{K}{1-\gamma} [V^{1-\gamma}]_{V_1}^{V_2} = \frac{K}{1-\gamma} (V_2^{1-\gamma} - V_1^{1-\gamma}) $$

Since $K = P_1 V_1^\gamma = P_2 V_2^\gamma$, we can write $K V_2^{1-\gamma} = P_2 V_2^\gamma V_2^{1-\gamma} = P_2 V_2$ and $K V_1^{1-\gamma} = P_1 V_1^\gamma V_1^{1-\gamma} = P_1 V_1$.

$$ W = \frac{1}{1-\gamma} (P_2 V_2 - P_1 V_1) $$

Using $PV = \mu RT$, $W = \frac{1}{1-\gamma} (\mu R T_2 - \mu R T_1) = \frac{\mu R}{\gamma-1} (T_1 - T_2)$.

In adiabatic expansion ($W>0$), $V_2 > V_1$. From $PV^\gamma=$ constant, $P_2 < P_1$. From $P_1 V_1 / T_1 = P_2 V_2 / T_2$, $T_2 = T_1 (P_2/P_1)(V_2/V_1) = T_1 (P_2/P_1) (P_1/P_2)^{1/\gamma} = T_1 (P_2/P_1)^{1-1/\gamma} = T_1 (P_2/P_1)^{(\gamma-1)/\gamma}$. Since $P_2 < P_1$, $T_2 < T_1$. Adiabatic expansion causes cooling. Similarly, adiabatic compression causes heating.

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Isochoric Process

In a quasi-static isochoric process, the volume $V$ of the system is kept constant. Since there is no change in volume ($\Delta V = 0$), no work is done ($W = P \Delta V = 0$). The First Law becomes $\Delta Q = \Delta U$. All heat supplied goes to increase the internal energy and temperature of the system.

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Isobaric Process

In a quasi-static isobaric process, the pressure $P$ of the system is kept constant. Work done by the system during a volume change from $V_1$ to $V_2$ is $W = \int_{V_1}^{V_2} P \, dV = P \int_{V_1}^{V_2} dV = P (V_2 - V_1) = P \Delta V$. Using the ideal gas law, $W = P \Delta V = \mu R \Delta T$. The First Law is $\Delta Q = \Delta U + P \Delta V$. Heat supplied partly increases internal energy and partly does work.

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Cyclic Process

A cyclic process is a series of processes that returns the system to its initial state. Since internal energy $U$ is a state variable, the change in internal energy over a complete cycle is zero ($\Delta U_{cycle} = 0$). According to the First Law, $\Delta Q_{cycle} = \Delta U_{cycle} + \Delta W_{cycle}$, so $\Delta Q_{cycle} = \Delta W_{cycle}$. The net heat absorbed by the system in a cyclic process equals the net work done by the system over the cycle.

Heat Engines

A heat engine is a device designed to convert thermal energy (heat) into mechanical work through a cyclic process.

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Basic components and operation of a heat engine:

1. It uses a working substance (e.g., gas, steam) that undergoes a thermodynamic cycle.
2. The working substance absorbs heat ($Q_1$) from a high-temperature reservoir (source) at temperature $T_1$.
3. The working substance releases heat ($Q_2$) to a low-temperature reservoir (sink) at temperature $T_2$.
4. In the process, the system performs a net amount of work ($W$) on its surroundings over one complete cycle.

According to the First Law of Thermodynamics, over one complete cycle, the net heat absorbed equals the net work done: $W = Q_1 - Q_2$.

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The efficiency ($\eta$) of a heat engine is defined as the ratio of the net work output ($W$) to the heat input ($Q_1$) in a cycle:

$$ \eta = \frac{W}{Q_1} = \frac{Q_1 - Q_2}{Q_1} = 1 - \frac{Q_2}{Q_1} $$

For a heat engine to be 100% efficient ($\eta = 1$), it would require $Q_2 = 0$, meaning all the absorbed heat is converted into work. However, experience shows that some heat is always released to the cold reservoir ($Q_2 > 0$), implying $\eta < 1$. The Second Law of Thermodynamics sets a fundamental limit on the maximum possible efficiency.

Refrigerators And Heat Pumps

A refrigerator (or heat pump) is a device that operates on a cycle, like a heat engine, but in reverse. It transfers heat from a cold reservoir to a hot reservoir by consuming external work.

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Operation of a refrigerator:

• Work ($W$) is done on the working substance by external means.
• The working substance extracts heat ($Q_2$) from a cold reservoir at temperature $T_2$ (e.g., the inside of the refrigerator).
• The working substance releases heat ($Q_1$) to a hot reservoir at temperature $T_1$ (e.g., the surroundings).

According to the First Law, for a complete cycle, the energy balance is $Q_1 = W + Q_2$.

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The performance of a refrigerator is measured by its coefficient of performance ($\alpha$), defined as the ratio of the heat extracted from the cold reservoir ($Q_2$) to the work done on the system ($W$) in a cycle:

$$ \alpha = \frac{Q_2}{W} = \frac{Q_2}{Q_1 - Q_2} = \frac{1}{Q_1/Q_2 - 1} $$

For a heat pump, the coefficient of performance is defined as $\alpha' = Q_1/W$. Since $Q_1 = Q_2 + W$, $\alpha' = (Q_2 + W)/W = Q_2/W + 1 = \alpha + 1$. A heat pump is designed to transfer heat into a warm space (the hot reservoir $Q_1$), while a refrigerator is designed to remove heat from a cold space (the cold reservoir $Q_2$).

Experience shows that a refrigerator cannot operate without external work ($W>0$), meaning $\alpha$ cannot be infinite.

Second Law Of Thermodynamics

The Second Law of Thermodynamics introduces a fundamental limitation on the direction of natural processes and the efficiency of energy conversions that are not imposed by the First Law (energy conservation) alone.

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The Second Law can be stated in various equivalent forms, often related to the impossibility of certain processes:

Kelvin-Planck Statement (related to heat engines):

No process is possible whose sole result is the absorption of heat from a single reservoir and the complete conversion of this heat into work.

This means that it is impossible to build a heat engine that is 100% efficient ($\eta = 1$). Some heat must always be rejected to a cold reservoir ($Q_2 > 0$).

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Clausius Statement (related to refrigerators):

No process is possible whose sole result is the transfer of heat from a colder object to a hotter object.

This means heat cannot spontaneously flow from a cold body to a hot body; external work is always required to achieve such a transfer (as in a refrigerator, $W>0$, $\alpha$ cannot be infinite).

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These statements are equivalent; violating one implies violating the other. The Second Law essentially introduces the concept that processes have a preferred direction (e.g., heat flows from hot to cold, not vice versa, spontaneously).

Reversible And Irreversible Processes

A thermodynamic process takes a system from an initial state to a final state. Processes can be classified based on whether they can be reversed.

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Reversible Process:

• An ideal thermodynamic process that can be exactly reversed, such that both the system and its entire surroundings are restored to their original initial states, with no other changes anywhere else in the universe.
• A process is reversible only if it is quasi-static (system always in equilibrium with surroundings) and non-dissipative (no friction, viscosity, etc.).
• Reversible processes are idealizations; no real process is perfectly reversible. They are useful for establishing theoretical limits (e.g., maximum engine efficiency).

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Irreversible Process:

• A real process that is not reversible. Most natural processes are irreversible.
• Irreversibility arises from:
  • Processes that take the system through non-equilibrium states (e.g., rapid expansion or compression, explosive reactions).
  • Presence of dissipative effects like friction, viscosity, electrical resistance, etc., which convert work or ordered energy into disordered heat energy that cannot be perfectly recovered.

The Second Law of Thermodynamics is closely related to irreversibility. Irreversible processes increase the total entropy of the universe (a measure of disorder), while reversible processes leave the total entropy unchanged.

Carnot Engine

The Carnot engine is a theoretical, ideal heat engine that operates on a reversible cycle called the Carnot cycle, working between two heat reservoirs at fixed temperatures $T_1$ (hot source) and $T_2$ (cold sink), with $T_1 > T_2$.

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The Carnot cycle for an ideal gas consists of four quasi-static (and reversible) processes:

1. Isothermal Expansion (1 $\to$ 2): The gas absorbs heat $Q_1$ from the hot reservoir at $T_1$ and expands, doing work. ($T=T_1$, $\Delta U=0$, $Q_1 = W_{12} = \mu RT_1 \ln(V_2/V_1)$).
2. Adiabatic Expansion (2 $\to$ 3): The gas expands further without heat exchange, doing work and cooling from $T_1$ to $T_2$. ($Q=0$, $W_{23} = -\Delta U = \mu R(T_1-T_2)/(\gamma-1)$).
3. Isothermal Compression (3 $\to$ 4): The gas is compressed at constant temperature $T_2$, releasing heat $Q_2$ to the cold reservoir. Work is done on the gas. ($T=T_2$, $\Delta U=0$, $Q_2 = W_{34} = \mu RT_2 \ln(V_3/V_4)$, Heat released $Q_2 = -W_{34}$). Magnitude $|Q_2| = \mu RT_2 \ln(V_4/V_3)$.
4. Adiabatic Compression (4 $\to$ 1): The gas is compressed further without heat exchange, work is done on it, and its temperature rises from $T_2$ back to $T_1$. ($Q=0$, $W_{41} = -\Delta U = \mu R(T_2-T_1)/(\gamma-1)$, work done on gas is $-W_{41}$).

The net work done by the engine in one cycle is $W_{cycle} = W_{12} + W_{23} + W_{34} + W_{41}$. Due to the adiabatic steps, $W_{23} + W_{41} = \mu R(T_1-T_2)/(\gamma-1) + \mu R(T_2-T_1)/(\gamma-1) = 0$. So $W_{cycle} = W_{12} + W_{34} = Q_1 - |Q_2|$.

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The efficiency ($\eta$) of the Carnot engine is $\eta = W_{cycle} / Q_1 = (Q_1 - |Q_2|)/Q_1 = 1 - |Q_2|/Q_1$.

From the adiabatic processes in the cycle, it can be shown that the ratio of the heats exchanged is equal to the ratio of the temperatures of the reservoirs:

$$ \frac{|Q_2|}{Q_1} = \frac{T_2}{T_1} $$

Thus, the efficiency of a Carnot engine is:

$$ \eta_C = 1 - \frac{T_2}{T_1} $$

where $T_1$ and $T_2$ must be in absolute temperature scale (Kelvin).

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Significance of the Carnot Engine:

• The Carnot engine is the most efficient possible heat engine operating between two given temperatures.
• Carnot's Theorem: No engine operating between two given temperatures can have efficiency greater than that of a Carnot engine operating between the same two temperatures.
• The efficiency of a Carnot engine depends only on the temperatures of the hot and cold reservoirs, not on the working substance or the details of the cycle steps (as long as they are reversible).

The Carnot cycle is reversible. If reversed, it operates as a Carnot refrigerator/heat pump, with coefficient of performance $\alpha_C = \frac{Q_2}{W} = \frac{Q_2}{Q_1 - Q_2} = \frac{T_2}{T_1 - T_2}$.

The relation $|Q_2|/Q_1 = T_2/T_1$ for a Carnot engine (or reversed engine) provides a basis for defining a thermodynamic temperature scale independent of any specific substance.

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