Heat, Work, and Internal Energy

In thermodynamics, we study the relationships between heat, work, and internal energy. These three concepts are fundamental to understanding how energy is transferred and transformed in various systems.

Internal Energy (U)

Internal energy is the total energy contained within a thermodynamic system. It is the sum of the kinetic energy of the molecules (due to their motion) and the potential energy of the molecules (due to intermolecular forces).

• For an ideal gas, the internal energy depends only on its temperature. The molecules of an ideal gas are assumed to have no intermolecular forces, so there is no potential energy component.
• For real gases, liquids, and solids, internal energy includes both kinetic and potential energy components.
• A change in internal energy ($\Delta U$) can occur due to heat transfer or work done on or by the system.

Heat (Q)

Heat is the transfer of thermal energy between systems due to a temperature difference. It is a form of energy in transit.

• Heat flows spontaneously from a region of higher temperature to a region of lower temperature.
• Heat is a path function, meaning the amount of heat transferred depends on the process by which the system changes from one state to another.
• Heat can be positive (added to the system) or negative (removed from the system).
• The SI unit of heat is the joule (J). Other common units include the calorie (cal) and kilocalorie (kcal).
• 1 calorie is the amount of heat required to raise the temperature of 1 gram of water by 1 degree Celsius (specifically from 14.5°C to 15.5°C).
• 1 calorie = 4.184 joules.

Work (W)

In thermodynamics, work is done when a force acts over a distance. In the context of gases, work is often done when the volume of the gas changes against an external pressure.

Consider a gas enclosed in a cylinder with a movable piston. If the gas expands, it pushes the piston outwards, doing work on the surroundings. If the gas is compressed, the surroundings do work on the gas.

The work done by the gas during an infinitesimal volume change $dV$ against a constant external pressure $P_{ext}$ is given by:

$dW = P_{ext} dV$

If the gas expands ($dV > 0$), the work done by the gas is positive. If the gas is compressed ($dV < 0$), the work done by the gas is negative (meaning work is done *on* the gas).

For a process where the pressure may vary, the total work done by the gas as its volume changes from $V_1$ to $V_2$ is the integral:

$W = \int_{V_1}^{V_2} P_{ext} dV$

If the process is slow and the pressure inside the cylinder is always equal to the external pressure ($P_{ext} = P$), then:

$W = \int_{V_1}^{V_2} P dV$

Work is also a path function. The amount of work done depends on the specific process.

The SI unit of work is the joule (J).

Relationship between Heat, Work, and Internal Energy

The First Law of Thermodynamics describes the relationship between these three quantities.

First Law of Thermodynamics

The First Law of Thermodynamics is essentially a statement of the conservation of energy. It states that the change in the internal energy of a system is equal to the net heat added to the system minus the net work done by the system.

Mathematically, it is expressed as:

$\Delta U = Q - W$

Where:

• $\Delta U$ is the change in internal energy of the system.
• $Q$ is the heat added to the system. If heat is removed from the system, $Q$ is negative.
• $W$ is the work done *by* the system. If work is done *on* the system, $W$ is negative.

Important Note on Sign Conventions: There are different sign conventions used for work. The convention $\Delta U = Q - W$ assumes $W$ is work done *by* the system. Another common convention is $\Delta U = Q + W$, where $W$ is work done *on* the system. Always be consistent with the convention you use. For NEET exams, the $\Delta U = Q - W$ (work done *by* the system) convention is more commonly used.

Implications of the First Law

The First Law has several important implications:

• Energy is conserved: Energy cannot be created or destroyed, only transferred or converted from one form to another.
• Perpetual Motion Machines of the First Kind are Impossible: A machine that produces work without any input of energy (heat or work) would violate the First Law.

Special Cases of the First Law

The First Law can be applied to specific thermodynamic processes:

1. Isochoric Process (Constant Volume)

In this process, the volume of the system remains constant ($V_1 = V_2$), so $dV = 0$.

Since $W = \int P dV$, the work done is zero ($W = 0$).

According to the First Law:

$\Delta U = Q - 0$

$\Delta U = Q$

This means that any heat added to the system goes entirely into increasing its internal energy, and any heat removed decreases its internal energy. For an ideal gas, this would lead to a temperature change.

2. Isobaric Process (Constant Pressure)

In this process, the pressure of the system remains constant ($P_1 = P_2 = P$).

The work done is:

$W = P \int_{V_1}^{V_2} dV = P (V_2 - V_1) = P \Delta V$

The First Law becomes:

$\Delta U = Q - P \Delta V$

Here, the heat added ($Q$) is used to both increase the internal energy ($\Delta U$) and do work ($P \Delta V$) by expanding the system.

3. Isothermal Process (Constant Temperature)

This process occurs at a constant temperature ($T_1 = T_2$).

For an ideal gas, the internal energy depends only on temperature. Therefore, if the temperature is constant, the change in internal energy is zero ($\Delta U = 0$).

Applying the First Law:

$0 = Q - W$

$Q = W$

This means that any heat added to the system is entirely converted into work done by the system (in case of expansion), or any work done on the system (in case of compression) is entirely removed as heat.

The work done during an isothermal expansion of an ideal gas from volume $V_1$ to $V_2$ at constant temperature $T$ is given by:

$W = \int_{V_1}^{V_2} P dV$

Using the ideal gas law, $PV = nRT$, so $P = \frac{nRT}{V}$.

$W = \int_{V_1}^{V_2} \frac{nRT}{V} dV = nRT \int_{V_1}^{V_2} \frac{1}{V} dV$

$W = nRT [\ln V]_{V_1}^{V_2} = nRT (\ln V_2 - \ln V_1)$

$W = nRT \ln \left(\frac{V_2}{V_1}\right)$

Since $T$ is constant and $PV$ is constant for an ideal gas during an isothermal process, $P_1V_1 = P_2V_2$, which means $\frac{V_2}{V_1} = \frac{P_1}{P_2}$.

So, the work done can also be expressed as:

$W = nRT \ln \left(\frac{P_1}{P_2}\right)$

4. Adiabatic Process (No Heat Transfer)

In an adiabatic process, there is no heat exchange between the system and its surroundings. This means $Q = 0$.

Applying the First Law:

$\Delta U = 0 - W$

$\Delta U = -W$

This implies that if the system does work (expands, $W > 0$), its internal energy must decrease ($\Delta U < 0$), leading to a drop in temperature. If work is done on the system (compression, $W < 0$), its internal energy increases ($\Delta U > 0$), leading to a rise in temperature.

Adiabatic processes are often rapid, preventing significant heat transfer. Examples include the rapid compression or expansion of gases in engines or the atmosphere.

For an ideal gas undergoing a reversible adiabatic process, the relationship between pressure ($P$) and volume ($V$) is given by:

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

Where $\gamma$ (gamma) is the adiabatic index or heat capacity ratio, defined as the ratio of the specific heat at constant pressure ($C_p$) to the specific heat at constant volume ($C_v$):

$\gamma = \frac{C_p}{C_v}$

The value of $\gamma$ depends on the nature of the gas:

• For monatomic gases (like He, Ne, Ar): $\gamma \approx 1.67$
• For diatomic gases (like O2, N2, H2): $\gamma \approx 1.40$
• For polyatomic gases (like CO2, SO2): $\gamma \approx 1.33$

The relationship can also be expressed in terms of temperature and volume, or temperature and pressure:

$TV^{\gamma-1} = \text{constant}$

$T^{1-\gamma} P^\gamma = \text{constant}$

The work done during an adiabatic expansion of an ideal gas from state $(P_1, V_1, T_1)$ to $(P_2, V_2, T_2)$ is:

$W = \frac{P_1V_1 - P_2V_2}{\gamma - 1}$

Since $PV = nRT$, we can also write:

$W = \frac{nR(T_1 - T_2)}{\gamma - 1}$

And since $\Delta U = -W$ for an adiabatic process:

$\Delta U = \frac{nR(T_2 - T_1)}{\gamma - 1}$

This is consistent with the relation $\Delta U = n C_v \Delta T$, because for an ideal gas, $C_v = \frac{R}{\gamma - 1}$.

Isothermal and Adiabatic Processes: A Deeper Dive

Isothermal Process

An isothermal process is a thermodynamic process in which the temperature of the system remains constant throughout. This is achieved by ensuring that the process occurs very slowly, allowing for continuous heat exchange with the surroundings to maintain a constant temperature.

Key Characteristics of Isothermal Process:

• Constant Temperature ($\Delta T = 0$): This is the defining feature.
• Constant Internal Energy for Ideal Gases ($\Delta U = 0$): As internal energy of an ideal gas is solely a function of temperature, $\Delta U = 0$.
• Heat Transferred Equals Work Done ($Q = W$): From the First Law ($\Delta U = Q - W$), if $\Delta U = 0$, then $Q = W$. Heat added to the system is completely converted into work done by the system (during expansion), or work done on the system is completely dissipated as heat (during compression).
• Pressure-Volume Relationship: For an ideal gas, $PV = nRT$. Since $n$, $R$, and $T$ are constant, the product $PV$ remains constant. This means $P_1V_1 = P_2V_2$. The process follows a hyperbolic curve on a P-V diagram.
• Work Done: $W = nRT \ln \left(\frac{V_2}{V_1}\right) = nRT \ln \left(\frac{P_1}{P_2}\right)$.

Example:

Imagine slowly compressing a gas in a cylinder fitted with a piston, where the cylinder is placed in a large water bath. As the gas is compressed, it tends to heat up. However, because the process is slow and the water bath maintains a constant temperature, the heat generated is transferred to the water bath, keeping the gas temperature constant.

Adiabatic Process

An adiabatic process is a thermodynamic process in which there is no heat transfer into or out of the system ($Q = 0$). This typically occurs when a process happens very rapidly, so there isn't enough time for significant heat exchange, or when the system is perfectly insulated from its surroundings.

Key Characteristics of Adiabatic Process:

• No Heat Transfer ($Q = 0$): This is the defining feature.
• Change in Internal Energy Equals Negative Work Done ($\Delta U = -W$): From the First Law ($\Delta U = Q - W$), if $Q = 0$, then $\Delta U = -W$. If the system expands and does work ($W > 0$), its internal energy decreases ($\Delta U < 0$), leading to a drop in temperature. If work is done on the system (compression, $W < 0$), its internal energy increases ($\Delta U > 0$), leading to a rise in temperature.
• Pressure-Volume Relationship: For a reversible adiabatic process of an ideal gas, $PV^\gamma = \text{constant}$.
• Temperature-Volume Relationship: $TV^{\gamma-1} = \text{constant}$.
• Temperature-Pressure Relationship: $T^{1-\gamma} P^\gamma = \text{constant}$.
• Work Done: $W = \frac{P_1V_1 - P_2V_2}{\gamma - 1} = \frac{nR(T_1 - T_2)}{\gamma - 1}$.

Example:

The rapid compression of air in a diesel engine cylinder is a good example of an adiabatic process. The compression happens so quickly that little heat escapes. The work done on the air increases its internal energy and thus its temperature, often high enough to ignite fuel. Similarly, the sudden expansion of gas when a soda bottle is opened can be considered approximately adiabatic; the gas cools down rapidly.

Comparison: Isothermal vs. Adiabatic Expansion

Let's consider the expansion of an ideal gas from an initial state $(P_1, V_1)$ to a final volume $V_2$.

• Work Done: For the same initial and final volumes, the work done in an isothermal expansion is always greater than the work done in an adiabatic expansion. This is because in the isothermal process, heat is continuously supplied to maintain temperature, allowing for more expansion (and thus more work). In the adiabatic process, the gas cools as it expands, reducing the pressure and limiting the amount of work done.
• Final Pressure: For the same initial state and final volume, the final pressure after an adiabatic expansion is lower than after an isothermal expansion. This is because the gas cools down during adiabatic expansion, reducing its pressure.
• P-V Diagram: On a P-V diagram, the curve for an adiabatic process is steeper than the curve for an isothermal process when both represent expansion from the same initial point. This reflects that for the same volume change, the pressure drops more significantly in an adiabatic process.

Summary of Key Concepts and Formulas

This section provides a quick recap of the essential definitions and formulas related to heat, work, internal energy, and the First Law of Thermodynamics, including isothermal and adiabatic processes.

Fundamental Concepts

• Internal Energy (U): Total energy of the molecules within a system. For an ideal gas, it depends only on temperature.
• Heat (Q): Energy transferred due to temperature difference. Positive when added to the system.
• Work (W): Energy transferred by mechanical means (force over distance). Positive when done *by* the system.

First Law of Thermodynamics

$\Delta U = Q - W$

(Where $\Delta U$ = change in internal energy, $Q$ = heat added to the system, $W$ = work done by the system)

Specific Processes

1. Isochoric Process (Constant Volume, $\Delta V = 0$)

• $W = 0$
• $\Delta U = Q$

2. Isobaric Process (Constant Pressure, $P = \text{constant}$)

• $W = P \Delta V$
• $\Delta U = Q - P \Delta V$

3. Isothermal Process (Constant Temperature, $\Delta T = 0$)

• For Ideal Gas: $\Delta U = 0$
• $Q = W$
• $PV = \text{constant}$
• $W = nRT \ln \left(\frac{V_2}{V_1}\right) = nRT \ln \left(\frac{P_1}{P_2}\right)$

4. Adiabatic Process (No Heat Transfer, $Q = 0$)

• $\Delta U = -W$
• $PV^\gamma = \text{constant}$
• $TV^{\gamma-1} = \text{constant}$
• $T^{1-\gamma} P^\gamma = \text{constant}$
• $W = \frac{P_1V_1 - P_2V_2}{\gamma - 1} = \frac{nR(T_1 - T_2)}{\gamma - 1}$
• $\gamma = \frac{C_p}{C_v}$ (Adiabatic Index)

Ideal Gas Properties

• Monatomic gas: $\gamma \approx 1.67$
• Diatomic gas: $\gamma \approx 1.40$
• Polyatomic gas: $\gamma \approx 1.33$
• For an ideal gas, $C_v = \frac{R}{\gamma - 1}$ and $C_p = \frac{\gamma R}{\gamma - 1}$.
• $\Delta U = n C_v \Delta T$ (This holds for any process for an ideal gas).