First Law of Thermodynamics and Internal Energy

The First Law of Thermodynamics is a fundamental principle that governs energy transformations. It is essentially a statement of the law of conservation of energy applied to thermodynamic systems. This law helps us understand how heat, work, and internal energy are related within a system.

Internal Energy (U)

Internal energy is the total energy contained within a thermodynamic system. It is the sum of the kinetic energy and potential energy of all the molecules that make up the system.

• Kinetic Energy: This includes translational, rotational, and vibrational energies of the molecules. The average kinetic energy of molecules is directly proportional to the absolute temperature of the system.
• Potential Energy: This arises from the forces of interaction between molecules and between the atoms within molecules. It depends on the state of the system, such as its volume and the presence of intermolecular forces.

For an ideal gas, the molecules are assumed to have no intermolecular forces. Therefore, the internal energy of an ideal gas depends only on its kinetic energy and hence, solely on its absolute temperature. This is a crucial point for many thermodynamic problems.

Mathematically, for an ideal gas:

U = f(T)

Where U is the internal energy and T is the absolute temperature. The function f(T) depends on the number of degrees of freedom of the gas molecules.

For a monatomic ideal gas (like Helium, Neon), the internal energy is given by:

U = (3/2) nRT

For a diatomic ideal gas (like Oxygen, Nitrogen), the internal energy is given by:

U = (5/2) nRT

For a polyatomic ideal gas, the internal energy is generally given by:

U = (f/2) nRT

Where:

• n is the number of moles of the gas.
• R is the universal gas constant (approximately 8.314 J/mol·K).
• T is the absolute temperature in Kelvin.
• f is the number of degrees of freedom per molecule (3 for monatomic, 5 for diatomic at moderate temperatures, higher for polyatomic gases and at higher temperatures due to vibrational modes).

It's important to note that the absolute value of internal energy is often difficult to determine and not practically useful. What matters in thermodynamics are the *changes* in internal energy (ΔU) during a process.

The First Law of Thermodynamics

The First Law of Thermodynamics states that when heat is added to a system, some of that energy increases the internal energy of the system, and some of it may be used by the system to do work on its surroundings. Conversely, if the system does work on the surroundings, its internal energy decreases, or heat must be supplied to the system.

The law can be expressed mathematically as:

ΔQ = ΔU + ΔW

Where:

• ΔQ is the heat added to the system.
• ΔU is the change in internal energy of the system.
• ΔW is the work done *by* the system on its surroundings.

This equation is a statement of energy conservation. The total energy exchange with the surroundings (heat added and work done) must balance the change in the system's internal energy.

Sign Conventions for the First Law

It is crucial to be consistent with sign conventions. The convention used above (ΔQ = ΔU + ΔW) is widely adopted in physics.

• Heat (ΔQ):
• +ΔQ: Heat supplied *to* the system (system gains energy).
• -ΔQ: Heat supplied *by* the system (system loses energy).
• Internal Energy (ΔU):
• +ΔU: Internal energy of the system increases.
• -ΔU: Internal energy of the system decreases.
• Work (ΔW):
• +ΔW: Work done *by* the system on the surroundings (system loses energy).
• -ΔW: Work done *on* the system by the surroundings (system gains energy).

An alternative convention is sometimes used in chemistry where work done *on* the system is considered positive (ΔQ = ΔU - ΔW', where ΔW' is work done *on* the system). Always clarify which convention is being used. For JEE Main Physics, the convention ΔQ = ΔU + ΔW (work done *by* the system is positive) is standard.

Work Done by a Thermodynamic System (ΔW)

In many thermodynamic processes, work is done due to a change in volume against an external pressure. Consider a gas in a cylinder fitted with a movable piston. If the gas expands, it pushes the piston outwards and does work on the surroundings. If the gas is compressed, the surroundings do work on the gas.

Let the pressure of the gas be P, the area of the piston be A, and the piston move by a small distance dx. The force exerted by the gas on the piston is F = P * A.

The small amount of work done dW by the gas is:

dW = Force × distance = F × dx = (P * A) × dx = P * (A × dx)

Since A × dx is the small change in volume dV, we have:

dW = P dV

For a process where the volume changes from an initial volume V₁ to a final volume V₂, the total work done by the system is the integral of P dV:

W = ∫_V1^V₂ P dV

This integral represents the area under the pressure-volume (P-V) curve on a P-V diagram.

Important Note on Work: Work done is a path-dependent quantity. The amount of work done depends on the specific process (how pressure and volume change) between the initial and final states. In contrast, internal energy is a state function; its change depends only on the initial and final states, not on the path taken.

Relationship between ΔQ, ΔU, and ΔW in Different Processes

Let's analyze the First Law in some common thermodynamic processes:

1. Isothermal Process (Constant Temperature, ΔT = 0)

In an isothermal process, the temperature of the system remains constant.

• For an ideal gas: Since internal energy depends only on temperature for an ideal gas, if ΔT = 0, then ΔU = 0.
• First Law becomes: 0 = 0 + ΔW, which means ΔQ = ΔW.
• Interpretation: All the heat added to the system is used to do work on the surroundings (during expansion), or all the work done on the system (during compression) is released as heat.

Example: Slow expansion or compression of a gas in contact with a heat reservoir.

2. Adiabatic Process (No Heat Exchange, ΔQ = 0)

In an adiabatic process, the system is perfectly insulated, so no heat can enter or leave the system.

• First Law becomes: ΔQ = ΔU + ΔW becomes 0 = ΔU + ΔW, which means ΔU = -ΔW.
• Interpretation: If the system does work on the surroundings (expansion, ΔW > 0), its internal energy must decrease (ΔU < 0), leading to a drop in temperature. If work is done on the system (compression, ΔW < 0), its internal energy must increase (ΔU > 0), leading to a rise in temperature.

Example: Rapid compression or expansion of a gas (like in a diesel engine cylinder or a gas escaping from a nozzle), where there isn't enough time for significant heat transfer.

3. Isobaric Process (Constant Pressure, ΔP = 0)

In an isobaric process, the pressure of the system remains constant.

• First Law: ΔQ = ΔU + ΔW
• Work done: Since P is constant, W = P ∫_V1^V₂ dV = P(V₂ - V₁) = PΔV.
• Interpretation: Heat added goes into increasing internal energy and doing expansion work. If the gas expands (ΔV > 0), both ΔU and ΔW are positive (assuming heat is added). If the gas is heated at constant pressure, it expands, and thus does work.

Example: Heating water to steam in an open container (pressure is atmospheric).

4. Isochoric/Isovolumetric Process (Constant Volume, ΔV = 0)

In an isochoric process, the volume of the system remains constant.

• Work done: Since ΔV = 0, W = ∫ P dV = 0. The system does no work on the surroundings.
• First Law becomes: ΔQ = ΔU + 0, which means ΔQ = ΔU.
• Interpretation: All the heat added to the system goes directly into increasing its internal energy, leading to a rise in temperature. If heat is removed, the internal energy decreases, and the temperature drops.

Example: Heating a gas in a rigid, sealed container.

Cyclic Process

A cyclic process is one in which a system undergoes a series of changes and eventually returns to its original state.

• Change in Internal Energy: Since the system returns to its original state, its properties (including internal energy) must return to their initial values. Therefore, for any cyclic process, ΔU = 0.
• First Law becomes: 0 = ΔQ + ΔW, which means ΔQ = -ΔW.
• Interpretation: In a cycle, the net heat absorbed by the system is equal to the net work done *on* the system (or, the net heat rejected equals the net work done *by* the system). If the net work done *by* the system over a cycle is positive (area enclosed by the cycle on P-V diagram is positive), then the net heat absorbed must be positive.

Example: The operation of an engine or a refrigerator involves cyclic processes.

Applications and Examples

Example 1: Heating a gas in a cylinder

Suppose 5000 J of heat is supplied to a gas in a cylinder. The gas expands and does 1500 J of work on the piston. Calculate the change in internal energy of the gas.

Given: ΔQ = +5000 J (heat supplied to the system) ΔW = +1500 J (work done by the system)

Using the First Law: ΔQ = ΔU + ΔW 5000 J = ΔU + 1500 J ΔU = 5000 J - 1500 J ΔU = 3500 J

The internal energy of the gas increased by 3500 J.

Example 2: Adiabatic compression

A gas is compressed adiabatically, and 800 J of work is done *on* the gas. What is the change in its internal energy?

Given: Adiabatic process, so ΔQ = 0. Work done *on* the gas is 800 J. This means the work done *by* the system is ΔW = -800 J.

Using the First Law: ΔQ = ΔU + ΔW 0 = ΔU + (-800 J) ΔU = +800 J

The internal energy of the gas increased by 800 J.

Example 3: Isothermal expansion of an ideal gas

An ideal gas is expanded isothermally from a volume V₁ to V₂. What is the change in internal energy?

For an ideal gas, internal energy U depends only on temperature T. In an isothermal process, T is constant. Therefore, ΔT = 0, which implies ΔU = 0.

From the First Law, ΔQ = ΔU + ΔW. Since ΔU = 0, we get ΔQ = ΔW. This means all the heat absorbed by the gas is converted into work done by the gas during expansion.

Internal Energy and Specific Heat Capacities

The concept of internal energy is closely related to specific heat capacities. Specific heat capacity is the amount of heat required to raise the temperature of a unit mass (or mole) of a substance by one degree Celsius (or Kelvin).

For gases, we distinguish between two principal specific heat capacities:

• Specific heat capacity at constant volume (C_v): The heat required to raise the temperature of one mole of gas by 1 K when its volume is kept constant.
• Specific heat capacity at constant pressure (C_p): The heat required to raise the temperature of one mole of gas by 1 K when its pressure is kept constant.

From the First Law: ΔQ = ΔU + ΔW.

Consider 1 mole of an ideal gas.

At constant volume (isochoric process): ΔV = 0, so ΔW = 0. ΔQ = ΔU. The heat supplied solely increases internal energy. So, C_v = (ΔU / ΔT) at constant V. For 1 mole, ΔU = C_v ΔT.

At constant pressure (isobaric process): P is constant. ΔQ = ΔU + PΔV. The heat supplied increases internal energy *and* does work due to expansion. So, C_p = (ΔQ / ΔT) at constant P. For 1 mole, ΔQ = C_p ΔT. Thus, C_p ΔT = ΔU + PΔV. Substituting ΔU = C_v ΔT: C_p ΔT = C_v ΔT + PΔV.

From the ideal gas law, PV = RT. For 1 mole, PΔV = RΔT. So, C_p ΔT = C_v ΔT + RΔT. Dividing by ΔT, we get: C_p = C_v + R.

This relation, C_p - C_v = R, is known as Mayer's relation and is fundamental for ideal gases.

The ratio of specific heat capacities, γ (gamma), is also important:

γ = C_p / C_v

For monatomic gases, γ ≈ 1.67. For diatomic gases, γ ≈ 1.40. For polyatomic gases, γ is typically around 1.33.

Summary of First Law and Internal Energy

• Internal Energy (U): Total energy within a system. For ideal gases, it depends only on temperature.
• First Law of Thermodynamics: ΔQ = ΔU + ΔW (Energy conservation).
• Sign Conventions: +ΔQ (heat in), -ΔQ (heat out), +ΔU (internal energy increases), -ΔU (internal energy decreases), +ΔW (work done by system), -ΔW (work done on system).
• Work (W): Path-dependent. For volume change, W = ∫ P dV.
• Special Processes:
• Isothermal (ΔT=0): ΔU=0 (ideal gas), ΔQ = ΔW.
• Adiabatic (ΔQ=0): ΔU = -ΔW.
• Isobaric (ΔP=0): W = PΔV.
• Isochoric (ΔV=0): W = 0, ΔQ = ΔU.
• Cyclic Process: ΔU = 0, ΔQ = -ΔW (net).
• Specific Heats: C_v relates to ΔU, C_p relates to ΔQ at constant pressure. Mayer's relation: C_p - C_v = R for ideal gases.