First Law of Thermodynamics: Explanation, Mathematical Derivations,important and Applications1

Thermodynamics is a vital branch of physics that deals with energy transfer and the relationships between different forms of energy. It explains how energy moves within a system and how it interacts with the surrounding environment. Thermodynamics plays a crucial role in engineering, chemistry, biology, and environmental science. Among its core principles, the First Law of Thermodynamics is fundamental, as it deals with the conservation of energy and provides the basis for energy management in various processes.

This article offers an in-depth, SEO-friendly, and comprehensive explanation of the First Law of Thermodynamics, including all essential mathematical derivations, concepts, and practical applications.

What Is the First Law of Thermodynamics?

The First Law of Thermodynamics, also known as the Law of Conservation of Energy, states:

“Energy cannot be created or destroyed in an isolated system. It can only be transformed from one form to another.”

In simpler terms, the total energy of an isolated system remains constant. When energy is transferred into or out of a system, it changes form—such as from heat to work—but the total quantity of energy stays unchanged.

Key Concepts:

• System: The part of the universe under study (e.g., gas in a cylinder).
• Surroundings: Everything external to the system.
• Boundary: Separates the system from the surroundings.
• Energy Transfer: Occurs in the form of heat (Q) and work (W).

The Mathematical Statement of the First Law of Thermodynamics

The First Law of Thermodynamics can be mathematically expressed as: ΔU=Q−W\Delta U = Q – WΔU=Q−W

Where:

• ΔU\Delta UΔU = Change in internal energy of the system.
• QQQ = Heat added to the system.
• WWW = Work done by the system.

Interpretation:

• If Q>0Q > 0Q>0: Heat is added to the system.
• If Q<0Q < 0Q<0: Heat is removed from the system.
• If W>0W > 0W>0: Work is done by the system on the surroundings.
• If W<0W < 0W<0: Work is done on the system by the surroundings.

Internal Energy (UUU)

Internal energy is the total energy contained within a system. It includes:

• Kinetic energy of molecules (due to motion).
• Potential energy of molecules (due to intermolecular forces).
• Vibrational, rotational, and electronic energies.

For an ideal gas, internal energy depends solely on temperature: U=nCvTU = n C_v TU=nCv​T

Where:

• nnn: Number of moles.
• CvC_vCv​: Molar specific heat at constant volume.
• TTT: Temperature in Kelvin.

Work Done by the System (WWW)

In thermodynamics, work typically refers to mechanical work done by expanding or compressing a gas. Mathematically: W=∫ViVfP dVW = \int_{V_i}^{V_f} P \, dVW=∫Vi​Vf​​PdV

Where:

• PPP: Pressure.
• ViV_iVi​, VfV_fVf​: Initial and final volumes.

For Different Processes:

1. Isothermal Process (T = constant):
   • Work done by ideal gas:
   W=nRTln⁡(VfVi)W = nRT \ln \left( \dfrac{V_f}{V_i} \right)W=nRTln(Vi​Vf​​)
2. Adiabatic Process (No heat exchange, Q=0Q = 0Q=0):
   • Work done:
   W=PiVi−PfVfγ−1W = \dfrac{P_i V_i – P_f V_f}{\gamma – 1}W=γ−1Pi​Vi​−Pf​Vf​​ Where γ=CpCv\gamma = \dfrac{C_p}{C_v}γ=Cv​Cp​​.

Heat Transfer (QQQ)

Heat is energy transferred due to temperature difference. It can be calculated as:

• At constant volume: Q=nCvΔTQ = n C_v \Delta TQ=nCv​ΔT
• At constant pressure: Q=nCpΔTQ = n C_p \Delta TQ=nCp​ΔT

Where CpC_pCp​ and CvC_vCv​ are molar specific heats at constant pressure and volume, respectively.

Derivation of the First Law of Thermodynamics

Step 1: Conceptual Understanding

Consider a system absorbing heat QQQ, doing work WWW, and having a change in internal energy ΔU\Delta UΔU.

By energy conservation: Energy supplied as heat=Increase in internal energy+Energy used to do work\text{Energy supplied as heat} = \text{Increase in internal energy} + \text{Energy used to do work}Energy supplied as heat=Increase in internal energy+Energy used to do work

This leads to: Q=ΔU+WQ = \Delta U + WQ=ΔU+W

Rearranging: ΔU=Q−W\Delta U = Q – WΔU=Q−W

Thermodynamic Processes and the First Law

1. Isothermal Process (Constant Temperature)

• T=constant  ⟹  ΔU=0T = \text{constant} \implies \Delta U = 0T=constant⟹ΔU=0
• First law becomes: Q=WQ = WQ=W
• Entire heat added is used to do work.

2. Adiabatic Process (No Heat Exchange)

• Q=0Q = 0Q=0
• First law: ΔU=−W\Delta U = -WΔU=−W
• Work done by the system results in a decrease in internal energy.

3. Isochoric Process (Constant Volume)

• dV=0  ⟹  W=0dV = 0 \implies W = 0dV=0⟹W=0
• First law: ΔU=Q\Delta U = QΔU=Q
• All heat added increases internal energy.

4. Isobaric Process (Constant Pressure)

• Work done: W=PΔVW = P \Delta VW=PΔV
• First law: ΔU=Q−PΔV\Delta U = Q – P \Delta VΔU=Q−PΔV

Graphical Representation (P-V Diagrams)

• Isothermal Expansion: Hyperbolic curve.
• Adiabatic Expansion: Steeper curve than isothermal.
• Isochoric Process: Vertical line (constant VVV).
• Isobaric Process: Horizontal line (constant PPP).

First Law of Thermodynamics for a Cyclic Process

In a cyclic process:

• The system returns to its original state.
• ΔU=0\Delta U = 0ΔU=0.

So, the first law becomes: Q=WQ = WQ=W

Heat absorbed equals the work done by the system over one complete cycle.

Applications of the First Law of Thermodynamics

1. Heat Engines

• Converts heat energy into work.
• Follows cyclic processes (e.g., Carnot, Otto, Diesel cycles).

2. Refrigerators and Heat Pumps

• Transfers heat from colder to hotter areas by consuming work.
• Follows reversed thermodynamic cycles.

3. Internal Combustion Engines

• Converts chemical energy of fuel into mechanical work.
• Follows adiabatic and isothermal expansions/compressions.

4. Biological Systems

• Human metabolism follows energy conservation.
• Chemical energy from food converts to mechanical work, heat, and internal energy.

Energy Conservation and Efficiency

Efficiency (η\etaη) of a heat engine:

η=WQin=Qin−QoutQin\eta = \dfrac{W}{Q_{in}} = \dfrac{Q_{in} – Q_{out}}{Q_{in}}η=Qin​W​=Qin​Qin​−Qout​​

No system can be 100% efficient due to energy losses (Second Law of Thermodynamics), but the First Law ensures total energy balance.

Limitations of the First Law of Thermodynamics

1. No Directionality:
   • It doesn’t specify the direction of processes.
   • Whether heat flows spontaneously or requires work isn’t addressed (covered by Second Law).
2. Does Not Explain Irreversibility:
   • Irreversible processes and entropy aren’t considered.
3. Doesn’t Predict Efficiency Limits:
   • It doesn’t set the maximum possible efficiency (Second Law does).

Real-Life Examples of the First Law of Thermodynamics

Example 1: Gas Compression in a Cylinder

• Gas compressed adiabatically by a piston.
• No heat exchange (Q=0Q = 0Q=0).
• Work done on the gas increases its internal energy, raising temperature.

Example 2: Boiling Water in an Open Vessel

• Heat supplied increases internal energy and does work by expanding steam.
• Internal energy increases as water turns to vapor.

Example 3: Refrigerator Operation

• Work done by compressor extracts heat from inside (cold region) and releases it outside (hot region).
• Energy conservation principle maintains energy balance.

First Law in Open and Closed Systems

Closed System:

• No mass exchange.
• Only energy exchange (heat, work).
• First Law: ΔU=Q−W\Delta U = Q – WΔU=Q−W.

Open System:

• Mass and energy can cross the boundary.
• Steady-flow energy equation: Q−W=ΔH+ΔKE+ΔPEQ – W = \Delta H + \Delta KE + \Delta PEQ−W=ΔH+ΔKE+ΔPE Where HHH is enthalpy, KEKEKE kinetic energy, PEPEPE potential energy.

Enthalpy (HHH) and the First Law

For open systems (constant pressure processes): H=U+PVH = U + PVH=U+PV

Change in enthalpy: ΔH=ΔU+PΔV\Delta H = \Delta U + P \Delta VΔH=ΔU+PΔV

At constant pressure: Qp=ΔHQ_p = \Delta HQp​=ΔH

Experimental Validation of the First Law of Thermodynamics

Joule’s Experiment:

• Mechanical work (paddles stirring water) increases internal energy (temperature rise).
• Demonstrated equivalence of heat and work.
• Led to the formulation of the mechanical equivalent of heat: 1 calorie=4.186 Joules1 \text{ calorie} = 4.186 \text{ Joules}1 calorie=4.186 Joules

Conclusion

The First Law of Thermodynamics establishes the principle of energy conservation, forming the foundation for all energy-related sciences. It explains how energy transfers between systems as heat and work and underlines the importance of internal energy changes in various processes.

Whether designing efficient engines, understanding natural phenomena, or explaining metabolic processes, the First Law remains essential. However, it must be supplemented with additional laws to describe directionality, irreversibility, and entropy in real-world scenarios.

Frequently Asked Questions (FAQs)

Q1: What does the First Law of Thermodynamics state?

Ans: It states that energy cannot be created or destroyed, only converted from one form to another.

Q2: What is the equation of the First Law of Thermodynamics?

Ans: ΔU=Q−W\Delta U = Q – WΔU=Q−W.

Q3: What are the practical applications of the First Law?

Ans: Heat engines, refrigerators, biological systems, and internal combustion engines.

Q4: What is internal energy?

Ans: Internal energy is the total energy of all molecules in a system, including kinetic and potential energies.

Q5: Why can’t the First Law predict efficiency?

Ans: It doesn’t account for entropy and irreversibility, which are handled by the Second Law. Click here