Thermodynamics: First Law
The First Law of Thermodynamics is a statement of conservation of energy: energy cannot be created or destroyed, only transformed from one form to another.
This law is critical for analyzing energy conversion devices: power plants, engines, refrigerators, and heat pumps.
1Introduction
The First Law states that for a closed system, the change in internal energy equals net heat added minus net work done by the system: ΔU = Q - W.
Imagine a steam engine. Heat from burning coal (Q) is added to water, creating steam that pushes a piston (work W). The change in the steam's internal energy accounts for the difference. Energy in = Energy out + Energy stored!
2Key Definitions
System
Quantity of matter or region chosen for study.
Closed System
Fixed mass, no mass crossing boundaries. Energy can cross.
Heat (Q)
Energy transfer due to ΔT. Positive when added TO system.
Work (W)
Energy via force × distance. Positive when done BY system.
Internal Energy (U)
Microscopic energy of molecules. State function.
Enthalpy (H)
H = U + PV. Useful for constant pressure processes.
3First Law for Closed Systems
ΔU = Q - W
Q > 0: Heat added to system | W > 0: Work done by system
Sign Convention Critical!
Always be consistent: Q positive = into system, W positive = out of system.
4Work and Heat Transfer
Boundary Work (P-V Work)
W = ∫P dV
Area under P-V curve
Isobaric (Constant P)
W = PΔV
Isothermal (Constant T)
W = mRT ln(V₂/V₁)
Adiabatic (Q=0)
W = mR(T₂-T₁)/(1-γ)
Isochoric (Constant V)
W = 0
5Internal Energy & Enthalpy
For Ideal Gases:
Δu = cᵥΔT Δh = cₚΔT
cₚ - cᵥ = R γ = cₚ/cᵥ
For ideal gases, U is solely a function of temperature (U = U(T)). Therefore, for isothermal processes, ΔU = 0.
6First Law for Open Systems (SFEE)
Q̇ - Ẇ = ṁ(h₂ + v₂²/2 + gz₂) - ṁ(h₁ + v₁²/2 + gz₁)
Turbine (Ẇ out)
Ẇ = ṁ(h₁ - h₂)
Compressor (Ẇ in)
Ẇ = ṁ(h₂ - h₁)
Nozzle
h₁ + v₁²/2 = h₂ + v₂²/2
Throttling
h₁ = h₂ (isenthalpic)
7Design Applications
- Power Plants: Calculate efficiency, heat input, work output
- Refrigeration: Determine COP, work input, heat rejection
- Internal Combustion Engines: Model combustion, expansion, compression
- Heat Exchangers: Analyze heat transfer between fluid streams
- Jet Engines: Track enthalpy and kinetic energy changes
8Worked Examples
Basic
Isothermal Expansion of Ideal Gas
0.5 kg air, T=300K, P₁=100kPa, expands isothermally to P₂=50kPa. Find W and Q.
Step 1: V₂ = P₁V₁/P₂ = 0.861 m³
Step 2: W = mRT ln(V₂/V₁) = 30.0 kJ
Step 3: ΔU = 0 (isothermal), so Q = W = 30.0 kJ
Intermediate
Adiabatic Compression
0.1 kg air, P₁=100kPa, T₁=298K, compresses to P₂=800kPa adiabatically.
Step 1: γ = 1.4, T₂ = T₁(P₂/P₁)^((γ-1)/γ) = 540K
Step 2: ΔU = mcᵥΔT = 17.4 kJ
Step 3: Q=0, so W = -17.4 kJ (work ON system)
Advanced
Steam Turbine (SFEE)
Steam enters turbine at 3MPa, 400°C, exits at 50kPa, 100°C. Find ṁ and Ẇ.
Step 1: ṁ = (A₁v₁)/v = 5.03 kg/s
Step 2: Ẇ = ṁ[(h₁-h₂) + (v₁²-v₂²)/2] = 2717 kW
9Key Formulas
ΔU = Q - W W = ∫PdV H = U + PV
Δu = cᵥΔT Δh = cₚΔT cₚ - cᵥ = R
MMemory Aids
"Q-W for ΔU" — Q comes in, W goes out, leaving internal energy change.
"U is Unique, Q and W are Quick and Winding" — State vs. path functions!
MCommon Mistakes
Using inconsistent sign convention for Q and W
Using cₚ for constant volume or vice versa
Mixing kJ and J, not converting temperature to Kelvin
Frequently Asked Questions
- What is the difference between heat and work?
- Heat is energy transfer driven by temperature difference. Work is energy transfer driven by force through distance. Both are path functions, but heat requires ΔT while work does not.
- Why is internal energy a state function?
- State functions depend only on initial and final states, not the path taken. Internal energy depends only on the state of the system (temperature, pressure, etc.).
- When is enthalpy (H) more useful than internal energy (U)?
- Enthalpy is particularly useful for constant pressure processes (Q_p = ΔH) and open systems (flow processes) where P-V work is significant.
- What is an adiabatic process?
- An adiabatic process has no heat transfer (Q = 0). If it's also reversible, it's isentropic. Rapid processes are often approximated as adiabatic.
- How do c_p and c_v relate for ideal gases?
- c_p - c_v = R, where R is the specific gas constant. Also, γ = c_p/c_v (ratio of specific heats).
- What is the Steady-Flow Energy Equation (SFEE)?
- For open systems: Q̇_net,in - Ẇ_net,out = Σ_out ṁ(h + v²/2 + gz) - Σ_in ṁ(h + v²/2 + gz). Used for turbines, compressors, heat exchangers.
Practice Quiz
Test your understanding — select the correct answer for each question.
1.What does the First Law of Thermodynamics state?
2.For a closed system, the First Law states:
3.For an isothermal process of an ideal gas:
4.Boundary work for constant pressure process equals:
5.In an adiabatic process:
6.Enthalpy H is defined as:
7.For ideal gases, c_p - c_v equals:
8.Which is a state function?
9.For a cycle, the net change in internal energy is:
10.Specific heat at constant volume (c_v) relates to:
Study Tips
- Master sign convention: Q > 0 into system, W > 0 out of system
- Remember: U is a state function, Q and W are path functions
- For ideal gas: U depends only on T
- Draw P-V diagrams to visualize work as area under curve
- Use SFEE for open systems: turbines, compressors, nozzles