Thermodynamics Class Notes 04: Internal Energy and First Law Calculations

Problem Solving: Isothermal Expansion and Internal Energy

• Scenario Analysis: A gas undergoes expansion under isothermal conditions against a constant external pressure ($P_{ext}$).
• Initial Volume ($V_1$): $10\,dm^3$
• Final Volume ($V_2$): $20\,dm^3$
• Heat Exchange ($q$): The system absorbs $800\,J$ of energy from its surroundings ($q = +800\,J$).
• Unit Conversions provided in Class Notes:
  • $1\,dm^3 = 1\,L$
  • $1\,atm \cdot L = 101.3\,J$ (The calculation shown in the notes uses the approximation $101\,J$).
• Work Calculation ($w$):
  • Formula: $w = -P_{ext} \Delta V$
  • Calculation: $w = -1 \times (20 - 10)$
  • Result in $atm \cdot L$: $w = -10\,atm \cdot L$
  • Conversion to Joules: $w = -10 \times 101 = -1010\,J$
• Internal Energy Calculation ($\Delta U$):
  • First Law of Thermodynamics Formula: $\Delta U = q + w$
  • Substitution: $\Delta U = +800 + (-1010)$
  • Final Calculation: $\Delta U = -210\,J$
• Options Provided:
  1. $-312\,J$
  2. $+123\,J$
  3. $-213\,J$ (Note: The slide's manual calculation results in $-210\,J$, while this is the closest option listed).
  4. $+231\,J$

Problem Solving: Change in Internal Energy via Work and Heat

• Problem Statement: Calculate the change in internal energy when specific quantities of work and heat are involved.
• Work ($w$): $4\,kJ$ of work is done on the system ($w = +4\,kJ$).
• Heat ($q$): $1\,kJ$ of heat is given out by the system ($q = -1\,kJ$).
• Calculation of $\Delta U$:
  • Formula: $\Delta U = q + w$
  • Substitution: $\Delta U = -1\,kJ + 4\,kJ$
  • Final Result: $\Delta U = +3\,kJ$
• Options Provided:
  1. $+1\,kJ$
  2. $-5\,kJ$
  3. $+5\,kJ$
  4. $+3\,kJ$ (Correct Answer Marked)

Concepts of the First Law of Thermodynamics

• Definition: The First Law of Thermodynamics is fundamentally established as the law of conservation of energy.
• Theoretical Context:
  • Option 1: The law of conservation of energy (Correct Identification).
  • Option 2: The law of conservation of mass.
  • Option 3: The law of conservation of momentum.
  • Option 4: Both (1) and (2).
• Key Takeaway: The primary focus of the first law is the principle that energy cannot be created or destroyed, only transformed from one form to another within a thermodynamic system.

Mathematical Notations and Unit Reference

• Standard Equation: $\Delta U = q + w$
• Expansion Work: $w = -P_{ext}(V_2 - V_1)$
• Unit Identity: $1\,dm^3 = 1\,L$
• Pressure-Volume Work Conversion: $1\,atm \cdot L \approx 101.3\,J$