#The First Law of Thermodynamics: Your Ultimate Guide 🚀

Hey there, future AP Physics 2 master! Let's dive into the First Law of Thermodynamics. Think of it as the universe's golden rule for energy: it's all about conservation! This guide will break down everything you need to know, making sure you're confident and ready for anything the exam throws your way. Let's get started!

#The First Law: Energy is Conserved!

The First Law of Thermodynamics is all about energy conservation. It tells us how a system's internal energy changes when heat is added or removed, and when work is done on or by the system. It's a cornerstone of thermodynamics, so let's get it down!

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Internal energy (U) is the total energy stored within a system. It's the sum of:

• Kinetic energy of all the particles (atoms, molecules) 🏃‍♂️
• Potential energy due to the interactions between these particles

#Ideal Gases: A Simpler Case

For ideal gases, things get simpler:

• No interactions between gas atoms/molecules (no potential energy)

• Internal structure is ignored

• Internal energy (U) is just the sum of kinetic energies: $$U = \frac{3}{2} n R T = \frac{3}{2} N k_B T$$

  Where:

  • n = number of moles
  • R = ideal gas constant
  • T = temperature (in Kelvin)
  • N = number of particles
  • $k_B$ = Boltzmann constant

#Thermodynamic Processes: How Systems Behave

The First Law restates the principle of energy conservation, considering energy transfers via heat and work:

• Isolated System: Total energy remains constant.

• Closed System: Change in internal energy (ΔU) equals heat added (Q) plus work done on the system (W).

  $$\Delta U = Q + W$$

  • Q is positive when heat is added to the system and negative when heat is removed from the system
  • W is positive when work is done on the system and negative when work is done by the system

#Work Done by a System

Work done by a system (like a gas expanding against a piston) is given by:

$$W = -P \Delta V$$

Where:

• P = pressure (constant or average)
• ΔV = change in volume

#PV Diagrams: Visualizing Processes

Pressure-Volume (PV) diagrams are super useful for visualizing thermodynamic processes:

• Isotherms: Lines of constant temperature.
• Work: The absolute value of the work done equals the area under the PV curve.

#Special Cases: Thermal Processes

These depend on the system's configuration and interactions with its surroundings:

• Isovolumetric (Isochoric): Constant volume (ΔV = 0), so W = 0. All energy transfer is through heat.
• Isothermal: Constant temperature (ΔT = 0). The system exchanges heat with its surroundings to maintain constant temperature.
• Isobaric: Constant pressure (ΔP = 0). Work is simply -PΔV.
• Adiabatic: No heat transfer (Q = 0). Changes in internal energy are due to work done.

#Final Exam Focus 🎯

Okay, let's zoom in on what's most important for the exam:

• First Law Equation: ΔU = Q + W. Know it inside and out!
• PV Diagrams: Be able to calculate work from the area under the curve and interpret different processes.
• Special Processes: Understand the conditions for isothermal, adiabatic, isovolumetric, and isobaric processes.
• Ideal Gas Law: Remember how temperature, pressure, and volume relate.

#Last-Minute Tips

• Time Management: Don't spend too long on one question. If you're stuck, move on and come back.
• Units: Always check your units. Use SI units (Pascals, cubic meters, Joules).
• Sign Conventions: Pay close attention to the signs of work and heat.
• Practice: Review your practice problems and focus on areas where you struggle.

#Practice Questions

Practice Question

#Multiple Choice Questions

1. A gas is compressed adiabatically. Which of the following is true? (A) The temperature of the gas increases. (B) The temperature of the gas decreases. (C) The temperature of the gas remains constant. (D) The internal energy of the gas remains constant.

2. In an isothermal process, a gas expands from $V_1$ to $V_2$ at temperature T. The work done by the gas is: (A) $P(V_2 - V_1)$ (B) $-P(V_2 - V_1)$ (C) $nRT \ln(V_2/V_1)$ (D) $-nRT \ln(V_2/V_1)$

3. A system undergoes a process where 50 J of heat is added to the system, and 20 J of work is done by the system. What is the change in internal energy of the system? (A) 70 J (B) 30 J (C) -30 J (D) -70 J

#Free Response Question

A cylinder contains 0.5 mol of an ideal monatomic gas initially at a pressure of 2.0 x 10^5 Pa and a volume of 5.0 x 10^-3 m^3. The gas undergoes the following reversible process:

I. The gas expands at constant pressure until its volume is 1.0 x 10^-2 m^3. II. The gas is then compressed adiabatically back to its initial volume of 5.0 x 10^-3 m^3. (a) Calculate the initial temperature of the gas.

(b) Calculate the work done by the gas during the constant pressure expansion.

(c) Calculate the final temperature of the gas after the adiabatic compression.

(d) Draw a PV diagram illustrating the entire process.

Scoring Rubric

(a) (2 points)

• 1 point for using the ideal gas law: PV = nRT
• 1 point for correct calculation: T = (2.0 x 10^5 Pa)(5.0 x 10^-3 m^3) / (0.5 mol * 8.31 J/mol·K) = 240.7 K

(b) (2 points)

• 1 point for using the work equation: W = -PΔV
• 1 point for correct calculation: W = - (2.0 x 10^5 Pa)(1.0 x 10^-2 m^3 - 5.0 x 10^-3 m^3) = -1000 J

(c) (3 points)

• 1 point for recognizing that Q = 0 for adiabatic process
• 1 point for using the adiabatic process equation: $P_1V_1^{\gamma} = P_2V_2^{\gamma}$ where $\gamma = 5/3$ for monatomic gas
• 1 point for correct calculation: $T_2 = T_1(V_1/V_2)^{\gamma - 1} = (240.7K)(10^-2/5*10^-3)^{2/3} = 382.4K$

(d) (2 points)

• 1 point for correct representation of isobaric expansion
• 1 point for correct representation of adiabatic compression

You've got this! Remember, the First Law is all about energy conservation. Review these notes, practice the problems, and you'll be ready to ace the exam. Good luck! 🍀