The First Law of Thermodynamics: Concepts, Applications, and Practice Problems

First Law of Thermodynamics

Introduction

The First Law of Thermodynamics is a fundamental principle of physics that relates changes in the internal energy of a system to the heat added to the system and the work done on or by the system. It is a statement of the conservation of energy, adapted for thermodynamic processes.

Review: Ideal Gas Law and Thermal Energy

Key Definitions

• Pressure (P): The force per unit area exerted by gas molecules colliding with the walls of a container. Equation: $P = \frac{F}{A}$

• Temperature (T): A measure of the average kinetic energy of each molecule in a substance. Equation: $\frac{1}{2} m v^2 = \frac{3}{2} k_B T$

• Thermal Energy ($E_{th}$): The total kinetic energy due to all the molecules in a system. Equation: $E_{th} = \frac{3}{2} N k_B T$

Example: Comparing Gases in Two Boxes

• Two boxes (A and B) contain the same type of ideal gas at the same temperature but different masses.

• Average kinetic energy per molecule is the same in both boxes (since temperature is the same).

• Thermal energy is greater in the box with more molecules (greater mass).

Expanding the Definition of Energy Changes

Work and Heat

• Energy can be transferred to or from a system as work or heat.

• Work ($W$): Energy transferred by a force acting through a distance.

• Heat ($Q$): Energy transferred due to a temperature difference.

First Law of Thermodynamics (General Form):

$\Delta E = W_{on\ system} + Q_{gained}$

or, more commonly in physics:

$\Delta E_{sys} = Q + W_{on\ system}$

or in chemistry notation:

$\Delta U = q + w$

Energy Accounting: Changes in Energy Tell a Story

Energy Transformations

• When analyzing a process, consider all forms of energy involved:

  • Kinetic Energy ($K$)

  • Gravitational Potential Energy ($U_g$)

  • Elastic Potential Energy ($U_s$)

  • Thermal Energy ($E_{th}$)

  • Chemical Energy ($E_{chem}$)

• The sum of all energy changes equals the work done:

$\Delta K + \Delta U_g + \Delta U_s + \Delta E_{th} + \Delta E_{chem} + \cdots = W$

Example: Walking at Constant Speed

• At constant speed on flat ground, kinetic and potential energies do not change, but chemical energy from food is converted into work and heat.

• Energy input (from food) is balanced by energy output (work done and heat transferred to surroundings).

Applications: The Human Body as a Thermodynamic System

Energy Input and Output

• Input: Chemical energy in food is metabolized to provide energy for bodily functions.

• Output: The body does work (e.g., muscle contraction) and releases heat to the surroundings.

Example Reaction: Glucose metabolism

$\text{C}_6\text{H}_{12}\text{O}_6 + 6\text{O}_2 \rightarrow 6\text{CO}_2 + 6\text{H}_2\text{O} + \text{energy}$

• Energy released is stored in ATP and used for muscle contraction and nerve signal propagation.

Comparing Physics and Chemistry Notation

• Ideal Gas Law in Physics: $PV = Nk_B T$

• Ideal Gas Law in Chemistry: $PV = nRT$

• First Law (Physics): $\Delta E_{sys} = Q + W_{on\ system}$

• First Law (Chemistry): $\Delta U = q + w$

• Note: Internal energy ($U$ or $E_{sys}$) is a form of kinetic energy at the molecular level.

Practice Problems with the First Law

Sample Problems

• Problem 1: A gas in a closed container increases its thermal energy by 400 J as a result of +100 J of work being done on the system. How much heat is transferred to the system?

• Solution: $\Delta E = Q + W_{on\ system} \implies 400 = Q + 100 \implies Q = 300\ \text{J}$

• Problem 2: A gas in a closed container decreases its thermal energy by 200 J as a result of +500 J of work being done on the system. How much heat is transferred?

• Solution: $-200 = Q + 500 \implies Q = -700\ \text{J}$ (heat leaves the system)

• Problem 3: A gas is heated by +500 J, but its internal energy does not increase. What must also be happening?

• Solution: The gas must be doing +500 J of work on the surroundings (expanding).

Conceptual Questions

• Rapid Compression in a Syringe: If air is compressed rapidly in a closed syringe, the temperature increases because work is done on the gas and little heat escapes.

• Temperature Increase in a Closed System: If the temperature of a gas increases, possible causes include heat transfer into the system or work done on the system.

• Constant Volume vs. Constant Pressure Heating:

  • At constant volume, all heat goes into increasing internal energy.

  • At constant pressure, some heat does work to expand the gas, so more heat is required to achieve the same temperature increase.

Work Done on a Gas

Definition and Calculation

• Work is done on a gas when its volume changes under pressure.

• Pressure is force per unit area: $P = \frac{F}{A}$

• For a piston moving a distance $dx$ with area $A$:

$dW = F dx = P A dx = P dV$

• Total work done during a volume change from $V_i$ to $V_f$:

$W = \int_{V_i}^{V_f} P dV$

Summary Table: Key Thermodynamic Quantities

Quantity	Symbol	Definition/Formula	Units
Pressure	P	$P = \frac{F}{A}$	Pa (N/m2)
Temperature	T	Average kinetic energy per molecule: $\frac{3}{2} k_B T$	K (Kelvin)
Thermal Energy	$E_{th}$	$E_{th} = \frac{3}{2} N k_B T$	J (Joules)
Work	W	$W = \int P dV$	J (Joules)
Heat	Q	Energy transferred due to temperature difference	J (Joules)
Internal Energy	U or $E_{sys}$	Sum of all microscopic kinetic and potential energies	J (Joules)

Key Takeaways

• The First Law of Thermodynamics is a statement of energy conservation for thermodynamic systems.

• Energy can be transferred as heat or work, and the internal energy of a system changes accordingly.

• Understanding the direction of energy transfer and the sign conventions is crucial for solving thermodynamics problems.

Additional info: In real systems, energy can also be transferred by radiation or mass flow, but these are not covered in this summary.