The First Law of Thermodynamics: Energy Conservation

The Universe's #1 Rule: You Can't Create Energy from Nothing

At its heart, the First Law of Thermodynamics is a simple, profound statement about the universe: Energy cannot be created or destroyed; it can only be converted from one form to another. It's like a universal accounting rule. The total amount of energy is constant. You can move it around and change its appearance, but you can't make it appear out of thin air or make it vanish.

To build our intuition, let's start with a simple mechanical example, which Sal from Khan Academy explains beautifully.

As you saw, when you throw a ball up, its kinetic energy (energy of motion) decreases, but its potential energy (energy of position) increases. At the very top, all the kinetic energy has been converted to potential energy. As it falls, that potential energy is converted back into kinetic energy.

But what about air resistance? The ball doesn't come back down with the exact same speed. Some energy seems to be "lost." But it's not lost at all! It was transferred to the air molecules, causing them to vibrate faster. In other words, the "lost" mechanical energy was converted into heat.

This is the core idea. Energy transforms between many different forms:

• Chemical energy in gasoline becomes mechanical energy to move a car and heat energy in the hot exhaust.
• Electrical energy from a wall socket becomes light and heat in a lightbulb.
• Chemical energy in the food you eat becomes mechanical energy to lift a weight and heat that your body radiates.

The System's Bank Account: Internal Energy (U)

Okay, so energy transforms. But when we analyze a system—like the gas inside a piston—we need to know how much energy it has stored inside it. We call this the Internal Energy, and we give it the symbol $\mathbf{U}$.

Think of internal energy as the total energy contained within the boundaries of your system. It's the sum of all the microscopic energies of all the atoms and molecules inside. This includes:

• Kinetic energy from the molecules zipping around (translational), spinning (rotational), and vibrating.
• Potential energy stored in the chemical bonds between atoms.
• Nuclear energy stored within the atoms themselves (though we usually ignore this in classical thermodynamics).

In short, internal energy is the grand total of all the energy "owned" by the system.

For a more detailed breakdown of what makes up internal energy, let's revisit Sal's explanation.

Deposits and Withdrawals: Heat (Q) and Work (W)

If a system's internal energy ($U$) is its "energy bank account," how can you change the balance? You can't create or destroy energy, so any change must come from energy crossing the system's boundary.

In thermodynamics, there are only two ways for energy to cross the boundary of a closed system: Heat and Work.

1. Heat ($Q$): This is energy transfer driven by a temperature difference. If you place a cold can of soda on a warm table, energy flows from the table to the can in the form of heat. It's a "disorganized" transfer of energy at the molecular level.

2. Work ($W$): This is an "organized" energy transfer. It involves a force acting over a distance. For example, when the expanding gas in a cylinder pushes a piston, the gas is doing work on the piston. When you use a pump to compress air into a tire, you are doing work on the air.

This is a crucial point: Heat and work are not things a system has; they are things that are done to or by a system. They are energy in transit. A system contains internal energy, not heat or work.

Think of it this way:

• Internal Energy ($U$) is the money in your bank account.
• Heat ($Q$) and Work ($W$) are the deposits and withdrawals. You don't "contain" a deposit; you make a deposit, and it changes your balance.

Recap Break

Let's pause. This is a big conceptual step.

• The total energy of the universe is constant. (Energy Conservation)
• The energy stored inside our system is called Internal Energy ($U$).
• Energy can enter or leave the system in only two ways: as Heat ($Q$) or Work ($W$). These are processes, not stored quantities.

Now, let's put this together into a famous equation.

The Energy Balance Equation

If the change in a system's internal energy can only be caused by heat and work crossing the boundary, we can write a simple energy balance:

Change in Internal Energy = (Energy Added) - (Energy Removed)

This gives us the formal equation for the First Law of Thermodynamics:

$\Delta U=Q-W$

Where:

• $\Delta U$ is the change in the system's internal energy ($U_{final}-U_{initial}$).
• $Q$ is the net heat transferred into the system.
• $W$ is the net work done by the system.

The All-Important Sign Convention

This is one of the most common trip-up points for students, so let's be very clear. By convention in engineering thermodynamics:

• Heat ($Q$)
  • Positive (+): Heat is added TO the system. (A deposit)
  • Negative (-): Heat is removed FROM the system.
• Work ($W$)
  • Positive (+): Work is done BY the system on its surroundings (e.g., an expanding gas pushing a piston). (A withdrawal)
  • Negative (-): Work is done ON the system by its surroundings (e.g., a piston compressing a gas).

Why this seemingly strange convention for work? It's historical. Early thermodynamics was all about building heat engines. The goal was to add heat ($Q_{in}$) to get useful work out ($W_{out}$). Both of these were seen as positive, desirable quantities, and the convention stuck.

Real-World Application: Human Metabolism

Let's apply the First Law to something you experience every day: your own body.

Let's break it down with our new equation, $\Delta U=Q-W$:

• Eating Food: You consume chemical energy. This isn't heat or work in the thermodynamic sense; it's a process that increases your body's stored internal energy ($U$). Let's call this $E_{food}$.
• Losing Body Heat: Your body is almost always warmer than your surroundings, so you are constantly losing heat. This is heat leaving the system, so $Q$ is negative.
• Doing Work: When you walk, run, or lift something, your body is doing work on the surroundings. $W$ is positive.

So, the change in your body's stored energy over a day is:
$\Delta U_{body}=E_{food}+Q-W$
(Here $Q$ is negative and $W$ is positive).

• If you eat exactly as much energy as you lose through heat and work, then $\Delta U_{body}=0$, and your weight stays constant.
• If you eat more than you use ($E_{food}>|Q|+W$), then $\Delta U_{body}$ is positive. Your body stores this excess internal energy as fat.
• If you eat less than you use ($E_{food}<|Q|+W$), then $\Delta U_{body}$ is negative. Your body makes up for the deficit by metabolizing its own stored fat. This is how dieting works!

Let's Practice

Time to put the equation to use with some straightforward problems.

Worked Problem 1

Problem: A closed system undergoes a process where 200 kJ of heat is added to it. During this process, the system does 50 kJ of work on its surroundings. What is the change in the internal energy of the system?

Solution:

1. Identify the given values and their signs:
   • Heat is added to the system, so $Q=+200\text{kJ}$.
   • Work is done by the system, so $W=+50\text{kJ}$.
2. Apply the First Law equation:
   $\Delta U=Q-W$
   $\Delta U=(+200\text{kJ})-(+50\text{kJ})$
   $\Delta U=150\text{kJ}$
   The internal energy of the system increased by 150 kJ.

Worked Problem 2

Problem: The air inside a rigid, insulated tank is stirred by a paddle wheel. The paddle wheel does 75 kJ of work on the air. What is the change in the internal energy of the air?

Solution:

1. Identify the given values and their signs:
   • The tank is insulated, which means no heat can get in or out. So, $Q=0$.
   • Work is done on the air by the paddle wheel, so $W=-75\text{kJ}$.
2. Apply the First Law equation:
   $\Delta U=Q-W$
   $\Delta U=(0)-(-75\text{kJ})$
   $\Delta U=75\text{kJ}$
   The internal energy of the air increased by 75 kJ. This makes sense—stirring the air (like Joule did in his famous experiment) makes it warmer.

Conceptual Questions

1. In your own words, explain the difference between internal energy, heat, and work.

   • Answer: Internal energy is the total energy stored inside a system. Heat and work are not stored; they are processes of energy transfer across the system boundary. Heat is energy transfer due to a temperature difference, while work is energy transfer via an organized force over a distance.

2. A gas in a piston-cylinder is compressed. During the process, 50 J of heat is removed. The internal energy of the gas increases by 20 J. Was work done on the gas or by the gas? How much?

   • Answer: We have $\Delta U=+20\text{J}$ and $Q=-50\text{J}$.
   • Using $\Delta U=Q-W$, we can solve for W:
     $+20\text{J}=-50\text{J}-W$
     $W=-50\text{J}-20\text{J}=-70\text{J}$.
   • Since $W$ is negative, 70 J of work was done on the gas. This makes sense for a compression process.

Conclusion

Excellent work today! We've laid down one of the most critical cornerstones of thermodynamics. The First Law isn't just an equation; it's a powerful statement about how energy behaves in our universe.

Key Takeaways:

• The First Law of Thermodynamics is the principle of energy conservation: energy cannot be created or destroyed, only transformed.
• The energy stored within a system is its Internal Energy ($U$).
• Energy crosses the boundary of a closed system only as Heat ($Q$) or Work ($W$).
• The First Law is an energy balance equation: $\mathbf{\Delta }\mathbf{U}=\mathbf{Q}-\mathbf{W}$.
• The sign convention is key: Heat in is positive, Work out is positive.

Preview of the Next Lesson

We've established that the First Law is an accounting principle for energy. But so far, internal energy ($U$) is still a bit of a vague concept—the "sum of all microscopic energies." This isn't very practical if we want to calculate it.

In our next lesson, we will dig deeper into internal energy as a measurable property of a system. We'll explore how it's related to things we can measure, like temperature, and distinguish it more formally from heat and work, which are not properties. This will move us one step closer to using the First Law for practical engineering analysis.