# First Law of Thermodynamics

Physics I · Principles of Waves, Fluids and Thermodynamics · https://tryals.app/learn/physics-i/first-law-of-thermodynamics

## The Energy Ledger

The **First Law of Thermodynamics** is the universe's unbreakable accounting rule: a system's internal energy changes only by heat added or work done.

$$\Delta U = Q - W$$

Here, $\Delta U$ is **internal energy change**, $Q$ is **heat added**, and $W$ is **work done by** the system. A pump warms as you compress it (work becomes energy); a spray can chills as gas expands.

**Internal energy** for an ideal gas is $U = nC_V T$, and **Mayer's relation** links heat capacities by $C_P - C_V = R$.

> **Key insight:** This law conserves energy but does not restrict direction—the second law handles that.

## Ideal Gas Processes

Gas processes transform energy in distinct ways. The table below outlines the core behaviors:

| Process | Condition | Work ($W$) | Heat ($Q$) |
| --- | --- | --- | --- |
| **Isothermal** | $T$ constant | $nRT\ln(V_f/V_i)$ | $Q = W$ |
| **Isobaric** | $P$ constant | $P\Delta V$ | $nC_P\Delta T$ |
| **Isochoric** | $V$ constant | $0$ | $nC_V\Delta T$ |
| **Adiabatic** | $Q = 0$ | $-\Delta U$ | $0$ |

For **adiabatic processes**, $PV^\gamma = \text{const}$ and $TV^{\gamma-1} = \text{const}$, where $\gamma = C_P/C_V$.

> **Common pitfall:** $Q = 0$ does not mean $\Delta T = 0$. Adiabatic compression raises temperature purely through work—this is how a diesel engine fires without a spark plug.

## Practice questions

7 of this lesson's 15 practice questions, with answers. The full set is in the app.

### 1. Complete the following passage by choosing the correct terms from the word bank.

**Answer:** The first law of thermodynamics, $\Delta U = Q - W$, states that the change in **internal** energy equals heat **added** to the system minus work **done** by the system.

**Why:** $\Delta U = Q - W$: U is internal energy, Q is heat into the system, W is work done by the system.

Page: https://tryals.app/practice/physics-i/first-law-of-thermodynamics/complete-the-following-passage-by-choosing-the-correct-terms-from-the

### 2. Sort each statement as applying to Internal energy (U), Heat (Q), or Work (W).

**Answer:**

- Internal energy U: A state function; depends only on current state, $\Delta U = 0$ for an isothermal ideal gas cycle
- Heat Q: Energy transferred due to temperature difference
- Work W: Energy transferred by a force through a displacement

**Why:** U is a state function; Q and W are path-dependent energy transfers.

Page: https://tryals.app/practice/physics-i/first-law-of-thermodynamics/sort-each-statement-as-applying-to-internal-energy-u-heat-q-or

### 3. A gas in an insulated cylinder (no heat exchange) is **compressed** by pushing the piston in. What happens to the gas temperature?

A. It cannot be determined without knowing the gas
B. It rises: the work done becomes internal energy
C. It falls, since compression removes energy
D. It stays constant because no heat was added

**Answer:** B. It rises: the work done becomes internal energy

**Why:** Adiabatic compression pumps energy in through work alone, so $U$, and with it the temperature, must rise. A bicycle pump warming up, and a diesel engine igniting fuel with no spark plug, are both this line of the first law in action.

Page: https://tryals.app/practice/physics-i/first-law-of-thermodynamics/a-gas-in-an-insulated-cylinder-no-heat-exchange-is-compressed-by

### 4. The first law says $\Delta U = Q - W$, and internal energy is a state function while heat and work are not. What does that distinction mean in practice?

A. That the first law applies only to reversible processes
B. That $\Delta U$ depends on the endpoints and $Q$, $W$ on the path
C. That internal energy is conserved while heat and work are not
D. That heat and work cannot be measured directly

**Answer:** B. That $\Delta U$ depends on the endpoints and $Q$, $W$ on the path

**Why:** The split between $Q$ and $W$ is a property of the journey; their difference is not. This is exactly what makes engine cycles worth designing.

Page: https://tryals.app/practice/physics-i/first-law-of-thermodynamics/the-first-law-says-u-q-w-and-internal-energy-is-a-state

### 5. Order these milestones in the development of thermodynamics.

**Answer:**

1. Carnot analyses ideal engine efficiency
2. Joule establishes mechanical equivalent of heat
3. Clausius formulates the first and second laws
4. Boltzmann connects entropy to statistical mechanics
5. Nernst proposes the third law (heat theorem)

**Why:** Carnot (1824), Joule (1843–49), Clausius (1850–65), Boltzmann (1870s–80s), Nernst (1906).

Page: https://tryals.app/practice/physics-i/first-law-of-thermodynamics/order-these-milestones-in-the-development-of-thermodynamics

### 6. For each process on an ideal gas, sort by what the **first law** says about it. ($Q$ = heat added, $W$ = work done by the gas.)

**Answer:**

- Q = 0: Adiabatic expansion (insulated)
- ΔU = 0: Isothermal compression (constant T), Any cycle that returns the gas to its starting state
- W = 0: Heating at fixed volume

**Why:** Each named process zeroes one term of $\Delta U = Q - W$, which is exactly what makes it solvable. Spotting the zero is always step one of a thermodynamics problem.

Page: https://tryals.app/practice/physics-i/first-law-of-thermodynamics/for-each-process-on-an-ideal-gas-sort-by-what-the-first-law-says

### 7. A gas absorbs **500 J** of heat and does **300 J** of work pushing its piston out. By how much does its internal energy change, in joules?

**Answer:** 200 (within ±1)

**Why:** $\Delta U = 500 - 300 = 200$ J. The first law is nothing but energy bookkeeping: heat and work are the only two currencies, and $U$ is the account balance.

Page: https://tryals.app/practice/physics-i/first-law-of-thermodynamics/a-gas-absorbs-500-j-of-heat-and-does-300-j-of-work-pushing-its-piston