First law - ΔU = q + w (chemist sign convention)

Core Concept

The First Law of Thermodynamics states that the change in internal energy of a system equals the sum of heat transferred to the system and work done on the system:

WHY This Form?

This is conservation of energy for thermodynamic processes. Energy of the universe is constant—if a system gains energy, surroundings lose it. The system can gain energy two ways:

1. Heat flowing in (q > 0): molecules collide with hotter surroundings, speed up
2. Work done on it (w > 0): compression makes molecules crowd and move faster

Sign Convention: Chemist vs Physicist

CHEMIST CONVENTION (what we use):

• q > 0: heat absorbed BY system (endothermic)
• q < 0: heat released BY system (exothermic)
• w > 0: work done ON system (compression)
• w < 0: work done BY system (expansion)

Physicist convention (don't use in chemistry): ΔU = q − w (work done BY system is positive). We avoid this because chemists care about the system's perspective: what happens to the system.

Derivation from First Principles

Starting point: Conservation of energy for an isolated system (system + surroundings).

$E_{\text{universe}} = E_{\text{system}} + E_{\text{surroundings}} = \text{constant}$

Taking differentials: $dE_{\text{system}} + dE_{\text{surroundings}} = 0$

So: $dE_{\text{system}} = -dE_{\text{surroundings}}$

For the system, internal energy can change via:

1. Heat transfer ($\delta q$): energy transfer due to temperature difference
2. Work transfer ($\delta w$): energy transfer due to organized force × displacement

WHY these two only? These are the ONLY ways energy crosses boundaries at the molecular level. Heat = random molecular collisions. Work = coordinated force (like a piston pushing).

For an infinitesimal process: $dU = \delta q + \delta w$

Integrating over a finite process: $\Delta U = \int \delta q + \int \delta w = q + w$

WHY is ΔU a state function but q and w are not?

• ΔU depends only on initial and final states (like elevation change hiking: only depends on start/end altitude, not path).
• q and w depend on the PATH taken (like distance walked: depends if you zigzag or go straight).

This is why we write $dU$ (exact differential) but $\delta q$ and $\delta w$ (inexact differentials).

Worked Examples

Common Mistakes

Deep Connection: Why This Matters

The first law is the foundation for:

• Enthalpy definition: At constant P, we define H = U + PV so that ΔH = q_p (heat at constant pressure). This only makes sense BECAUSEΔU = q + w.
• Hess's Law: ΔU is path-independent, so we can add reaction steps algebraically.
• Calorimetry: We measure heat (q) under controlled conditions (constant V or P) to find ΔU orΔH.
• Carnot cycle: Efficiency limits arise from first law applied to cyclic processes (ΔU_cycle = 0).

Philosophical point: This law says energy is conserved but transferable. The universe's total energy is fixed since the Big Bang. Every chemical reaction, every engine, every living cell operates under this constraint. We can't create energy; we can only move it around and change its form.

Connections

• Internal Energy U - the state function being changed
• Heat q - thermal energy transfer mechanism
• Work w - mechanical energy transfer mechanism
• PV work - the main type of work in chemistry: w = −Pext·ΔV
• Enthalpy H - defined using first law: H = U + PV
• State functions vs path functions - why ΔU is exact but q, w are not
• Bomb calorimetry - measures ΔU directly (constant V)
• Adiabatic processes - q = 0, so ΔU = w
• Isothermal processes - ΔU = 0 for ideal gas
• Hess's Law - relies on ΔU being state function

#flashcards/chemistry

What is the First Law of Thermodynamics in chemist sign convention? :: ΔU = q + w, where ΔU is change in internal energy, q is heat absorbed by system, w is work done on system.

In chemist sign convention, what is the sign of q when heat is absorbed by the system? :: Positive (+). Heat flowing INTO the system is positive.

Concept Map

Hinglish (regional understanding)