Heat, Work, Internal Energy and First Law

The First Law and Energy Conservation

The First Law of Thermodynamics states that energy added to a System as heat minus the work done by the system equals the change in its Internal Energy.

$$\Delta U = Q - W$$

Energy conservation statement for thermodynamic processes.

$$\Delta U$$=Change in internal energy(J)

$$Q$$=Net heat added to the system(J)

$$W$$=Work done BY the system(J)

Adiabatic process ($Q = 0$)

→

$$\Delta U = -W$$ (work done at expense of internal energy)

Isothermal process ($\Delta U = 0$)

→

$$Q = W$$ (all heat converts to work)

Isochoric (constant volume, $W = 0$)

→

$$\Delta U = Q$$ (all heat stored internally)

Sign Convention: Heat entering the system is positive ($+Q$); work done by the system is positive ($+W$). Work done ON the system makes $W$ negative.

State vs Path: Internal Energy depends only on the current state (State Function), while $Q$ and $W$ are Path Function variables.

Dimensional Check: All three quantities ($\Delta U$, $Q$, $W$) have units of joules — verify unit consistency before substituting.

Isothermal and Adiabatic Processes

An isothermal process occurs at constant temperature. For an ideal gas, Internal Energy depends only on temperature, so $\Delta U = 0$ and all heat converts to work.

$$P_1 V_1 = P_2 V_2$$

Boyle's Law holds during isothermal changes — pressure and volume are inversely proportional.

$$P_1, V_1$$=Initial pressure and volume(Pa, m³)

$$P_2, V_2$$=Final pressure and volume(Pa, m³)

Volume doubles ($V_2 = 2V_1$)

→

Pressure halves ($P_2 = P_1/2$)

Slow Process: Isothermal changes must occur slowly to allow continuous heat exchange and maintain constant temperature.

First Law Reduction: Since $\Delta U = 0$, the first law gives $Q = W$ — heat absorbed equals work done by the gas.

PV Diagram: The curve is a rectangular hyperbola called an isotherm.

An adiabatic process is one where no heat enters or leaves the system ($Q = 0$). Work done by the gas comes entirely from its internal energy, causing the temperature to change.

$$PV^\gamma = \text{constant}$$

The adiabatic condition — pressure drops faster than in isothermal expansion because temperature also decreases.

$$\gamma$$=Ratio of molar specific heats $C_p / C_v$(dimensionless)

First Law Reduction: With $Q = 0$, the first law gives $W = -\Delta U$. Expansion cools the gas; compression heats it.

Rapid Process: Adiabatic changes occur rapidly (no time for heat exchange) or in insulated systems.

Real Examples: Burst tyre (rapid expansion → cooling), bicycle pump (rapid compression → heating), cloud formation (rising air expands adiabatically).

Steeper Curve: On a PV diagram the adiabat falls more steeply than the isotherm because both pressure and temperature decrease during expansion.

Reversible and Irreversible Processes

A reversible process can be retraced in exactly reverse order without producing any change in the surroundings. All real processes are irreversible to some degree.

Reversible Examples: Slow compression of a gas in a cylinder, slow evaporation and condensation of a substance.

Irreversible Examples: Explosions, friction, rapid heat transfer, free expansion — any process involving dissipation.

Cycle: A succession of events that returns the system to its initial state. A reversible cycle has all reversible steps.

Key Distinction: Reversible processes are idealizations that set the theoretical upper bound on efficiency. Real processes always fall short.

Heat Engines and Thermal Efficiency

A heat engine converts thermal energy into mechanical work by absorbing heat $Q_H$ from a high-temperature source and rejecting waste heat $Q_L$ to a cold Heat Reservoir.

$$\eta = \frac{W}{Q_H} = 1 - \frac{Q_L}{Q_H}$$

Efficiency compares the useful work output to the total heat energy paid for.

$$\eta$$=Thermal efficiency(dimensionless (0 to 1))

$$W$$=Net work output(J)

$$Q_H$$=Heat absorbed from hot reservoir(J)

$$Q_L$$=Heat rejected to cold reservoir(J)

Cyclic Process: The engine must return to its initial state each cycle, so $\Delta U_{\text{cycle}} = 0$ and $W = Q_H - Q_L$.

Energy Balance: By the First Law, $Q_H = W + Q_L$ — input heat splits into useful work and waste heat.

Real-world Limits: Petrol engines ≈ 25-30%, diesel engines ≈ 35-40%, steam turbines ≈ 35-46%.

The Second Law of Thermodynamics

The Second Law restricts the direction of natural processes: heat flows spontaneously only from hot to cold, and no engine can convert heat entirely into work.

Kelvin Statement: It is impossible to convert heat extracted from a single reservoir entirely into work without leaving any change in the working system.

Practical Implication: Two reservoirs at different temperatures are essential for converting heat into work. Oceans contain vast thermal energy, but without a colder reservoir, none can be extracted as work.

Perpetual Motion: The second law forbids perpetual motion machines of the second kind (100% heat-to-work conversion).

Refrigerator: A heat engine running in reverse — it requires external work to transfer heat from cold to hot surroundings.

The Carnot Cycle

The Carnot Cycle is an ideal reversible cycle consisting of two Isothermal and two Adiabatic steps. No engine can be more efficient than a Carnot engine operating between the same two temperatures.

$$\eta_{\text{Carnot}} = 1 - \frac{T_L}{T_H}$$

The theoretical maximum efficiency depends only on reservoir temperatures in Kelvin.

$$T_L$$=Cold reservoir temperature(K (must be Kelvin))

$$T_H$$=Hot reservoir temperature(K (must be Kelvin))

$T_L = 0$ K

→

$\eta = 100\%$ (impossible — absolute zero cannot be reached)

$T_L = T_H$

→

$\eta = 0\%$ (no temperature gradient, no work possible)

$T_H$ doubled while $T_L$ fixed

→

$\eta$ increases but never reaches 100%

Carnot's Theorem: All Carnot engines between the same two temperatures have the same efficiency, regardless of the working substance.

Proportionality Shortcut: $\eta = (T_H - T_L)/T_H$. If $T_H = 2T_L$, then $\eta = 50\%$. If $T_H = 3T_L$, then $\eta = 66.7\%$.

Raising Efficiency: Since $T_L$ is usually near room temperature, the only practical way to increase efficiency is to raise $T_H$.

Four Steps of the Carnot Cycle (PV Diagram)

A→B: Isothermal expansion at $T_H$ — gas absorbs $Q_H$, does work, volume increases

B→C: Adiabatic expansion — temperature drops from $T_H$ to $T_L$, no heat exchange

C→D: Isothermal compression at $T_L$ — gas rejects $Q_L$, work done on gas

D→A: Adiabatic compression — temperature rises from $T_L$ back to $T_H$

Petrol and Diesel Engines

Practical heat engines like petrol and diesel engines operate on four-stroke cycles based on the Carnot principle, but with irreversible processes that reduce their efficiency.

Petrol Engine (4 strokes): (1) Intake — fuel-air mixture drawn in, (2) Compression — adiabatic compression, (3) Power — spark ignites mixture, rapid expansion drives piston, (4) Exhaust — waste gases expelled.

Diesel Engine Difference: No spark plug needed. Air compressed to very high temperature, then diesel sprayed in and ignites on contact (compression ignition).

Efficiency Comparison: Diesel engines (35-40%) are more efficient than petrol engines (25-30%) because of higher compression ratios.

Multi-cylinder Design: Cars typically use 4+ cylinders on the same crankshaft, firing in sequence for smooth power delivery.

Entropy: The Measure of Disorder

Entropy ($S$) is a thermodynamic State Function that measures the unavailability of a system's energy to do work — often interpreted as molecular disorder.

$$\Delta S = \frac{\Delta Q}{T}$$

Entropy change equals heat transferred divided by absolute temperature (for a reversible process).

$$\Delta S$$=Change in entropy(J/K (or J K⁻¹))

$$\Delta Q$$=Heat transferred (positive if added, negative if removed)(J)

$$T$$=Absolute temperature at which transfer occurs(K)

Reversible process

→

$\Delta S_{\text{total}} = 0$ (entropy of universe unchanged)

Irreversible process

→

$\Delta S_{\text{total}} > 0$ (entropy of universe always increases)

Sign Convention: $\Delta S > 0$ when heat is added to the system; $\Delta S < 0$ when heat is removed.

Net Entropy of Heat Transfer: When $Q$ flows from $T_H$ to $T_L$: $\Delta S_{\text{net}} = Q/T_L - Q/T_H > 0$ (always positive since $T_L < T_H$).

Proportionality Insight: Same $Q$ at low $T$ creates MORE disorder than at high $T$ — warming ice by 1 J changes entropy more than warming steam by 1 J.

Phase Changes: Melting and boiling increase entropy (more disorder); freezing and condensation decrease local entropy.

Degradation of Energy: Increasing entropy means energy degrades from useful, ordered forms to random thermal motion.

The entropy statement of the Second Law: any natural process proceeds in the direction that increases the total entropy of the system plus its environment.

Arrow of Time: Entropy increase defines the direction of natural processes — you can scramble an egg but never unscramble it.

Mixing Example: When hot and cold water are mixed, the warm result cannot spontaneously separate. Energy is conserved but is no longer available for work.

Free Expansion: Gas expanding into a vacuum increases entropy — molecules gain greater randomness of position.

Environmental Impact: Thermal pollution is an inevitable consequence of the Second Law. All energy processes increase the entropy of the environment, and even small temperature changes can disrupt ecological balance.

Thermodynamic Temperature Scale

The Carnot cycle provides a temperature scale independent of any material's properties, using the ratio of heat transfers at two temperatures.

$$T = 273.16 \frac{Q}{Q_{\text{tr}}}$$

Unknown temperature determined by comparing heat exchange to that at the triple point of water.

$$T$$=Unknown temperature(K)

$$Q$$=Heat absorbed/rejected at temperature $T$(J)

$$Q_{\text{tr}}$$=Heat absorbed/rejected at triple point of water(J)

Triple Point: The unique state where ice, liquid water, and water vapour coexist in equilibrium at exactly 273.16 K.

Material-Independent: Unlike mercury or alcohol thermometers, this scale does not depend on any substance's expansion properties.

Kelvin Unit: 1 K is defined as 1/273.16 of the thermodynamic temperature of the triple point of water.