Thermodynamics

A thermodynamic system is any part of the universe under study (for example, a gas enclosed in a cylinder), while everything else is called the surroundings. The wall separating a system from its surroundings can be adiabatic (does not allow heat flow even after a long time) or diathermic (allows heat to flow readily).

When two systems are connected by a diathermic wall, energy flows from the hotter to the colder system until no further net exchange of heat takes place. The systems are then said to be in thermal equilibrium. A system is in thermal equilibrium if its macroscopic variables (pressure, volume, temperature, mass, composition) do not change with time.

Zeroth law of thermodynamics: If two systems A and B are separately in thermal equilibrium with a third system C, then A and B are also in thermal equilibrium with each other.

The zeroth law is stated separately from the first and second laws because it was recognised only after those laws were named, but logically it comes first, since it defines the very idea of temperature: the equilibrium value of the common property (temperature) is the same for all three systems A, B and C. Temperature is thus a state variable that decides the direction of net heat flow — heat always flows from a body at higher temperature to one at lower temperature until thermal equilibrium is reached.

The Celsius and Kelvin temperature scales are related by T (K) = t (°C) + 273.15. The Kelvin scale is the SI absolute temperature scale used in all thermodynamic formulas.

Worked Example (Principle of thermal equilibrium — calorimetry):

Given: Mass of hot water, m₁ = 0.2 kg at 80°C; mass of cold water, m₂ = 0.3 kg at 20°C, mixed in an insulated container (specific heat capacity same for both, container heat capacity neglected).

Formula: At thermal equilibrium, heat lost by the hot water equals heat gained by the cold water: m₁s(80 − T) = m₂s(T − 20), where T is the common final (equilibrium) temperature and s is the specific heat capacity of water (cancels out).

Substitution: 0.2(80 − T) = 0.3(T − 20) ⟹ 16 − 0.2T = 0.3T − 6 ⟹ 22 = 0.5T

Result: T = 44°C. This common final temperature is the thermal equilibrium temperature reached by the two masses of water, illustrating the zeroth law in action.

Heat (Q) is energy transferred between a system and its surroundings because of a temperature difference. Work (W) is energy transferred when a system changes its volume against an external pressure, or more generally when a macroscopic force acts through a displacement. Both heat and work are path-dependent quantities — their values depend on the process followed to go from an initial to a final state, not just on the two states themselves.

Internal energy (U) is the sum of the kinetic and potential energies of all the molecules of a system due to their motion and mutual interactions. Unlike heat and work, internal energy is a state variable (state function): its value depends only on the state of the system, described by variables such as pressure, volume and temperature, and not on the path by which that state was reached. For an ideal gas, internal energy depends only on its temperature.

When a gas expands quasi-statically (slowly enough to remain in equilibrium at every stage) against a pressure P, the small work done by the gas for a small change in volume dV is dW = P dV. For a finite change from volume V₁ to V₂, the total work done by the gas equals the area under the curve on a P-V diagram:

W = ∫ (from V₁ to V₂) P dV

Sign convention (as used in the first law): heat absorbed by the system and work done BY the system are both taken as positive; heat given out by the system and work done ON the system are taken as negative.

Worked Example (Work done at constant pressure):

Given: A gas is expanded at a constant pressure P = 2 × 10⁵ Pa. Initial volume V₁ = 2 × 10⁻³ m³, final volume V₂ = 5 × 10⁻³ m³.

Formula: For an isobaric (constant pressure) process, W = P(V₂ − V₁).

Substitution: W = 2 × 10⁵ × (5 × 10⁻³ − 2 × 10⁻³) = 2 × 10⁵ × 3 × 10⁻³

Result: W = 600 J. The gas does 600 J of work on its surroundings as it expands.

The first law of thermodynamics is simply the law of conservation of energy applied to a thermodynamic system. If a quantity of heat ΔQ is supplied to a system, part of it may increase the internal energy of the system by ΔU, and the remaining part may appear as work ΔW done by the system on its surroundings:

ΔQ = ΔU + ΔW

Here ΔQ is heat supplied to the system, ΔW is work done by the system, and ΔU is the resulting change in internal energy. The first law holds for every process, reversible or irreversible, and for every kind of system; it is essentially a restatement of the principle that energy can neither be created nor destroyed, only converted from one form to another.

Applying the first law to a fixed process defines the molar specific heat capacity, C, the heat required to raise the temperature of one mole of a substance by one degree: ΔQ = nCΔT. Because ΔW depends on the process, the specific heat capacity of a gas also depends on the process:

At constant volume (ΔW = 0, since ΔV = 0), all the heat goes into increasing internal energy, defining the molar specific heat at constant volume C_v: ΔQ = nC_vΔT = ΔU.

At constant pressure, some heat also does work as the gas expands, defining the molar specific heat at constant pressure C_p: ΔQ = nC_pΔT. Since work is also done, C_p is always greater than C_v for an ideal gas.

Using the ideal gas equation PV = nRT together with the first law gives Mayer's relation:

C_p − C_v = R

where R is the universal gas constant (R = 8.31 J mol⁻¹ K⁻¹). For an ideal monatomic gas (3 translational degrees of freedom per molecule), C_v = (3/2)R and C_p = (5/2)R, so the ratio of specific heats γ = C_p/C_v = 5/3. For a diatomic gas (with 2 additional rotational degrees of freedom at ordinary temperatures), C_v = (5/2)R, C_p = (7/2)R, and γ = 7/5.

Worked Example (Heat absorbed at constant volume):

Given: n = 2 mol of a monatomic ideal gas is heated at constant volume; its temperature rises by ΔT = 10 K. (R = 8.31 J mol⁻¹ K⁻¹)

Formula: At constant volume, ΔW = 0, so ΔQ = ΔU = nC_vΔT, with C_v = (3/2)R for a monatomic gas.

Substitution: C_v = 1.5 × 8.31 = 12.465 J mol⁻¹ K⁻¹; ΔQ = 2 × 12.465 × 10

Result: ΔQ = ΔU = 249.3 J. All the heat supplied goes into raising the internal energy since no work is done at constant volume.

A thermodynamic system in equilibrium is described by state variables such as pressure P, volume V, temperature T and the amount of substance n. State variables can be extensive (depend on the size of the system, e.g. volume, internal energy) or intensive (independent of size, e.g. pressure, temperature). For an ideal gas, the state variables are connected by the equation of state:

PV = nRT

A thermodynamic process is any change that takes a system from one equilibrium state to another. A quasi-static process is one carried out infinitely slowly, so the system passes through a continuous succession of equilibrium states; pressure and temperature are then uniform throughout the system at every stage. Two important idealised quasi-static processes are:

Isothermal process (constant temperature, T = constant): The system is in contact with a large heat reservoir so its temperature does not change. Since internal energy of an ideal gas depends only on T, ΔU = 0 for any isothermal change, and the first law gives ΔQ = ΔW — all the heat absorbed is converted into work. The equation of state gives PV = constant (Boyle's law) along an isothermal curve, and the work done by the gas expanding from V₁ to V₂ at temperature T is:

W = nRT ln(V₂/V₁) = nRT ln(P₁/P₂)

Adiabatic process (no heat exchange, ΔQ = 0): The system is thermally insulated, or the process happens fast enough that no heat can flow in or out. The first law then gives ΔU = −ΔW: the work done by the gas comes entirely at the expense of its internal energy (the gas cools on expansion and heats up on compression). An adiabatic process obeys:

PV^γ = constant, and equivalently TV^γ−1 = constant

The work done by the gas in an adiabatic change from (P₁, V₁, T₁) to (P₂, V₂, T₂) is:

W = (P₁V₁ − P₂V₂)/(γ − 1) = nR(T₁ − T₂)/(γ − 1)

Because PV^γ falls off faster than PV = constant, an adiabatic curve is always steeper than an isothermal curve through the same point on a P-V diagram.

Worked Example (Isothermal work):

Given: n = 1 mol of an ideal gas expands isothermally and reversibly at T = 300 K from V₁ to V₂ = 2V₁ (R = 8.31 J mol⁻¹ K⁻¹, ln 2 = 0.693).

Formula: W = nRT ln(V₂/V₁)

Substitution: W = 1 × 8.31 × 300 × ln 2 = 2493 × 0.693

Result: W ≈ 1728 J. This work is done entirely at the expense of heat absorbed from the surroundings, since ΔU = 0.

Worked Example (Adiabatic work):

Given: n = 1 mol of a monatomic ideal gas (γ = 5/3) is adiabatically compressed so that its temperature rises from T₁ = 300 K to T₂ = 400 K.

Formula: Work done by the gas, W = nR(T₁ − T₂)/(γ − 1)

Substitution: W = 1 × 8.31 × (300 − 400)/(5/3 − 1) = 8.31 × (−100)/0.667

Result: W ≈ −1246.5 J. The negative sign shows the surroundings do 1246.5 J of work on the gas during compression, which raises its internal energy and hence its temperature.

A heat engine is a device that converts heat into work, operating in a repeating cycle so it returns to its initial state after every cycle (making the internal energy change zero over one complete cycle). In each cycle, the working substance absorbs heat Q₁ from a hot reservoir (the source) at temperature T₁, does external work W, and rejects the remaining heat Q₂ to a cold reservoir (the sink) at temperature T₂. By the first law, applied over a full cycle (ΔU = 0):

W = Q₁ − Q₂

The efficiency of a heat engine is defined as the ratio of the work output to the heat input:

η = W/Q₁ = 1 − Q₂/Q₁

No heat engine can have an efficiency of 100%, since some heat must always be rejected to a sink — this is one form of the second law of thermodynamics. The theoretical maximum efficiency of any engine operating between temperatures T₁ (hot) and T₂ (cold) is achieved by a perfectly reversible (Carnot) engine:

η_max = 1 − T₂/T₁

A refrigerator or heat pump is essentially a heat engine run in reverse: external work W is done on the working substance to extract heat Q₂ from a cold reservoir (the space to be cooled) and reject a larger amount of heat Q₁ = Q₂ + W to a hot reservoir (the surroundings). Its performance is measured by the coefficient of performance (COP) rather than efficiency, because the quantity of interest is the heat removed (for a refrigerator) or the heat delivered (for a heat pump), not just work:

For a refrigerator: α = Q₂/W = Q₂/(Q₁ − Q₂)

For a heat pump: COP = Q₁/W = Q₁/(Q₁ − Q₂)

Unlike efficiency, the coefficient of performance can be, and usually is, greater than 1. For an ideal (Carnot) refrigerator operating between T₁ and T₂, α = T₂/(T₁ − T₂).

Worked Example (Efficiency of a heat engine):

Given: A heat engine absorbs Q₁ = 2000 J of heat from its source and performs W = 500 J of useful work per cycle.

Formula: η = W/Q₁; heat rejected Q₂ = Q₁ − W.

Substitution: η = 500/2000 = 0.25; Q₂ = 2000 − 500 = 1500 J.

Result: The engine's efficiency is 25%, and it rejects 1500 J of heat to the sink every cycle.

The first law of thermodynamics only requires that energy be conserved; it does not forbid processes that never actually occur in nature, such as heat flowing spontaneously from a cold body to a hot one. The second law of thermodynamics supplies the missing rule about the direction in which natural processes can proceed. It has several equivalent statements:

Kelvin-Planck statement: No process is possible whose sole result is the absorption of heat from a single reservoir and the complete conversion of that heat into work. (No heat engine can have 100% efficiency.)

Clausius statement: No process is possible whose sole result is the transfer of heat from a colder object to a hotter object without external work being done on the system. (Heat does not flow spontaneously from cold to hot.)

These two statements can be shown to be logically equivalent — violating one allows the other to be violated as well.

A process is reversible if it can be reversed such that both the system and its surroundings return exactly to their original states, with no other change left anywhere in the universe. A reversible process must be quasi-static (infinitely slow, so the system stays arbitrarily close to equilibrium throughout) and free of dissipative effects such as friction, viscosity or electrical resistance. Real processes always involve some finite rate of change and some dissipation, so all natural processes are, strictly speaking, irreversible. Common examples of irreversibility include free expansion of a gas into vacuum, heat flow across a finite temperature difference, and any process involving friction.

Carnot's theorem, a consequence of the second law, states that no engine operating between two given temperatures can be more efficient than a reversible (Carnot) engine operating between the same two temperatures, and that all reversible engines operating between the same two temperatures have exactly the same efficiency, η_max = 1 − T₂/T₁, regardless of the working substance used. This places a fundamental, substance-independent upper limit on how efficient any real heat engine can be.

Worked Example (Applying Carnot's limit):

Given: An inventor claims to have built a heat engine that operates between a source at T₁ = 600 K and a sink at T₂ = 300 K with an efficiency of 60%.

Formula: Maximum possible (Carnot) efficiency between these two temperatures, η_max = 1 − T₂/T₁.

Substitution: η_max = 1 − 300/600 = 1 − 0.5

Result: η_max = 0.5 = 50%. Since the claimed efficiency of 60% exceeds this theoretical maximum, the claim violates the second law of thermodynamics and is therefore impossible.