Exercises: Fundamentals of thermodynamics

Explain what a thermodynamic system is and classify these examples as open, closed, or isolated: (a) an uncovered pot on a stove, (b) a gas cylinder with a piston, (c) an ideal vacuum flask. Indicate in each case what it exchanges with its surroundings (matter, heat, or work).

State the first law of thermodynamics, $\Delta U = Q - W$, explaining what each term represents and the sign convention used (heat absorbed by the gas: positive; work done by the gas: positive). Give an everyday example of a system that receives heat and does work at the same time.

Explain why heat and work are not 'properties' of the system but transfers: a gas at $300\ \text{K}$ does not 'have' heat; it has internal energy. Reason which quantities depend on the path taken between two states ($Q$ and $W$) and which does not ($\Delta U$).

Match each transformation of a gas with the quantity that remains constant: (a) isobaric, (b) isochoric, (c) isothermal, (d) adiabatic; with: pressure, volume, temperature, and absence of heat exchange. Give an approximate real-life example of each (pressure cooker, piston, bicycle pump).

State whether it is true or false, justifying your answer using the second law: 'If I leave a glass of cold water on the table, it can cool down even further by transferring heat to the room, which is warmer.' Explain in which direction heat flows spontaneously and what would need to be provided to reverse this flow.

Write the ideal gas equation of state, $p \cdot V = n \cdot R \cdot T$, identify each quantity with its usual units, and explain why temperature must be expressed in kelvins and not in degrees Celsius (what would happen if $T = 0\ ^\circ\text{C}$ were used in the formula).

The figure shows, starting from the same initial state, the four basic transformations of an ideal gas on a p-V diagram, but one is unnamed. a) State which curve corresponds to each transformation (isobaric, isochoric, isothermal, and adiabatic). b) Explain which quantity remains constant in each one. c) Reason why the curve with the question mark falls faster than the isothermal curve as the gas expands.

Explain what a heat engine is and what its three energy elements are (hot reservoir, cold reservoir, and work produced). Give two real-life examples other than a car engine (thermal power station, steam engine), identifying what acts as the hot reservoir and cold reservoir in each.

A piston compresses $2\ \text{L}$ of gas at atmospheric pressure ($101\,325\ \text{Pa}$) to $0{,}5\ \text{L}$ without changing temperature. a) Identify the type of transformation. b) Calculate the final pressure using the corresponding law. c) Reason, using the first law, whether the gas releases or absorbs heat during the process.

A gas occupies $3\ \text{L}$ at $27\ ^\circ\text{C}$ and constant pressure. It is heated to $127\ ^\circ\text{C}$. a) Convert the temperatures to kelvins. b) Apply the isobaric transformation law, $\frac{V_1}{T_1} = \frac{V_2}{T_2}$, to calculate the final volume. c) Explain the result in terms of the motion of the molecules.

An aerosol can is at $2\ \text{atm}$ and $20\ ^\circ\text{C}$. It is left in the sun and reaches $50\ ^\circ\text{C}$ without its volume changing. a) Identify the transformation. b) Calculate the final pressure in atmospheres. c) Reason why aerosol cans warn against exposing them to heat.

Calculate the moles of air contained in a room of $30\ \text{m}^3$ at $1\ \text{atm}$ and $27\ ^\circ\text{C}$ using $p \cdot V = n \cdot R \cdot T$ with $R = 0{,}082\ \text{atm}\cdot\text{L}/(\text{mol}\cdot\text{K})$ (be careful with the volume units). Comment on whether the result seems large or small to you by comparing it with the molar mass of air ($\approx 29\ \text{g/mol}$).

A gas expands at a constant pressure of $2 \cdot 10^5\ \text{Pa}$ from $1\ \text{L}$ to $3\ \text{L}$. a) Calculate the work done by the gas, $W = p \cdot \Delta V$ (remember that $1\ \text{L} = 10^{-3}\ \text{m}^3$). b) If it absorbs $900\ \text{J}$ of heat during the process, calculate the change in its internal energy using the first law. c) State whether the gas heats up or cools down.

The figure shows an isothermal expansion on the p-V diagram from state 1 to state 2, with the area under the curve shaded. a) Explain what this shaded area represents physically. b) Reason why the work done in an isochoric transformation is zero. c) If the gas returned from state 2 to state 1 along a horizontal isobar, would the net work of the cycle be positive or negative? Justify your answer by comparing the areas.

A Carnot engine works between a hot reservoir at $600\ \text{K}$ and a cold reservoir at $300\ \text{K}$. a) Calculate its efficiency, $\eta = 1 - \frac{T_f}{T_c}$. b) If in each cycle it absorbs $1\,000\ \text{kJ}$ from the hot reservoir, calculate the work it produces and the heat it releases to the cold reservoir. c) Comment on where the unused energy goes.

Explain why in an adiabatic compression (without heat exchange) the gas heats up even though it receives no heat: relate the work done on the gas to its internal energy using the first law. Connect your explanation with a bicycle pump heating up when inflating and with the auto-ignition of air in a Diesel engine.

In the attached p-V diagram, the theoretical Carnot cycle followed by an ideal gas operating between temperatures of $600\ \text{K}$ and $300\ \text{K}$ is represented.

1. a) To double the theoretical efficiency of the cycle while keeping one of the temperatures fixed, one engineer proposes doubling $T_c$ (up to $1200\ \text{K}$), while another proposes halving $T_f$ (down to $150\ \text{K}$). Evaluate both proposals, demonstrating analytically whether either of them achieves it, and deduce which of the two reservoirs has a greater influence on increasing thermal efficiency.
2. b) If during the hot isothermal expansion ($A \to B$) the gas absorbs $1200\ \text{J}$, calculate the theoretical net work of the cycle and compare it with that of a real prototype that, operating between the same reservoirs, loses $40\,\%$ of the Carnot net work due to irreversibilities and friction. What total heat does this real engine reject to the cold reservoir?
3. c) Justify how the interior area enclosed by the curves in the p-V diagram would qualitatively change if the maximum volume reached at point C is increased while keeping the extreme pressures and temperatures fixed.

A thermal power plant operates between $550\ ^\circ\text{C}$ (superheated steam at the turbine inlet) and $35\ ^\circ\text{C}$ (cooling water in the condenser). The electrical power generated is $400\ \text{MW}$ with a fuel consumption of $950\ \text{MW}$ thermal.

1. a) Calculate the real thermal efficiency of the plant and compare it with the maximum theoretical Carnot efficiency. Using energy and entropy balances, justify why, even if all mechanical friction were eliminated, the real efficiency would never reach $100\,\%$.
2. b) During the summer, the temperature of the cooling river rises and condensation changes from $35\ ^\circ\text{C}$ to $48\ ^\circ\text{C}$. Assuming that the ratio between real efficiency and Carnot efficiency remains constant, predict the reduction in delivered electrical power (in $\text{MW}$) if fuel consumption is kept constant.
3. c) To compensate for this summer loss, the operating team considers superheating the steam up to $600\ ^\circ\text{C}$. Analyse whether this temperature increase is sufficient to recover the original electrical power and reason what material and design limitations prevent raising the boiler temperature indefinitely.

A pneumatic cylinder contains an ideal gas at $100\ \text{kPa}$, occupying $2\ \text{L}$ at a temperature of $300\ \text{K}$. To perform a mechanical actuation maneuver, the gas needs to be brought to a volume of $4\ \text{L}$ and a pressure of $200\ \text{kPa}$. Two possible control paths are compared:

• Path 1: Isochoric heating until reaching $200\ \text{kPa}$ and, subsequently, isobaric expansion up to $4\ \text{L}$.

• Path 2: Isobaric expansion up to $4\ \text{L}$ and, subsequently, isochoric heating until reaching $200\ \text{kPa}$.

1. a) Represent both processes on a single qualitative p-V diagram and calculate the mechanical work developed by the gas in joules for each of the paths.
2. b) Determine the final temperature reached by the gas in the final state and reason, applying the first law of thermodynamics, in which of the two paths more heat from the heating system is consumed.
3. c) If the piston returns to the initial state from the final state via a straight-line transformation on the p-V diagram, complete the resulting closed cycle with Path 1 and calculate the net work per cycle, stating with justification whether the device operates as a heat engine or as a heat pump.

A startup company is seeking investors for a marine propulsion system that aims to extract thermal energy from seawater at $20\ ^\circ\text{C}$ ($293\ \text{K}$) and discharge water at $15\ ^\circ\text{C}$, claiming that it can propel the ship by entirely converting that heat into mechanical work "thanks to the use of superconducting materials that eliminate friction and resistance".

1. a) Formally refute the inventor's proposal based on the Kelvin-Planck and Clausius statements of the second law of thermodynamics, justifying why the absence of friction does not eliminate the need for a thermal reservoir at a lower temperature.
2. b) The designer rectifies and proposes using atmospheric air at $20\ ^\circ\text{C}$ as the hot reservoir and deep seawater at $15\ ^\circ\text{C}$ as the cold reservoir. If the ship's propellers require $150\ \text{kW}$ of power and the real engine achieves $35\,\%$ of the Carnot limit between those reservoirs, calculate the minimum mass flow rate of seawater (in $\text{kg}/\text{s}$) that must circulate through the heat exchanger if the water can only heat up by $1\ ^\circ\text{C}$ as it passes through ($c_e = 4180\ \text{J}/(\text{kg}\cdot\text{K})$).
3. c) Based on the flow rate obtained, give a reasoned assessment of the constructive and operational viability of the proposed commercial vessel.

To optimize a reversible heat engine operating between a hot reservoir at $T_c = 600\ \text{K}$ and the ambient at $T_{\text{amb}} = 300\ \text{K}$, a technician suggests interposing an auxiliary refrigeration machine that actively cools the cold reservoir of the heat engine to keep it at $T_f = 200\ \text{K}$, discharging the extracted heat to the ambient at $300\ \text{K}$. Their reasoning is that the increase in thermal efficiency ($\ eta = 1 - T_f/T_c$) will more than compensate for the work consumed by the refrigerator.

1. a) Assuming that both machines operate according to reversible Carnot cycles, analytically demonstrate whether the total net work of the coupled system (engine work minus refrigerator work) is greater than, less than, or equal to the work that the heat engine would produce working directly between $600\ \text{K}$ and the ambient at $300\ \text{K}$.
2. b) Interpret the previous result in light of the second law and deduce why no artificial active cooling system makes it possible to bypass the Carnot limit of the coldest natural reservoir available.
3. c) From the expression for the coefficient of performance of a Carnot refrigerator ($\text{COP} = T_f / (T_c - T_f)$), deduce what happens to the electrical power required to extract a constant amount of heat as the cold reservoir approaches $0\ \text{K}$, and relate this conclusion to the third law of thermodynamics.

In the preliminary design phase of a thermoelectric power plant that will operate with a hot reservoir at $T_c = 850\ \text{K}$ and open-circuit cooling at $T_f = 290\ \text{K}$, the engineering team evaluates two technical energy improvement options with the same implementation cost:

• Option A: Improve combustion to raise $T_c$ by $40\ \text{K}$, keeping $T_f$ constant.

• Option B: Install a high-performance cooling system to reduce $T_f$ by $40\ \text{K}$, keeping $T_c$ constant.

1. a) Calculate the increase in the theoretical Carnot thermal efficiency for each option and analytically demonstrate which of the two measures is thermodynamically more effective for equal thermal variations ($\Delta T$).
2. b) Interpret how each of the two modifications translates into the area enclosed in the cycle's p-V diagram, and calculate what percentage of waste heat discharged to the surroundings is saved for each megawatt-hour generated in the most favorable option.
3. c) Argue two technical, geographical, or environmental constraints that in practice prevent lowering the temperature of the cold reservoir as much as thermodynamic engineers would desire.

A patent office submits for your evaluation the technical report of an industrial air conditioner which, according to its promoter, captures heat from an industrial room at $25\ ^\circ\text{C}$ at a rate of $12\ \text{kW}$, delivers $4\ \text{kW}$ of mechanical power to an air compressor, and returns the remaining $8\ \text{kW}$ of thermal energy to an adjacent hall at $40\ ^\circ\text{C}$, all without requiring any electricity supply or fuel in steady-state operation.

1. a) Check whether the device complies with the first law of thermodynamics and demonstrate whether it violates the second law by calculating the net rate of entropy variation generated in the universe ($\dot{S}_{\text{univ}}$).
2. b) Reasonably classify this device among perpetual motion machines (perpetual motion machine of the first or second kind) and draft the formal justification with which you would reject the patent.
3. c) If the machine were reversible and operated by absorbing $12\ \text{kW}$ from the reservoir at $40\ ^\circ\text{C}$ ($313\ \text{K}$) and rejecting heat to the reservoir at $25\ ^\circ\text{C}$ ($298\ \text{K}$), calculate the maximum theoretical mechanical power it could supply and the heat dissipated to the cold reservoir.

In an oil refinery, two waste heat streams are generated and released into the atmosphere ($T_0 = 15\ ^\circ\text{C} = 288\ \text{K}$):

• Stream 1: Flue gas at $420\ ^\circ\text{C}$ carrying a thermal power of $600\ \text{kW}$.

• Stream 2: Process water stream at $65\ ^\circ\text{C}$ also carrying a thermal power of $600\ \text{kW}$.

1. a) Quantitatively demonstrate, by calculating the maximum theoretical mechanical power (exergy) recoverable from each stream through an ideal Carnot cycle coupled to the ambient, that two forms of energy with the same quantity in joules can have radically different thermodynamic qualities.
2. b) To harness Stream 1, an engineer proposes an uninsulated heat exchanger that cools the flue gas down to $180\ ^\circ\text{C}$ while preheating steam to $150\ ^\circ\text{C}$. Justify why any heat transfer across a finite temperature difference entails an irreversible degradation of energy, and calculate the rate of entropy change associated with the transfer of $250\ \text{kW}$ between both streams if they are modeled at effective average temperatures of $300\ ^\circ\text{C}$ and $150\ ^\circ\text{C}$.
3. c) Relate the concept of energy degradation and irreversibility to real losses in power cycles compared to ideal reversible cycles.