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).

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).

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 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.

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.

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 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.

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.

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.