Thermodynamic variables of a system

In a technology laboratory, three systems are studied: an uncovered saucepan of boiling soup, a tightly closed soft drink bottle, and a stainless steel thermos flask with hot food, also tightly closed.

1. a) Classify each system as open, closed, or isolated.
2. b) Indicate, in each case, whether the system can exchange matter, energy, both, or neither with the surroundings.

In the cylinder of a motorcycle engine, the mixture of petrol and air is compressed and ignited. We take the gases enclosed in the cylinder as the system.

1. a) State which bodies form the surroundings of the system.
2. b) Which elements act as the boundary between the system and its surroundings?
3. c) Classify the system while the valves remain closed: can it exchange matter with the outside? What about energy?

Consider the following thermodynamic quantities: pressure $p$, volume $V$, temperature $T$, internal energy $U$, heat $Q$ and work $W$.

1. a) State which of these are state variables of the system.
2. b) Explain the criterion you used: what is the difference between a state variable and energy transfer?

A gas is in a state A described by $p_{\text{A}} = 1,5\ \text{atm}$, $V_{\text{A}} = 2,5\ \text{L}$ and $t_{\text{A}} = 27\ \celsius$. Data: $1\ \text{atm} = 1,013\cdot10^{5}\ \text{Pa}$, $1\ \text{L} = 10^{-3}\ \text{m}^3$ and $T\,(\text{K}) = t\,(\celsius) + 273$.

1. a) Express the three variables in SI units.
2. b) Of the three variables, state which is extensive (depends on the amount of gas enclosed) and which are intensive (do not depend on it).

After a long motorway journey, the air inside the tyres has warmed up, while the internal volume of the tyre has barely changed.

1. a) Which thermodynamic variable of the enclosed air remains practically constant?
2. b) What is this type of process called?
3. c) Predict with reasoning what will have happened to the air pressure and explain why manufacturers recommend measuring tyre pressures 'cold'.

A laboratory cylinder contains a gas enclosed by a frictionless sliding piston, always subject to external atmospheric pressure. We heat the gas slowly.

1. a) Which thermodynamic variable remains constant during the process? What is this type of process called?
2. b) Explain what happens to the volume of the gas as its temperature increases: does the piston move up or down?

We want to compress a gas while keeping its temperature constant, that is, by an isothermal process. To do this, we submerge the cylinder in a water bath thermostatically controlled at $25\ \celsius$ and push the piston very slowly.

1. a) Explain why the compression must be carried out slowly so that the temperature of the gas does not change.
2. b) What role does the water bath play? What exchange does it allow between the gas and the surroundings?

A rigid camping gas cylinder contains gas at $p_1 = 2\cdot10^{5}\ \text{Pa}$ when its temperature is $T_1 = 300\ \text{K}$. On a very hot day, the gas temperature rises to $T_2 = 450\ \text{K}$. Since the cylinder is rigid, the volume does not change and, at constant volume, the ratio $p/T$ remains constant (with $T$ expressed in kelvins). The process is shown in the graph in the figure.

1. a) Before calculating, predict whether the pressure of the gas will increase or decrease.
2. b) Calculate the final pressure $p_2$ of the gas.

A gas moves from state 1 ($p_1 = 2\cdot10^{5}\ \text{Pa}$, $V_1 = 1\cdot10^{-3}\ \text{m}^3$, $T_1 = 300\ \text{K}$) to state 2 ($p_2 = 4\cdot10^{5}\ \text{Pa}$, $V_2 = 1\cdot10^{-3}\ \text{m}^3$, $T_2 = 600\ \text{K}$). The diagram shows the two states on a p-V diagram and two possible paths to go from one to the other.

1. a) State which variables have changed and which has remained constant. What is such a process called?
2. b) If the gas had reached state 2 via a different path, for example expanding first and compressing afterwards, would the change in internal energy $\Delta U$ between both states have changed? Justify your answer using the concept of a state function.

A small laboratory engine makes a gas undergo closed thermodynamic cycles: at the end of each cycle, the gas returns exactly to its initial state.

1. a) Justify, using the fact that internal energy is a state function, that the change in internal energy in a complete cycle is zero.
2. b) Applying the first law $\Delta U = Q + W$ to a complete cycle, deduce the relationship between the net heat and the net work exchanged.
3. c) If in each cycle the engine delivers $300\ \text{J}$ of net work to the surroundings (that is, $W = -300\ \text{J}$ with the sign convention), calculate the net heat that the gas must absorb per cycle.
4. d) How much net heat does the gas absorb in $50$ cycles?

The cabin of the International Space Station receives oxygen from tanks, sends carbon dioxide to purification systems and transfers heat to the outside solely by radiation into space.

1. a) Classify the cabin as a thermodynamic system (open, closed or isolated), justifying your answer using its exchanges of matter and energy.
2. b) Assign the sign of the heat that the cabin exchanges with outer space.
3. c) State what would need to change for us to consider it a closed system.
4. d) Reason whether it would be possible, in practice, to turn it into a truly isolated system regarding energy exchange.