Exercises: 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.

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?

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?

A physiotherapy heating pad at $40\ \celsius$ is applied to a patient's back, whose skin is at $34\ \celsius$.

1. a) In which direction is heat transferred initially: from the heating pad to the skin or vice versa?
2. b) Describe what happens to the temperatures of both bodies until thermal equilibrium is reached.
3. c) Define in your own words what thermal equilibrium is.

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 classmate claims: 'The water in this pot has a lot of heat.'

1. a) Explain why the phrase is incorrect from a thermodynamic point of view: heat is not "possessed", it is transferred.
2. b) Reformulate the phrase correctly using the concepts of internal energy and temperature.
3. c) State two ways in which a system can exchange energy with its surroundings.

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: $1\ \text{atm} = 1.013\times10^{5}\ \text{Pa}$, $1\ \text{L} = 10^{-3}\ \text{m}^3$ 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).

A gas moves from state 1 ($p_1 = 2\times10^{5}\ \text{Pa}$, $V_1 = 1\times10^{-3}\ \text{m}^3$, $T_1 = 300\ \text{K}$) to state 2 ($p_2 = 4\times10^{5}\ \text{Pa}$, $V_2 = 1\times10^{-3}\ \text{m}^3$, $T_2 = 600\ \text{K}$). 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 rigid camping gas cylinder contains gas at $p_1 = 2\times10^{5}\ \text{Pa}$ 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 expands at constant pressure $p = 1\times10^{5}\ \text{Pa}$ from $V_1 = 1\times10^{-3}\ \text{m}^3$ to $V_2 = 3\times10^{-3}\ \text{m}^3$, pushing a piston. The work exchanged in an isobaric process is calculated as $W = -p \times \Delta V$, where the minus sign accounts for the sign convention: $W > 0$ when the surroundings do work on the system. The process is represented in the p-V diagram in the figure.

1. a) Calculate the change in volume $\Delta V$.
2. b) Calculate the work $W$ and interpret its sign: who does work on whom?

A gas enclosed in a cylinder with a piston expands at constant pressure $p = 1\times10^{5}\ \text{Pa}$ and its volume changes from $1\times10^{-3}\ \text{m}^3$ to $3\times10^{-3}\ \text{m}^3$. Afterwards, the reverse process is carried out on the same gas: at the same constant pressure $p = 1\times10^{5}\ \text{Pa}$, the volume changes from $V_1 = 3\times10^{-3}\ \text{m}^3$ to $V_2 = 1\times10^{-3}\ \text{m}^3$. This second process is represented in the p-V diagram in the figure.

1. a) Calculate $\Delta V$ and the work $W = -p \times \Delta V$ with its sign in this reverse process (from $V_1$ to $V_2$).
2. b) Interpret the result: does the gas gain or lose energy by this route? Also calculate the work of the expansion process described at the beginning (from $1\times10^{-3}\ \text{m}^3$ to $3\times10^{-3}\ \text{m}^3$ at the same pressure) and compare it with the one obtained in part a).

A system moves from state A to state B. Along path 1 it absorbs heat $Q_1 = 500\ \text{J}$ and receives work $W_1 = 100\ \text{J}$ from the surroundings. Along path 2, between the same initial and final states, it receives work $W_2 = 300\ \text{J}$. The diagram shows the two paths between states A and B with their energy transfers.

1. a) Calculate the change in internal energy along path 1 using $\Delta U = Q + W$.
2. b) Since $U$ is a state function, $\Delta U$ is the same along path 2: calculate the heat $Q_2$ exchanged along that path.
3. c) Conclude: which quantities depend on the path followed between two states and which does not?

Classify the following systems as open, closed or isolated, justifying your answer based on their exchanges of matter and energy with the surroundings:

1. a) A person doing sport outdoors.
2. b) A butane cylinder with its valve closed.
3. c) An ideal vacuum flask that does not allow any heat to pass through its walls.
4. d) The cabin of the International Space Station, which receives oxygen from its tanks and gives off heat to space.

On a $p$-$V$ diagram, three processes for the same gas are represented: process 1 is a horizontal line segment traversed to the right; process 2 is a vertical line segment traversed upwards; process 3 is a downward curve traversed at constant temperature.

1. a) Name each of the three processes (isobaric, isochoric or isothermal).
2. b) State in which of them the volume increases while the pressure remains constant.
3. c) State in which the pressure increases without the volume changing.
4. d) In which of the three does the temperature not vary?

A gas undergoes the following cycle inside a cylinder with a piston:

• A → B: isobaric expansion at $p = 2\times10^{5}\ \text{Pa}$, from $V_{\text{A}} = 1\times10^{-3}\ \text{m}^3$ to $V_{\text{B}} = 3\times10^{-3}\ \text{m}^3$.

• B → C: isochoric process until the pressure is $1\times10^{5}\ \text{Pa}$.

• C → D: isobaric compression to $V_{\text{D}} = 1\times10^{-3}\ \text{m}^3$.

• D → A: isochoric process returning the gas to its initial state.

1. a) Draw the cycle on a $p$-$V$ diagram in your notebook.
2. b) Calculate the work in stages A → B and C → D using $W = -p \times \Delta V$, taking care with the signs.
3. c) Justify why the work is zero in the isochoric stages.
4. d) Calculate the net work of the cycle and reason whether, overall, the gas has received work from the surroundings or transferred work to them.

You can do it on paper and send a photo with your answer.

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?

When spraying with an aerosol can for a few seconds, the propellant gas expands very quickly through the valve and the container cools down noticeably. The expansion is so fast that there is hardly any time for the gas to exchange heat with the surroundings.

1. a) Explain why we can consider that, during the expansion, the heat exchanged is practically zero ($Q \approx 0$).
2. b) State the sign of the work $W$ of the propellant gas, taking into account that it expands.
3. c) Applying $\Delta U = Q + W$, deduce the sign of $\Delta U$ and, for a gas whose internal energy increases with its temperature, the sign of its temperature change.
4. d) Using all the above, explain why the container cools down.

When inflating a bicycle tyre with a hand pump, the trapped air is compressed very quickly with each stroke; so fast that heat exchange with the ambient environment is negligible.

1. a) State the sign of the work $W$ on the air during compression.
2. b) Applying $\Delta U = Q + W$ with $Q \approx 0$, deduce the sign of $\Delta U$ and predict what happens to the air temperature.
3. c) Explain why the barrel of the pump heats up after several strokes.
4. d) Compare this process with the aerosol expansion: in both cases $Q \approx 0$, so reason why in one the gas heats up and in the other it cools down.

In an industrial compressor, the compressed gas heats the cylinder walls, which in turn transfer heat to the air in the building. Analyse the process choosing two different systems: system 1 is only the gas, and system 2 is the gas plus the cylinder.

1. a) For system 1, state which energy transfers cross its boundary (the work from the piston and the heat towards the walls) and assign their sign.
2. b) For system 2, explain why the heat between the gas and the walls no longer appears in the energy balance as heat exchanged with the surroundings.
3. c) Has the physics of the process changed by choosing another system? Conclude why it is essential to define the system before setting up any energy balance.

Inside a perfectly insulating vacuum flask there is cold water and a hot metal block is introduced into it. We consider the system to be the water + metal combination. The situation is shown in the diagram in the figure.

1. a) Justify why, for the complete system, $Q = 0$ and $W = 0$, and deduce the value of the change in internal energy $\Delta U$ of the whole system during the process.
2. b) State the sign of the heat exchanged by the metal and the sign of that exchanged by the water, considering each as a subsystem.
3. c) If the metal releases heat $Q_{\text{metal}} = -40\,000\ \text{J}$, what is the value of the heat received by the water?
4. d) Describe the final state of the system: what equilibrium condition is reached between the water and the metal?

For an ideal gas, the internal energy depends solely on its temperature. An ideal gas expands slowly inside a thermostatic bath, so that the process is isothermal, and during the expansion it absorbs heat $Q = +800\ \text{J}$ from the bath. The diagram shows the energy transfers of the gas during the process.

1. a) Justify why $\Delta U = 0$ in this process.
2. b) Calculate the work $W$ exchanged, using the IUPAC sign convention ($\Delta U = Q + W$, with $W$ the work done on the gas), and interpret its sign.
3. c) Explain where the energy comes from for the gas to perform the expansion work, if its internal energy has not changed.
4. d) Reason why the process must be carried out slowly and in thermal contact with the bath to be considered isothermal.

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.