Exercises: Energy transfers with the surroundings

In thermodynamics we will use the following sign convention: $Q > 0$ when the system absorbs heat from the surroundings and $Q < 0$ when it gives off heat; $W > 0$ when the surroundings do work on the system (compression) and $W < 0$ when the system does work on the surroundings (expansion). With this convention, the first law is written as $\Delta U = Q + W$.

Write with its sign the value of $Q$ or $W$ in each case:

1. a) The system absorbs $400\ \text{J}$ of heat.
2. b) The system gives off $250\ \text{J}$ of heat to the surroundings.
3. c) The surroundings compress the system, doing $100\ \text{J}$ of work.
4. d) The system expands and does $150\ \text{J}$ of work on the piston.

A gas inside a cylinder absorbs heat $Q = +600\ \text{J}$ from a flame while the piston compresses it, doing work $W = +150\ \text{J}$ on it. The diagram shows the energy transfers to the gas.

1. a) Calculate the change in internal energy $\Delta U$ of the gas.
2. b) Interpret the result: has the internal energy of the gas increased or decreased?

The hot water in a home heating circuit passes through a radiator and gives off $500\ \text{J}$ of heat to the room, without any work being done.

1. a) Write the values of $Q$ and $W$ with their sign according to the standard convention.
2. b) Calculate $\Delta U$ of the water in that section and interpret its sign.

State the first law of thermodynamics in your own words.

1. a) Explain what each of the three terms in the expression $\Delta U = Q + W$ represents.
2. b) Reason why this law is, at heart, the principle of conservation of energy applied to thermodynamic systems.

An electric kettle heats $m = 2\ \text{kg}$ of water and raises its temperature by $\Delta T = 10\ \celsius$. The specific heat capacity of water is $c = 4180\ \text{J}/(\text{kg}\times\celsius)$.

Calculate the heat absorbed by the water using the expression $Q = m \times c \times \Delta T$.

The specific heat capacity of water is $c_{\text{agua}} = 4180\ \text{J}/(\text{kg}\times\celsius)$ and that of ethanol is $c_{\text{etanol}} = 2400\ \text{J}/(\text{kg}\times\celsius)$.

1. a) Explain in your own words what the value $4180\ \text{J}/(\text{kg}\times\celsius)$ means: how many joules are needed and to achieve what, exactly.
2. b) For equal masses and the same temperature increase, reason using formula $Q = m \times c \times \Delta T$ which of the two substances requires more heat.

An ice pack containing $m = 0.5\ \text{kg}$ of ice at $0\ \celsius$ is applied to a sports injury until all the ice melts, remaining at $0\ \celsius$ throughout the process. The latent heat of fusion of ice is $L_{\text{f}} = 3.34\times10^{5}\ \text{J/kg}$.

1. a) Calculate the heat absorbed by the ice using $Q = m \times L_{\text{f}}$ and express the result in standard form.
2. b) State where that heat comes from and, therefore, explain the cooling effect on the treated area.

A pot of boiling water remains on the hob, but the thermometer reads $100\ \celsius$ throughout the process, even though the flame continues to supply energy. The graph shows how the temperature of the water changes over time.

1. a) Does the water continue to receive energy while it is boiling? Justify your answer.
2. b) If the temperature does not rise, what is that energy being used for at a microscopic level?
3. c) Explain, using the concept of latent heat, why the temperature remains constant during a change of state.

A gas gives off $300\ \text{J}$ of heat to the surroundings while a piston compresses it, doing work of $150\ \text{J}$ on it.

1. a) Write $Q$ and $W$ with their sign according to the standard convention.
2. b) Calculate $\Delta U$ and interpret the result: despite the compression, does the gas gain or lose internal energy?
3. c) For a gas whose internal energy increases with temperature, predict whether its temperature will have risen or fallen.

A closed pressure cooker, before its valve opens, maintains the volume of steam practically constant while receiving heat $Q = +4000\ \text{J}$ from the hob.

1. a) Justify why the work exchanged is zero in this stage (think about the expression $W = -p \times \Delta V$).
2. b) Calculate the change in internal energy $\Delta U$ of the contents in that stage.
3. c) Generalise the result by completing the sentence: "in any isochoric process, the change in internal energy is equal to $\dots$".

An ideal gas expands very slowly inside a thermostatic bath (isothermal process) and absorbs heat $Q = +500\ \text{J}$ from the bath. Remember that, in an ideal gas, internal energy depends only on temperature.

1. a) State the value of $\Delta U$ in this process and justify your answer.
2. b) Calculate the work $W$ exchanged, with its sign, and interpret it.
3. c) Explain, using the energy balance, what the absorbed heat has been converted into.

A gas expands at constant pressure $p = 1\times10^{5}\ \text{Pa}$ from $V_1 = 1\times10^{-3}\ \text{m}^3$ to $V_2 = 3\times10^{-3}\ \text{m}^3$ and, during the expansion, absorbs heat $Q = +700\ \text{J}$ from a burner.

1. a) Calculate the work $W = -p \times \Delta V$ with its sign.
2. b) Calculate the change in internal energy $\Delta U$ of the gas.
3. c) Verify that the heat absorbed is divided between the increase in internal energy and the work done by the gas.

In the laboratory, $750\ \text{mL}$ of water is heated from $20\ \celsius$ to $60\ \celsius$. Data: the density of water is $1\ \text{kg/L}$ and its specific heat capacity is $c_{\text{agua}} = 4180\ \text{J}/(\text{kg}\times\celsius)$.

1. a) Express the mass of water in kilograms and calculate the temperature change.
2. b) Calculate the heat absorbed by the water and express it in standard form.

Heat $Q = 11\,500\ \text{J}$ is supplied to a metal block of $m = 0.5\ \text{kg}$, initially at $20\ \celsius$, and its temperature rises to $70\ \celsius$.

1. a) Rearrange the formula $Q = m \times c \times \Delta T$ for the specific heat capacity $c$.
2. b) Calculate its value in $\text{J}/(\text{kg}\times\celsius)$.
3. c) By checking the values $c_{\text{plomo}} = 130\ \text{J}/(\text{kg}\times\celsius)$, $c_{\text{hierro}} = 460\ \text{J}/(\text{kg}\times\celsius)$ and $c_{\text{aluminio}} = 900\ \text{J}/(\text{kg}\times\celsius)$, identify which metal it could be.

In an industrial boiler, $m = 0.2\ \text{kg}$ of water is heated from $80\ \celsius$ to $100\ \celsius$ and then that mass is completely vaporised at $100\ \celsius$. Data: $c_{\text{agua}} = 4180\ \text{J}/(\text{kg}\times\celsius)$ and $L_{\text{v}} = 2.26\times10^{6}\ \text{J/kg}$.

1. a) Calculate the heat required for the first stage (heating the water).
2. b) Calculate the heat required for complete vaporisation using $Q = m \times L_{\text{v}}$.
3. c) Calculate the total heat of the process and comment on which of the two stages turned out to be much more costly in terms of energy.

In a thermal power plant, the engine driving the generator absorbs heat $Q_1 = 2000\ \text{J}$ per cycle from the hot reservoir (the boiler) and gives off heat $Q_2 = 1200\ \text{J}$ to the cold reservoir (the cooling water). In a complete cycle, the change in internal energy of the engine is zero. The heat engine diagram is shown in the figure.

1. a) Calculate the useful work $W_{\text{useful}}$ delivered by the engine in each cycle from the energy balance.
2. b) The efficiency of a heat engine is defined as $\eta = W_{\text{useful}}/Q_1$. Calculate the efficiency of this engine and express it as a percentage.
3. c) Interpret the result by completing the sentence: "for every $100\ \text{J}$ of heat absorbed, the engine uses $\dots$ and transfers $\dots$ to the cold reservoir".

We want to convert $m = 0.5\ \text{kg}$ of ice at $-10\ \celsius$ into liquid water at $40\ \celsius$. Data: $c_{\text{hielo}} = 2100\ \text{J}/(\text{kg}\times\celsius)$, $c_{\text{agua}} = 4180\ \text{J}/(\text{kg}\times\celsius)$ and $L_{\text{f}} = 3.34\times10^{5}\ \text{J/kg}$.

1. a) Calculate the heat required to heat the ice from $-10\ \celsius$ to $0\ \celsius$.
2. b) Calculate the heat required to melt all the ice at $0\ \celsius$.
3. c) Calculate the heat required to heat the resulting water from $0\ \celsius$ to $40\ \celsius$.
4. d) Calculate the total heat of the process, express it in standard form and draw the graph of temperature as a function of heat supplied in your notebook, marking the different stages.

In an insulated container there is $0.3\ \text{kg}$ of water at $60\ \celsius$ and $0.3\ \text{kg}$ of ice at $0\ \celsius$ is added. Data: $c_{\text{agua}} = 4180\ \text{J}/(\text{kg}\times\celsius)$ and $L_{\text{f}} = 3.34\times10^{5}\ \text{J/kg}$.

1. a) Calculate the maximum heat that the hot water can give off if it cooled down to $0\ \celsius$.
2. b) Calculate the heat needed to melt all the ice.
3. c) Comparing both results, predict whether all the ice will melt and deduce the final temperature of the mixture.
4. d) Calculate the mass of ice that melts and the mass that remains unmelted.

A heat engine undergoes closed cycles. In each cycle it absorbs heat $Q_1 = +3000\ \text{J}$ from the hot reservoir and delivers work of $900\ \text{J}$ to the surroundings, that is, $W = -900\ \text{J}$ with the standard sign convention. The diagram of the engine is shown in the figure.

1. a) Justify why the change in internal energy of the engine in a complete cycle is zero.
2. b) Apply $\Delta U = Q + W$ to the cycle and calculate the net heat $Q$ exchanged per cycle.
3. c) Taking into account that $Q = Q_1 + Q_2$, calculate the heat $Q_2$ exchanged with the cold reservoir, with its sign, and interpret it.
4. d) Calculate the efficiency $\eta = W_{\text{useful}}/Q_1$ of the engine.

A refrigerator operates as a heat engine in reverse: in each cycle, it extracts heat $Q_2 = +600\ \text{J}$ from the inside and the electric motor does work $W = +200\ \text{J}$ on the system. In a complete cycle, $\Delta U = 0$. The diagram of the refrigerator is shown in the figure.

1. a) Apply the first law of thermodynamics to the cycle and calculate the net heat $Q$ exchanged.
2. b) Calculate the heat $Q_1$ that the refrigerator releases to the kitchen, including its sign, and interpret it: why is it greater than the heat extracted from the inside?
3. c) Use the result to explain why leaving the refrigerator door open does not serve to cool down the kitchen, but actually ends up heating it up.

Two heat engines work between the same reservoirs. Engine A absorbs $Q_1 = 5000\ \text{J}$ from the hot reservoir and delivers useful work of $1500\ \text{J}$. Engine B absorbs the same $5000\ \text{J}$ and releases heat of $2500\ \text{J}$ to the cold reservoir.

1. a) Calculate the efficiency of engine A.
2. b) Calculate the useful work of engine B from its energy balance and then its efficiency.
3. c) Compare both engines: which one makes better use of the fuel's heat?
4. d) An inventor claims that they can build an engine with an efficiency of $100\ \%$, which releases no heat to the cold reservoir. Using the energy balance and the concept that every real heat engine must release heat to a cold reservoir, explain why such an engine is not possible.

A gas absorbs heat $Q = +1000\ \text{J}$ from a burner and its internal energy increases by $600\ \text{J}$ while pushing the piston of a mechanism.

1. a) Calculate the work $W$ exchanged by the gas, including its sign, and state how much work the gas has done.
2. b) The piston experiences friction, so out of that work only $300\ \text{J}$ is delivered as useful work to the external mechanism. Explain what happened to the rest.
3. c) Verify that the balance $\Delta U = Q + W$ still holds and explain why friction does not "break" the conservation of energy even though it reduces useful work.

On a clear day, the Sun transfers the same energy $Q = 4180\ \text{J}$ to $1\ \text{kg}$ of sea water ($c_{\text{agua}} = 4180\ \text{J}/(\text{kg}\times\celsius)$) and to $1\ \text{kg}$ of beach sand ($c_{\text{arena}} = 836\ \text{J}/(\text{kg}\times\celsius)$).

1. a) Calculate the temperature increase of the water.
2. b) Calculate the temperature increase of the sand.
3. c) Use the results to explain why the sand is burning hot at midday while the sea water is barely lukewarm.
4. d) Relate this to daytime sea breezes: reason over which surface the air heats up first and in which direction the breeze blows.

A thermal power station receives $5\times10^{6}\ \text{J}$ of energy from combustion per unit of time considered and delivers $1.5\times10^{6}\ \text{J}$ in the form of electrical work; the rest is released as heat to the cooling water of a river.

1. a) Calculate the heat released to the cooling water.
2. b) Calculate the current efficiency of the power station.
3. c) Following a technological upgrade, the power station achieves an efficiency of $45\ \%$ with the same combustion input. Calculate the new work delivered and the new heat released to the river.
4. d) Using your results to support your answer, explain why increasing the efficiency of power stations reduces the environmental impact of thermal pollution on river ecosystems.