Exercises: Energy, heat and work

In a laboratory practical, an aluminium block of mass $m = 150.0\,\text{g}$ is heated from an initial temperature of $18.0\,^{\circ}\text{C}$ to $68.0\,^{\circ}\text{C}$. Given that the specific heat capacity of aluminium over this range is $c = 0.897\,\text{J}\times\text{g}^{-1}\times\text{K}^{-1}$ and assuming that heat losses to the surroundings are negligible:

1. a) Calculate the quantity of heat absorbed by the metal block, expressed in joules and in kilojoules.
2. b) Justify conceptually why the heat absorbed is not an intrinsic property possessed by the block, clearly distinguishing between heat and temperature.

A cylinder fitted with a movable piston contains a certain amount of nitrogen gas. The gas expands against a constant external pressure of $1.20\times 10^{5}\,\text{Pa}$, undergoing an increase in volume of $\Delta V = 2.50\times 10^{-3}\,\text{m}^{3}$. Data: $1\,\text{J} = 1\,\text{Pa}\times\text{m}^{3}$.

1. a) Calculate the thermodynamic work exchanged by the system, expressed in joules, applying the IUPAC sign convention ($w = -P_{\text{ext}}\,\Delta V$).
2. b) State the physical meaning of the sign obtained for the work according to the convention in which work done on the system is positive.

A closed gaseous system changes from an initial state $1$ to a final state $2$. During the transformation, the system absorbs $850\,\text{J}$ of heat from the surroundings and at the same time does expansion work on its surroundings of $320\,\text{J}$.

1. a) Calculate the change in internal energy ($\Delta U$) of the system according to the first law of thermodynamics formulated as $\Delta U = q + w$.
2. b) Explain whether internal energy, heat and work are state functions or quantities that depend on the path of the process.

A monatomic ideal gas is enclosed in a cylinder-piston device like the one illustrated below, with an initial volume of $V_{1} = 2.0\,\text{L}$. The gas expands quasi-statically against a constant external pressure $P_{\text{ext}} = 100\,\text{kPa}$ until it reaches a final volume of $V_{2} = 5.0\,\text{L}$, while receiving a heat flow from the surroundings of $q = 750\,\text{J}$. Data: $1\,\text{kPa} = 10^{3}\,\text{Pa}$, $1\,\text{L} = 10^{-3}\,\text{m}^{3}$, $1\,\text{J} = 1\,\text{Pa}\times\text{m}^{3}$. Look at the illustration of the process:

1. a) Calculate the thermodynamic work $w$ exchanged by the gas during the expansion, expressed in joules, using the convention $\Delta U = q + w$.
2. b) Determine the change in internal energy $\Delta U$ undergone by the gas in the process and interpret the result physically.

In a constant-pressure mixing calorimeter whose own heat capacity is negligible, $250.0\,\text{g}$ of liquid water at a temperature of $20.0\,^{\circ}\text{C}$ are introduced. Next, a metal piece of mass $80.0\,\text{g}$, taken from an oven at $95.0\,^{\circ}\text{C}$, is added. The mixture is stirred until a thermal equilibrium temperature of $24.2\,^{\circ}\text{C}$ is reached. Data: specific heat capacity of liquid water $c_{\text{agua}} = 4,184\,\text{J}\times\text{g}^{-1}\times\text{K}^{-1}$. Assume that there are no heat losses to the surroundings.

1. a) Set up the calorimetry balance and determine the specific heat capacity of the metal, expressed in $\text{J}\times\text{g}^{-1}\times\text{K}^{-1}$.
2. b) If a small heat loss to the surroundings had occurred during the experiment, explain whether the experimental value obtained for the specific heat capacity of the metal would be greater or less than the true value.

A monatomic ideal gas undergoes a transformation in two consecutive stages inside a container:

Stage 1: The gas is heated at constant volume ($V = 4.0\times 10^{-3}\,\text{m}^{3}$) by absorbing $620\,\text{J}$ of heat from a heat reservoir.

Stage 2: Then, keeping the pressure constant at the value reached, $1.50\times 10^{5}\,\text{Pa}$, the gas expands until it doubles its initial volume while additionally absorbing $1500\,\text{J}$ of heat.

1. a) Calculate the thermodynamic work $w$ and the change in internal energy $\Delta U$ for each of the two individual stages.
2. b) Determine the total heat $q_{\text{total}}$, the total work $w_{\text{total}}$ and the net change in internal energy $\Delta U_{\text{total}}$ of the whole combined process.

In a calorimetric beaker open to a constant atmospheric pressure of $1.00\times 10^{5}\,\text{Pa}$, $0.050\,\text{mol}$ of metallic zinc turnings are reacted with an excess of aqueous hydrochloric acid according to the chemical equation:

$\mathrm{Zn(s)} + 2\,\mathrm{HCl(aq)} \rightarrow \mathrm{ZnCl_{2}(aq)} + \mathrm{H_{2}(g)}$

The total mass of the solution is $200.0\,\text{g}$ and its temperature rises from $20.0\,^{\circ}\text{C}$ to $27.8\,^{\circ}\text{C}$. Take the specific heat capacity of the solution to be equal to that of pure water ($c = 4,184\,\text{J}\times\text{g}^{-1}\times\text{K}^{-1}$), neglect the heat capacity of the container and assume ideal behaviour for the hydrogen gas produced at a mean temperature of $297\,\text{K}$. Data: $R = 8,314\,\text{J}\times\text{mol}^{-1}\times\text{K}^{-1}$.

1. a) Calculate the heat transferred at constant pressure $q_{P}$ exchanged by the reacting system, expressed in joules, and the expansion work $w$ associated with the release of the gas.
2. b) Determine the change in internal energy $\Delta U$ for the reaction of the $0.050\,\text{mol}$ of $\mathrm{Zn}$.

One mole of ideal gas is in an initial state $A$ ($P_{A} = 2.00\times 10^{5}\,\text{Pa}$, $V_{A} = 1.00\times 10^{-2}\,\text{m}^{3}$) and is taken to a final state $B$ ($P_{B} = 1.00\times 10^{5}\,\text{Pa}$, $V_{B} = 2.00\times 10^{-2}\,\text{m}^{3}$) through two alternative thermodynamic paths:

• Path 1: Isobaric expansion to $V_{B}$ followed by isochoric cooling to $P_{B}$, in which a net heat absorbed of $q_{1} = 2000\,\text{J}$ is measured.

• Path 2: Isochoric decompression to $P_{B}$ followed by isobaric expansion to $V_{B}$.

1. a) Calculate the thermodynamic work done by the system in Path 1 and in Path 2.
2. b) Using the first law of thermodynamics, show numerically the heat exchanged $q_{2}$ along Path 2 and explain why the change in internal energy $\Delta U$ is identical in both paths while $q$ and $w$ differ.

The combustion of $0.020\,\text{mol}$ of liquid ethanol ($\mathrm{C_{2}H_{5}OH(l)}$) with stoichiometric oxygen at $298\,\text{K}$ is analysed according to the reaction:

$\mathrm{C_{2}H_{5}OH(l)} + 3\,\mathrm{O_{2}(g)} \rightarrow 2\,\mathrm{CO_{2}(g)} + 3\,\mathrm{H_{2}O(l)}$

In a constant-volume experiment (rigid bomb calorimeter with no possibility of expansion), the reaction releases heat of $q_{V} = -27.34\,\text{kJ}$. Assume ideal gas behaviour for all gaseous species, negligible volume of the condensed phases and a standard atmospheric pressure of $1.00\times 10^{5}\,\text{Pa}$. Data: $R = 8,314\,\text{J}\times\text{mol}^{-1}\times\text{K}^{-1}$.

1. a) Calculate the change in internal energy $\Delta U$ and the work $w_{V}$ associated with the process in the bomb calorimeter.
2. b) If the same reaction were carried out in a cylinder fitted with a movable piston at a constant external pressure of $1.00\times 10^{5}\,\text{Pa}$ and at the same temperature of $298\,\text{K}$, determine the mechanical work $w_{P}$ and the heat exchanged $q_{P}$ by applying the first law.