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Thermodynamics, the First Law: the concepts
Thermodynamics – concerned with the studies of transformation of energy from heat to work and vice versa. Originally formulated by physicists (the efficiency of steam engines). Thermodynamics – immense importance in chemistry – energy output of chemical reactions, why reactions reach equilibrium, what their composition is at equilibrium, how reactions in electrochemical (and biological) cells can generate electricity. Classical thermodynamics – does not depend on any models of the internal constitution of matter, can be developed and used without ever mentioning atoms and molecules. Properties of atoms and molecules – ultimately responsible for observable properties of bulk matter. Connection between atomic and bulk thermodynamic properties – statistical thermodynamics. Branches of thermodynamics Thermochemistry – deals with the heat output of chemical reactions. Electrochemistry – the interaction between electricity and chemistry. Bioenergetics – the deployment of energy in living organisms. Equilibrium chemistry – the formulation of equilibrium constants, the equilibrium composition of solutions of acids and bases.
Conservation of energy
Almost every argument and explanation in chemistry boils down to a consideration of some aspect of a single property: the energy. The energy determines what molecules may form what reaction may occur, how fast they may occur, in which direction a reaction has a tendency to occur. Energy – the capacity to do work. Work – motion against opposing force. A gas at high temperature has more energy than at low temperature – it has a higher pressure and can do more work in driving out a piston. Conservation of energy – the energy can be neither created nor destroyed but merely converted from one form to another or moved from place to place. Chemical reactions release or absorb energy – the conversion of energy or its transfer.
System and surroundings: system – the part of the world in which we have a special interest. Surroundings – where we make observations, so huge they have either constant volume or constant pressure regardless of any changes in the system. The surroundings remain effectively the same size.
Work and heat
Energy can be exchanged between a closed system and its surroundings as work or as heat.
Work – a transfer of energy that can cause motion against an opposing force. A process produces work if it can be used to change the height of a weight somewhere in the surroundings. Heat – a transfer of energy as a result of a temperature difference between the system and its surroundings. Walls that permit the passage of energy as heat – diathermic. (metal container). Walls that do not permit heat to pass through though there is a difference in temperature – adiabatic. (vacuum flask).
Example of different ways to transfer energy:
Zn(s) + 2 HCl(aq) → ZnCl 2 (aq) + H 2 (g)
- The reaction takes place inside a cylinder fitted with a piston – then the gas produced drives out the piston and raises its weight in the surroundings. Energy has migrated to the surroundings as a work. Some energy also migrates as heat
- if the reaction vessel is in ice, some amount of ice melts.
- A piston is locked in position. No work is done. More ice melts – more energy has migrated to the surroundings as heat. A process that releases heat into surroundings – exothermic. (Combustion). A process that absorbs energy from the surroundings – endothermic. (Dissolution of ammonium nitrate in water).
Molecular nature of work
When a weight is raised, all its atoms move in the same direction.
Work is the transfer of energy that achieves or utilizes uniform motion in the surroundings.
Whenever we think of work, we can always think of it in terms of uniform motion of some kind. Electrical work – electrons are pushed in the same direction through the circuit. Mechanical work – atoms are pushed in the same direction against an opposing force.
Molecular nature of heat
The motion stimulated by the arrival of energy from the system is disorderly, not uniform as in the case of work.
Heat is the transfer of energy that achieves or utilizes disorderly motion in the surroundings.
Fuel burning – generates disorderly molecular motion in its vicinity.
Molecular interpretation: The internal energy of gas A ‘quadratic contribution’ to the energy – a contribution that can be expressed as the square of a variable, such as the position or the velocity. The kinetic energy of a moving atom:
EK = 1
mvx^2 + 1 2
mvy^2 + 1 2
mvz^2 The equipartition theorem: for a collection of particles at thermal equilibrium at a temperature T, the average value of each quadratic contribution to the energy is the same and equal to (1/2)kT. k – Boltzmann’s constant. The equipartition theorem is a conclusion from classical mechanics. In practice, it can be used for molecular translation and rotation but not vibration. If we consider the moving atom, according to the equipartition theorem, the average energy of each term in the expression for kinetic energy is (1/2)kT. Therefore, the mean energy of the atom is (3/2)kT and the total energy of the gas (without potential energy contribution) is (3/2)NkT = (3/2)nRT.
Um = Um ( (^0) ) + 3 2
RT
Um – the molar internal energy at T = 0, when all translational motion ceased and the sole contribution to the internal energy arises from the internal structure of the atoms. We can see that the internal energy of a perfect gas increases linearly with temperature. At 25°C, (3/2)RT = 3.7 kJ mol-1.
Because the potential energy of interactions between the atoms or molecules of a perfect gas is zero, the internal energy is independent of how close they are together, i.e., of volume:
€
∂ U ∂ V ^ ^ ^ T
= 0. The internal energy of interacting molecules in addition has a contribution from the potential energy. However, no simple expressions can be written down in general. Nevertheless, as a temperature of a system is raised, the internal energy increases as the various modes of motion become more highly excited. The conservation of energy Experimental finding: the internal energy of a system may be changed either by doing work on the system or by heating it. Heat and work are equivalent ways of changing a system’s internal energy. It is also found experimentally that, if a system is isolated from the surroundings, then no change in internal energy takes place. Suppose we consider an isolated system. It can neither do work nor supply heat – the internal energy cannot change. First Law of thermodynamics: The internal energy of an isolated system is constant.
The measurement of work
Work = distance opposing force The work needed to raise the mass through a height h on the surface of Earth: Work = h mg = mgh Work= 1.0 kg 9.81 m s-2^ 0.75 m = 7.4 kg m^2 s-2^ 1 J = 1 kg m^2 s-
When energy leaves the system, i.e., the system does work in the surroundings, w < 0. w = -100 J When energy enters the system as work, w > 0. w = +100 J
Energy leaves the system as heat, q < 0. q = -100 J Energy enters the system as heat, q > 0. q = +100 J
Expansion work
The work arising from a change in volume; includes the work done by a gas as it expands and drives back the atmosphere. Many chemical reactions result in the generation or consumption of gases.
The general expression for work dw = -F dz The negative sign: when the system moves an object against an opposing force, the internal energy of the system doing the work will decrease. F = pex A dw = - pex A dz dV = A dz dw = - pex dV
w = − pex dV
Vi
Vf
Free expansion Expansion against zero opposing force: pex = 0 w = 0 No work is done when a system expands freely. Expansion of this kind occurs when a system expands into a vacuum.
i
f
V
V w = − nRT ln
Reversible isothermal expansion work of a perfect gas at a temperature T For a gas to expand reversibly, the external pressure must be adjusted to match the internal pressure at each stage of the expansion. This matching is represented in this illustration by gradually unloading weights from the piston as the piston is raised and the internal pressure falls. The procedure results in the extraction of the maximum possible work of isothermal expansion.
- A system does maximum expansion work when the external pressure is equal to that of the system (pex = p).
- A system does maximum expansion work when it is in mechanical equilibrium with its surroundings.
- Maximum expansion work is achieved in a reversible change. More work is obtained when the expansion is reversible because matching the external pressure to the internal pressure at each stage of the process ensures that none of the system’s pushing power is wasted. We cannot obtain more work than for the reversible process because increasing external pressure even infinitesimally at any stage results in compression. Also, some pushing power is wasted when p > pex.
In an expansion Vf > Vi, Vf/Vi > 1 – the logarithm is positive and w < 0. Energy leaves the system as the system does expansion work.
For a given change in volume, we get more work the higher the temperature of the confined gas. At high temperatures, the pressure of gas is high, so we have to use a high external pressure, and therefore a stronger opposing force, to match the internal pressure at each stage.
i
f
V
V w = − nRT ln
Example 1. Calculating the work of gas production
Calculate the work done when 50 g of iron reacts with hydrochloric acid to produce hydrogen gas in (a) a closed vessel of a fixed volume, (b) an open beaker at 25°C. In (a) the volume cannot change, no expansion work is done and w = 0. In (b) the gas drives back the atmosphere and therefore w = -pexΔV. We can neglect the initial volume because the final volume after the gas production is so much larger: ΔV = Vf – Vi ≈ Vf = nRT/pex n – the amount of H 2 produced w = -pexΔV = -pex × (nRT/pex) = -nRT The reaction equation is Fe(s) + 2 HCl(aq) → FeCl 2 (aq) + H 2 (g) Therefore, 1 mol H 2 is generated when 1 mol Fe is consumed and n can be taken as the amount of reacting Fe atoms. MFe = 55.85 g mol-1^ nFe = 50 g / 55.85 g mol- w ≈ -(50 g / 55.85 g mol-1) × 8.31451 J K mol-1^ × 298.15 K ≈ -2.2 kJ The reaction mixture does 2.2 kJ of work driving back the atmosphere. Note that for this perfect gas system the external pressure does not affect the final result: the lower the pressure, the larger the volume occupied by the gas, so the effects cancel.
Heat capacity The internal energy of a substance increases when its temperature is raised. Let’s suppose that the sample is confined to a constant volume. If the internal energy is plotted against temperature, the slope of the tangent to the curve at any temperature is called the heat capacity of the system at that temperature. The heat capacity at constant volume is denoted CV and is defined formally as
CV =^ ∂ U
∂ T
^ ^
^ V
A partial derivative is slope calculated with all variables except one held constant. In this case, the internal energy varies with the temperature and with the volume of the system, but we are interes- ted only in its variation with the temperature, the volume being held constant.
Heat capacity – an extensive property – depends on the size of the sample. More convenient to report the heat capacity as an intensive property. Specific heat capacity – Cs = C/m ~4 J K-1^ g-1^ for water at room T Molar heat capacity at constant volume – CV,m = CV/n ~25 J K-1^ mol-1^ for polyatomic gases
