May the mole be with you
The equipartition theorem states that the total energy of a system of particles at thermal equilibrium is equally divided among all the energy components of the particles’ motion, resulting in each energy component having the same average value.
We classify the components of motion into three types: translational, rotational and vibrational motions. The theorem is applicable to systems where no electronic transitions occur, usually at .
Heat capacity, , is the ratio of heat transferred to a system from its surroundings to the temperature change of the system due to the transfer. In other words, the heat capacity of a system or substance is the amount of heat the system or substance can hold per Kelvin. The transfer of heat to a system can take place at either constant pressure or constant volume, resulting in two types of heat capacities,
and
respectively.
To define the heat capacity of a system at constant pressure, we begin with eq30 of the previous article or its differential form:
where the symbol is to signify that
is a path function.
Does eq34 say that a state function is equal to a path function?
No, eq34 means that the value of is equal to the change in
along a path, but is always the same regardless of the path taken .
Dividing both sides of eq34 by ,
For a process at constant pressure, we rewrite eq35 as:
The difference in symbols on the LHS versus the RHS of eq36 for indicating the process at constant pressure is due to being a state function and
, a path function. According to the definition of heat capacity in the first paragraph of the article,
Since the quantities and
are easily found for different systems (or substances) via experiments like the neutralisation reaction mentioned in the previous article, the heat capacities for various systems or substances at constant pressure are easily determined. The molar heat capacity of a substance at constant pressure is:
and the specific heat capacity of a substance at constant pressure, , is found by dividing eq37 with the mass,
, of the substance:
Eq39 is a useful equation to calculate the change in enthalpy of a system or substance. From eq37, the heat capacity of a substance at constant pressure is the gradient of a curve on a plot of enthalpy against temperature at constant pressure. If the curve is a straight line, is a constant at all temperatures. In reality, it is a function of temperature.
For experiments over a small temperature range, may be assumed to be constant to simplify calculations. For experiments that are carried out over a large temperature range, we need to find an expression for
. This is done by using a power series to generate curves that fit experimental values of
for different substances on
versus temperature plots:
The set of constants , ,
,
,
etc. are specific to each substance and are the outcome of the polynomial regression. As the contribution of higher powers of
to
is small, the expression is fairly well represented by:
Eq40 is then substituted in eq39 and the resultant equation is integrated throughout to obtain:
Using the same logic, to define the heat capacity of a system at constant volume, we begin with eq31 of the previous article or its differential form , resulting in
and
Since the value of the heat capacity of a system depends on whether we increase the amount of heat of the system at constant volume or at constant pressure, heat capacity is a path function. However, if the path is chosen, then
or
are state functions.
Finally, we shall show that and
for a system containing an ideal gas.
The internal energy of a gas is the sum of the molecules’ translational energy , rotational energy
, vibrational energy
, electronic transition energy
, intermolecular forces of interaction
and rest-mass energy of electrons and nuclei
:
is constant for any gas, while
is constant at
and if no chemical reaction takes place. For an ideal gas,
is zero, leaving the internal energy of an ideal gas as:
It can be shown that ,
and
are functions of temperature (independent of volume). Since the internal energy of an ideal gas is only dependent on temperature,
. With
, we write, for an ideal gas:
Eq44 is a useful relation that is applicable to all reversible processes that involves an ideal gas (not just constant volume processes because the internal energy of an ideal gas is only dependent on temperature). Irreversible processes have systems with poorly defined and therefore have internal energy changes that cannot be accurately predicted by eq44. For the enthalpy of an ideal gas,
As of an ideal gas is only dependent on temperature, the enthalpy of an ideal gas according to eq45 is also only dependent on temperature. Using the same logic as above,
and
Show that for a perfect gas (one that obeys the ideal gas law and exhibits a heat capacity that is independent of temperature) using the above diagram where
AC: reversible isothermal process
AB: reversible isochoric process
BC: reversible isobaric processes
The change in internal energy from point A to C along the isotherm AC is zero for an ideal gas. Since
is a state function,
for the process AC and ABC is the same:
The change in internal energy for path ABC is:
Assuming and
are constant over the temperature range,
Since and
, we have, after applying the ideal gas law,
Rearranging gives
The enthalpy of a thermodynamic system is the sum of the system’s internal energy and the product of its pressure and volume.
In the previous article, we showed that the change in internal energy of the system at constant volume can be experimentally determined by quantifying the transfer of heat to the system. However, many chemical reactions are carried out under constant pressure instead of constant volume.
It would be useful to identify another state function that has a relationship with analogous to eq27, i.e. one that analyses the energy change of a system at constant pressure by quantifying the transfer of heat. From eq25, the 1st law of thermodynamics of a closed system that does only pV work at constant pressure can be expressed as
Since the system is at constant pressure, where , and we can rewrite eq28 as:
Let’s define . Eq29 becomes
or
We have identified a new state function called enthalpy. Therefore, the change in enthalpy of a closed system at constant pressure is a change in energy of the system that equals to the transfer of heat to the system. We can now determine the change in energy of a chemical reaction in a closed system at constant pressure that does only pV work by experimentally measuring the amount of heat transferred to the system.
Explain why is a state function.
As explained in an earlier article, the product of two state functions is a state function and the sum of two state functions is again a state function. Since ,
and
are all state functions,
is a state function.
A process that, as a whole, absorbs heat from the surroundings, resulting in a positive change in the enthalpy of the system, is called an endothermic process. Conversely, a process that, as a whole, releases heat to its surroundings resulting in a negative change in the enthalpy of the system is called an exothermic process.
Consider the neutralisation reaction of hydrochloric acid and sodium hydroxide in a styrofoam cup calorimeter at (see diagram above).
Let’s regard the system as the reacting acid and alkali, and the surroundings as water, with both system and surroundings maintained at constant atmospheric pressure. We assume that heat and material are neither transferred from the solution to the styrofoam cup nor to the atmosphere. This means that the system is considered a closed one, as the reactants and products are confined within the cup throughout the reaction. Since any heat evolved by the reaction (system) is released to the water (surroundings),
where and
.
The enthalpy change of neutralisation is:
where
is the number of moles of water
is molar heat capacity of water at constant pressure
is the change in temperature of the solution
If ,
for the reaction, i.e. an exothermic reaction. The volume of the system may change, but for reactions in solutions, the change is usually very small. Therefore, we can also regard the above experiment as measuring the energy change in a constant volume system with negligible pressure changes. With reference to eq27 and eq30,
for such a reaction.
Let’s look at another common chemical reaction, combustion.
The change in enthalpy of a combustion reaction is often measured using a bomb calorimeter (see diagram above). However, the system in the device is not subject to constant pressure but constant volume. Therefore, we need to compute the change in internal energy of the process and convert the value to the change in enthalpy. The sample is placed in a crucible, which is enclosed in a chamber that is made of a thermally conductive material. Oxygen is pumped into the chamber and the reaction is initiated by a current flowing through a wire that is embedded in the sample. The energy released by the combustion reaction flows across the chamber walls to heat up the water in the reservoir, which is the surroundings.
For a process at constant volume, the first law of thermodynamics becomes:
can be estimated to be
where
is the specific heat capacity of water at constant volume and
is the mass of water.
is therefore negative for combustion reactions. Consider the combustion of naphthalene:
The general change in enthalpy is:
At constant volume,
The change in pressure of the system is mainly due to the change in gaseous content in the system as solids and liquids in most experiments do not occupy significant volume in the calorimeter. Substituting the ideal gas law in eq32,
If the change in temperature is small,
with its integrated form as
where for the combustion of naphthalene.
Substituting the value of obtained via eq31 in eq33, we have the change in enthalpy for the reaction. Since both
and
are negative,
is negative, validating that combustion reactions are exothermic. The change in enthalpy of a reaction can also be determined by other methods, e.g. via the van’t Hoff equation.
The first law of thermodynamics states that the change in internal energy of a system is the sum of the energy transferred to the system, due to a difference in temperature between the system and its surroundings, and the work done on the system by the surroundings.
The internal energy of a system is defined as the sum of all the energies of molecules in the system. These energies include:
is a thermodynamic property of a system. However, judging from the composition of
, it is very difficult to measure
directly. An alternate representation of
therefore needs to be established to characterise the internal energy of a system.
According to the theory of conservation of energy, energy can neither be created nor destroyed but can be transformed from one form to another. Energy can also be transferred between thermodynamic systems or from a system to its surroundings.
Consider a closed piston-cylinder system containing a gas (see diagram above). The internal energy of the system increases when thermal energy flows into it, or when a force is applied on the piston, compressing the gas. By the theory of conservation of energy,
where
Eq24 is known as the first law of thermodynamics. Instead of considering the absolute internal energy of a system when we study processes, we analyse the change in internal energy of the system, which is easily quantified by accounting for the amount of energy transferred to and from the system in the form of and
(refer to the next article for quantification of
). From an earlier article, reversible work done on the system is given by
. Therefore, for a system undergoing a reversible PV process, eq24 becomes
If the walls of the cylinder and piston are adiabatic, the system is isolated and . Eq24 becomes
which states that work done on an isolated system is exactly equal to the increase in the internal energy of the system.
Describe the work-energy theorem and show that it is consistent with eq26 for work done on a monoatomic ideal gas in an adiabatic piston-cylinder system.
The work-energy theorem states that the net work done on a particle is equal to the change in the kinetic energy of the particle. This can be shown as follows:
Since the internal energy of a monoatomic ideal gas is entirely in the form of the kinetic energy of the gas, the work done on the gas in an adiabatic system is equal to the change in the internal energy of the gas.
Suppose the piston is now rendered immovable and the walls of the cylinder are thermally conducting, we have a closed constant volume system. Eq25 becomes
The change in internal energy of the system at constant volume can then be experimentally determined by quantifying the heat transferred to the system.
The zeroth law of thermodynamics states that two thermodynamic systems, each in thermal equilibrium with a third, are in thermal equilibrium with each other.
Consider two systems, A and B, in contact with each other via an immovable thermally conducting boundary, achieving thermal equilibrium over time.
B is then brought in contact with a third system, C, via the same type of boundary.
If no net energy and matter transfer is observed between B and C, we can deduce that A is also in thermal equilibrium with C. When this happens, there must be at least one measurable thermodynamic property that is common among the three systems.
We call this common thermodynamic property, temperature, and the above-mentioned principle that allows us to define temperature, the zeroth law of thermodynamics. Temperature is therefore defined as the thermodynamic property that has the same magnitude in two systems that are in thermal equilibrium.
The principle that eventually became the zeroth law of thermodynamics was first mentioned in 1871 by James Maxwell, who said, “bodies whose temperatures are equal to that of the same body have themselves equal temperatures”. A modern version of the zeroth law of thermodynamics utilises the term ‘thermal equilibrium’ and states that
Two thermodynamic systems, each in thermal equilibrium with a third, are in thermal equilibrium with each other.
Which of the three systems described above is the thermometer?
B, as it indicates that A and C have the same temperature. If B contains a substance with relatively high coefficient of thermal expansion, e.g., helium, it can be used to assess A and C at various temperatures by monitoring its volume at constant pressure. Thus, the zeroth law of thermodynamics forms the basis of thermometry.
A path function is a mathematical function that describes thermodynamic processes that are involved in a change of equilibrium states. Unlike a state function, whose output is independent of the path taken to reach it, the output of a path function is path-dependent.
An example of a path function is the work done by an ideal gas. Consider an ideal gas in a piston-cylinder device immersed in a water bath (see diagram below).
It is evident from the above PV diagram that there are three different paths for work done in bringing the system from one equilibrium state to another equilibrium state
:
Amongst the three paths, A to B to C requires the greatest work, while A to D to C requires the least amount of work. This is because the system is expanding reversibly against a higher constant pressure throughout the process from A to B at , versus that from D to C at
. The path from A to C requires intermediate work done as the system is expanding reversibly against a decreasing pressure from
to
. Relative work done by the three paths can be inspected visually by estimating the areas under the respective PV curves. The precise values are calculated using eq6 and eq7. Since work done has values that are dependent on the path taken from one thermodynamic state to another, it is a path function.
Lastly, the differential form of the reversible isobaric expansion of an ideal gas is
We have used the symbol to emphasise that
is a path function but the symbol
is also acceptable. Eq23 is called an inexact differential because its integral is not path independent. Even though work has the same units (Joules) as energy, it is misleading to say that work is a form of energy, which if it is, will be described by a thermodynamic function that is path-independent. Work is rather, a process of energy transfer between a system and its surroundings.
Show that is an inexact differential, while
is an exact differential.
Comparing the first equation to the general form of a differential equation , we have
and
. We need to show that
for an inexact differential:
and
For , its second cross partial derivatives are
and
. Therefore
is an exact differential.
A state function is a mathematical function that describes the thermodynamic state of a system at equilibrium.
The ideal gas equation is a state function because it describes the state of a gas at equilibrium with a specific set of values: volume, temperature and pressure. In other words, a state function for a system at equilibrium relates one or more input thermodynamic properties to an output thermodynamic property.
Since a particular state of a system at equilibrium is characterised by a set of numbers, the output value of a state function for that state is independent on the path (i.e. process) taken to reach that value.
Show that the output of the state function is independent on the path taken to reach it.
The total differential of is
The second cross partial derivatives of are
and
Eq19 refers to the path where the function V is changed with respect T at constant p, followed by a change with respect to p at constant T, whereas eq20 refers to another path where the function V is changed with respect to p at constant T, followed by a change with respect to T at constant p. If eq19 is equal to eq20, the change of V is independent of the path taken. Substituting in eq19 and eq20 gives
for both equations. Therefore, the change of V is path-independent and the differential given by eq18 is called an exact differential.
Finally, if the output value of the function of pressure and the output value of the function of volume are path-independent, then the output value of the product of the two functions or the output value of the sum of the two functions are also path-independent.
An exact differential is a differential equation , for instance of two variables, of the form
, where
.
Consider the function , where
is a constant. Its total differential is
. Comparing with the general form of an exact differential, we have
and
. Since
,
is an exact differential. The equality of the mixed partials
implies that the change in
is independent of the path taken.
Conversely, an inexact differential is a differential equation of the form
, where
. The change in
, in this case, is dependent on the path taken.
Show that is an inexact differential.
We have and
. So,
.
Another difference between an exact differential and an inexact differential is that an exact differential integrates directly to give the function , whereas an inexact differential does not.
Using the integral criterion, show that is an exact differential, while
is an inexact differential.
The integrated form of the first differential is evidently . The detailed analysis involves two steps. First, integrating
with respect to
, treating
as a constant, yields
The function accounts for terms in
involving
or constants, which differentiate to zero when differentiating
with respect to
. Secondly, integrating
with respect to
, with
as a constant, gives
Comparing eq17a and eq17b, we find , where
. In other words,
integrates directly from
.
For the second differential, integrating with respect to yields
, and integrating with respect to
gives
. Clearly,
, for all
and
, indicating that the second differential does not directly integrate to give a function.
The fact that an exact differential integrates directly to give the function but an inexact differential does not, implies that
and
for a differentiable function must be of the appropriate forms of
and
, respectively. In other words, the total differential of a differentiable function must be an exact differential.
A partial derivative of a multi-variable function is its derivative with respect to one of those variables, with the other variables held constant. For example, the partial derivatives of with respect to
and
are defined as
respectively, where the symbol means that the variable
is held constant (the symbol may be omitted for simplicity).
If , then
.
The total differential of a multi-variable function is its change with respect to the changes in all the independent variables. For example, the total differential of the function is
How is eq14 derived?
The total change in is
, which is equivalent to
Multiplying the 1st and 2nd terms on the RHS of the above equation by and the 3rd and 4th terms by
,
Taking the limits and
Since the 1st term on the RHS of the above equation is with respect to a change in ,
is a constant with
and
Substituting eq12 and eq13 in the above equation, we have eq14.
In general, the total differential of the function is
If the variables themselves depend on another variable , i.e.
and
, we divide eq14a throughout by
to give
Since and
as
, if we take the limit
, we have
Eq14b is known as the multivariable chain rule, which is also known as the total derivative of .
Next, we shall derive some of useful identities. With respect to eq14, if is a constant,
, which when divided throughout by
gives
or
If is a constant, eq14 becomes
, which when divided by
gives
. Using the reciprocal identity of eq15, we have
If in eq14b, we have the chain rule:
Finally, the second partial derivative of is defined as
.
Irreversible PV work falls under non-equilibrium thermodynamics, which is hard or sometimes impossible to calculate using simple equations. This is due to the difficulty of defining the properties of the system during the process. For example, during an irreversible expansion of a gas against a piston, the piston accelerates away from the system, resulting in regions of varying pressures in the system. To overcome this problem, broad assumptions are made to expresss irreversible PV work mathematically.
Consider the irreversible expansion of a system consisting of a gas in an isolated vertical cylinder. The frictionless piston, which is part of the surroundings, has mass and is held stationary with catches (see diagram above).
If the force exerted by the gas on the bottom surface of the piston is greater than the weight of the piston, the gas expands and pushes the piston up when the catches are removed. The piston moves over a distance , until the force exerted by the expanded gas on the bottom surface of the piston equals to the weight of the piston. This implies that the gas is expanding against a constant force
, which is due to the weight of the piston.
The change in energy of the surroundings is
where is acceleration due to gravity and
is the kinetic energy of the piston.
Since the initial and final are both zero,
and eq8 becomes
Noting that energy in the universe is conserved, where
, and that
, eq9 becomes
Since the system is isolated, there is no transfer of heat. The change in energy of the system, according to the first law of thermodynamics, is therefore the change in work done on the system:
The integral form is:
If , the gas expands freely into the vacuum. In this case,
and therefore
. In general, irreversible PV work against constant pressure is estimated using eq11.
Expansion work by a system on its surroundings is always greater when the process is carried out reversibly than irreversibly. This can be seen by plotting eq6 from the previous article and eq11 on the same PV graph (see upper diagram above), where the area under AC (reversible) is greater than the area under BC (irreversible). for the irreversible process is made equal to the final pressure at
when the piston stops, similar to our piston illustration above.
Conversely, compression work by the surroundings on the system is always greater when the process is carried out irreversibly (BA) than reversibly (CA, see lower diagram above). for the irreversible process is now made equal to the final pressure at
when the piston stops.
It is important to remember that the above is a crude attempt to associate an irreversible process with an equation.