354
9 Bioenergetics
is all about. However, it is the most fundamental way to define temperature of any
system, and it does not depend on the particular properties of a thermometer.
We now take advantage of the extensive nature of certain thermodynamic
functions. The internal energy, the entropy, the volume, the total particle number, are
all extensive quantities. In fact, if a system is reproduced λ times, then the energy
of the combined system satisfies
E(λS, λV , λN) = λE(S, V , N) ,
(9.73)
i.e. λ appears as an overall factor on the right, and to the first power. The ‘degree of
homogeneity’ is this power, so the internal energy is ‘homogeneous to degree one’.
As we will see, a consequence is that we can express the internal energy in terms of
its slopes. The argument goes as follows.
If a function is homogeneous to degree h, then:
f ({λX k }) = λ
h f ({X k }) ,
(9.74)
is called a homogeneous function of degree h. By differentiation with respect to λ,
and then setting λ = 1, we will have
hf ({X k }) =
k
∂f
∂X k
X k .
(9.75)
This is called the Euler relation. 36 It applies to both intensive (h = 0) and extensive
(h = 1) thermodynamics variables.
Applying the Euler relation to the internal energy, we have
E(S, V , N) = T S − pV +
k
μ k N k .
(9.76)
We now see that if the functions T (S, V , N), p(S, V , N) and μ(S, V , N) are
known, then the complete thermodynamic properties of the system are known. 37
These functions are called equations of state.
Given the equations of state, or given the function E(S, V , N), all other
measurable thermodynamic quantities can be expressed. For example, the heat
capacity at constant volume becomes
36 Actually, Euler discovered many important relations! Perhaps the most profound is exp (iθ) =
cos θ + i sin θ. The latter contains arguably the most magical formula in all mathematics,
(2.71828 · · · )
√ −1 (3.14159··· ) = −1.000 · · · .
37 These three functions determining the equations of state are not independent (see Sect. 9.11.2),
so that only two of the three are needed.
9 Bioenergetics
is all about. However, it is the most fundamental way to define temperature of any
system, and it does not depend on the particular properties of a thermometer.
We now take advantage of the extensive nature of certain thermodynamic
functions. The internal energy, the entropy, the volume, the total particle number, are
all extensive quantities. In fact, if a system is reproduced λ times, then the energy
of the combined system satisfies
E(λS, λV , λN) = λE(S, V , N) ,
(9.73)
i.e. λ appears as an overall factor on the right, and to the first power. The ‘degree of
homogeneity’ is this power, so the internal energy is ‘homogeneous to degree one’.
As we will see, a consequence is that we can express the internal energy in terms of
its slopes. The argument goes as follows.
If a function is homogeneous to degree h, then:
f ({λX k }) = λ
h f ({X k }) ,
(9.74)
is called a homogeneous function of degree h. By differentiation with respect to λ,
and then setting λ = 1, we will have
hf ({X k }) =
k
∂f
∂X k
X k .
(9.75)
This is called the Euler relation. 36 It applies to both intensive (h = 0) and extensive
(h = 1) thermodynamics variables.
Applying the Euler relation to the internal energy, we have
E(S, V , N) = T S − pV +
k
μ k N k .
(9.76)
We now see that if the functions T (S, V , N), p(S, V , N) and μ(S, V , N) are
known, then the complete thermodynamic properties of the system are known. 37
These functions are called equations of state.
Given the equations of state, or given the function E(S, V , N), all other
measurable thermodynamic quantities can be expressed. For example, the heat
capacity at constant volume becomes
36 Actually, Euler discovered many important relations! Perhaps the most profound is exp (iθ) =
cos θ + i sin θ. The latter contains arguably the most magical formula in all mathematics,
(2.71828 · · · )
√ −1 (3.14159··· ) = −1.000 · · · .
37 These three functions determining the equations of state are not independent (see Sect. 9.11.2),
so that only two of the three are needed.