Properties of Absolute Temperature

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(i) Absolute Temperature is Always Positive

From the definition of temperature given by Eqs. (12) and (13) of Section 1.4,

\[ \beta = \frac{\partial \ln \Omega(E)}{\partial E} = \frac{1}{k_B T} \tag{1} \]

where \( \Omega(E) \) represents the number of accessible microstates corresponding to the energy \(E\).

For a general system, \( \Omega(E) \) is a rapidly increasing function of energy \(E\). Therefore,

\[ \frac{\partial \ln \Omega(E)}{\partial E} > 0. \]

Hence, \( \beta \) is positive. Since

\[ \beta = \frac{1}{k_B T}, \]

and the Boltzmann constant \(k_B\) is positive, the absolute temperature \(T\) is also positive.

Therefore, \[ \boxed{T>0} \] for ordinary physical systems.

(ii) Magnitude of Absolute Temperature

To estimate the magnitude of temperature, the approximate dependence of \( \Omega(E) \) on energy may be written as

\[ \Omega(E) \propto (E-E_0)^f \tag{2} \]

where \(f\) is the number of degrees of freedom and \(E_0\) is the energy of the ground state of the system.

Taking logarithm on both sides of Eq. (2),

\[ \ln \Omega(E) \approx f\ln(E-E_0)+\text{constant}. \]

Differentiating with respect to \(E\),

\[ \beta = \frac{\partial \ln \Omega(E)}{\partial E} \approx \frac{f}{E-E_0} \tag{3} \]

At thermal equilibrium, \(E\) may be regarded approximately as the average energy \( \langle E\rangle \). Since

\[ \beta=\frac{1}{k_B T}, \]

we obtain

\[ \frac{1}{k_B T} \approx \frac{f}{\langle E\rangle-E_0} \tag{4} \]

Therefore,

\[ \boxed{ k_B T \approx \frac{\langle E\rangle-E_0}{f} } \]

It is clear that at absolute temperature \(T\), the quantity \(k_B T\) is nearly equal to the excess of the average energy over the ground-state energy per degree of freedom.

In thermal equilibrium, systems having the same temperature have, approximately, the same average excess energy per degree of freedom. Energy is exchanged between the systems until thermal equilibrium is established.

(iii) Temperature Increases with Energy

From Eq. (3),

\[ \beta \approx \frac{f}{E-E_0}. \]

Differentiating with respect to energy \(E\),

\[ \frac{\partial\beta}{\partial E} = -\frac{f}{(E-E_0)^2} \tag{5} \]

Since \(f>0\),

\[ \boxed{ \frac{\partial\beta}{\partial E}<0 }. \]

Now, from the definition

\[ \beta=\frac{1}{k_B T}, \]

differentiating with respect to \(E\),

\[ \frac{\partial\beta}{\partial E} = -\frac{1}{k_B T^2} \frac{\partial T}{\partial E}. \]

Comparing this result with Eq. (5),

\[ -\frac{1}{k_B T^2} \frac{\partial T}{\partial E} = -\frac{f}{(E-E_0)^2}, \]

Hence,

\[ \boxed{ \frac{\partial T}{\partial E} = \frac{f k_B T^2} {(E-E_0)^2} } \tag{6} \]

All the quantities on the right-hand side are positive. Therefore,

\[ \boxed{ \frac{\partial T}{\partial E}>0 }. \]

Thus, an increase in energy \(E\) produces an increase in absolute temperature \(T\). Hence, temperature is an increasing function of energy.

(iv) Direction of Heat Flow

Equation (6) shows the relationship between energy and temperature and helps us understand the direction of heat flow.

Consider two systems \(A\) and \(B\), which are separately in thermal equilibrium and have temperatures \(T_A\) and \(T_B\), respectively.

Let these two systems be brought into thermal contact. Energy will be exchanged between them until both systems reach a common equilibrium temperature \(T_f\).

Suppose

\[ T_A

System \(A\), being at the lower temperature, absorbs heat. Consequently, its energy increases and its temperature also increases.

Therefore,

\[ T_f>T_A. \]

On the other hand, system \(B\) releases heat. Its energy decreases and hence its temperature decreases. Therefore,

\[ T_f

Combining these two relations,

\[ \boxed{ T_A

Thus, when two systems are in thermal contact:

• The system at lower temperature absorbs heat and is called the cold system.
• The system at higher temperature releases heat and is called the hot system.
• Heat flows spontaneously from the system at higher temperature to the system at lower temperature.

Key Result: Heat flows from a body at higher temperature to a body at lower temperature until thermal equilibrium is established.

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