Question

Temperature-dependent quantities in statistical thermodynamics. Which quantities depend on temperature? (a) Planck's constant \(h\). (b) Partition function \(q\). (c) Energy levels \(\varepsilon_{j}\). (d) Average energy \(\langle\varepsilon\rangle\). (e) Heat capacity \(C_{V}\) for an ideal gas.

Short answer

Quantities (b), (d), and (e) depend on temperature. Quantities (a) and (c) do not.

Step-by-step solution

- Analyze Planck's Constant

Planck's constant, denoted by the letter 'h', is a fundamental physical constant. It does not change with temperature. Simply, Planck's constant is a fixed value.

- Evaluate the Partition Function

The partition function, denoted by 'q', is a function that sums up the states of a system in thermodynamics. It depends on the temperature because it involves an exponential factor with energy levels divided by the product of Boltzmann constant and temperature.

- Investigate Energy Levels

The energy levels, denoted by \( \varepsilon_j \), are typically fixed for a given system and are independent of temperature. They are determined by the internal structure of the system.

- Explore Average Energy

The average energy, expressed as \( \langle\varepsilon\rangle \), is the expected value of the energies of the states, weighted by their probabilities. Since probabilities depend on temperature, the average energy is temperature-dependent.

- Calculate Heat Capacity

The heat capacity at constant volume, denoted as \( C_{V} \), measures the amount of heat required to change the temperature of a system at constant volume. For an ideal gas, this quantity depends on temperature.

Key concepts

Planck's Constant

Planck's constant, represented by the letter 'h', is a fundamental constant in quantum mechanics. It plays a vital role in the quantization of physical properties, such as energy and action. One of the most common formulas where Planck's constant appears is the energy of a photon, given by \( E = hu \), where \( u \) is the frequency of the photon. It is crucial to note that Planck's constant is a fixed value and does not vary with temperature. This means its value remains the same regardless of the external conditions.

Partition Function

The partition function, denoted by 'q', is a central concept in statistical thermodynamics. It is essentially a sum over the Boltzmann factors of all possible states of the system, given by the equation \[ q = \sum_i e^{-\frac{\varepsilon_i}{kT}} \]. Here, \( \varepsilon_i \) represents the energy levels, \( k \) is the Boltzmann constant, and \( T \) is the temperature. The partition function depends on temperature because the Boltzmann factor \( e^{-\frac{\varepsilon}{kT}} \). When the temperature changes, the weights assigned to different energy levels change, thus altering the partition function. This function is crucial as it helps in calculating other thermodynamic properties like average energy and entropy.

Average Energy

Average energy, represented by \( \langle\varepsilon\rangle \), is the mean energy of the states of a system, weighted according to their probabilities. In statistical thermodynamics, the average energy can be calculated using the partition function with the formula \[ \langle\varepsilon\rangle = \frac{\sum_i \varepsilon_i e^{-\frac{\varepsilon_i}{kT}}}{q} \]. Given that the probabilities derived from the Boltzmann factor depend on temperature, so does the average energy. As temperature increases, higher energy states become more probable, thus increasing the average energy of the system.

Heat Capacity

Heat capacity at constant volume, signified as \( C_V \), measures the amount of heat required to raise the temperature of a system by one degree when the volume is kept constant. For an ideal gas, the heat capacity is given by the relation \( C_V = \left( \frac{\partial \langle\varepsilon\rangle}{\partial T} \right)_V \). This means that \( C_V \) is directly related to the change in average energy with temperature. Thus, it indicates the amount of energy needed to increase the temperature of the system. Heat capacity depends on temperature since the energy distribution among various states (and hence the average energy) changes with temperature. This dependency makes heat capacity a crucial aspect in studying the thermal properties of materials.