Question

Binding to a surface. Consider a particle that has two states: bonded to a surface, or non-bonded (released). The non-bonded state is higher in energy by an amount \(\varepsilon_{0}\). (a) Explain how the ability of the particle to bond to the surface contributes to the heat capacity, and why the heat capacity depends on temperature. (b) Compute the heat capacity \(C_{V}\) in units of \(N k\) if \(T=300 \mathrm{~K}\) and \(\varepsilon_{0}=1.2 \mathrm{kcal} \mathrm{mol}^{-1}\) (which is about the strength of a weak hydrogen bond in water).

Short answer

At high T, heat capacity contributes mainly due to large Boltzmann weighting, where bonding affects < U >, thereby influencing \( C_V \approx Nk \).

Step-by-step solution

Understanding Heat Capacity Contributions

Heat capacity, or the heat required to change the temperature of a system by one degree, depends on the energy states of the particles in the system. For our particle, which can be bonded or non-bonded to the surface, bonding reduces the energy, thus affecting the heat capacity. At different temperatures, the proportion of particles that are in the bonded or non-bonded states changes, influencing how much energy is needed to change the system's temperature.

Determine the Partition Function

The partition function, Z, is crucial for calculating thermodynamic quantities. It sums over all the accessible states: \[ Z = e^{0} + e^{-\frac{\beta \boldsymbol{\text{\text{ε}}_0}}{kT}} = 1 + e^{-\frac{\boldsymbol{\text{\text{ε}}_0}}{kT}} \] where \( k \) is Boltzmann's constant, \( T \) is the temperature in Kelvin, \( \boldsymbol{\text{\text{ε}}_0} \) is the energy difference between non-bonded and bonded states.

Define the Average Energy

The average energy < U > of the system is given by: \[ \text{} = \frac{\text{\text{ε}}_0 e^{-\frac{\text{\text{ε}}_0}{kT}}}{1 + e^{-\frac{\text{\text{ε}}_0}{kT}}} \]

Calculate the Heat Capacity

Heat capacity at constant volume \( C_V \) is the derivative of the average energy with respect to temperature: \[ C_V = \frac{\text{d} \text{}}{\text{dT}} \] To find this, we first express the derivative of < U >: \[ C_{V} = N k \frac{((\frac{\text{\text{ε}}_0}{kT})^2 e^{-\frac{\text{\text{ε}}_0}{kT}})}{(1 +e^{-\frac{\text{\text{ε}}_0}{kT}})^2} \] where N is the number of particles.

Plug in the Given Values

Given: \( T = 300 \text{ K} \) \( \text{\text{ε}}_0 = 1.2 \text{ kcal/mol} \) (1 kcal/mol = 4184 J/mol) Convert \( \text{\text{ε}}_0 \) to Joules: \( \text{\text{ε}}_0 = 1.2 \times 4184 \text{ J/mol} = 5020.8 \text{ J/mol} \) Plug these values into the expression for \( C_V \): \[ \frac{\text{\text{ε}}_0}{k \times T} = \frac{5020.8}{1.3806 \times 10^{-23} \times 300} = 1.211 \times 10^4 \] Now, calculate \( C_V \) by substituting this value back in: \[ C_V \text{in units of} Nk = N k \frac{((1.211 \times 10^4)^2 e^{-1.211 \times 10^4})}{(1 + e^{-1.211 \times 10^4})^2} \] Since \( e^{-1.211 \times 10^4} \approx 0 \) for large exponents: \[ C_V \text{in units of} Nk = N k \text{(neglecting the denominator effect)} \] Hence, when temperature is not extremely high, \( C_V \text{in units of} Nk \), can be approximated directly under Boltzmann expectation.

Key concepts

Understanding Heat Capacity

Heat capacity represents the amount of heat required to change a system's temperature by one degree. For a particle that can switch between bonded and non-bonded states, this switching significantly influences the heat capacity. When particles bind to a surface, they lower their energy, thereby altering how much energy is needed to heat the system. At higher temperatures, more particles have the energy to remain in the non-bonded state, leading to variations in the heat capacity. Thus, the heat capacity depends on the temperature because the distribution of particles between bonded and non-bonded states changes with temperature.
The equation for heat capacity at constant volume, often denoted as \( C_V \), is:
\[ C_V = \frac{d}{dT} \] where is the average energy of the system and T is the temperature.

Thermodynamic States

In thermodynamics, the state of a system can be described by various properties such as energy, temperature, and pressure. For our particle, we consider two thermodynamic states: bonded and non-bonded. The non-bonded state possesses higher energy. At any given temperature, the particles can occupy both states, but the proportion of particles in each state depends on the temperature. This concept is crucial in thermodynamics, as it helps us understand how energy is distributed within a system and how changes in physical conditions (like temperature) affect this distribution. Using the Boltzmann factor, we can describe the probability of a particle being in a specific state:
\[ P_i = \frac{e^{-\frac{E_i}{kT}}}{Z} \] where \({ E_i }\) is the energy of a state, \({ k }\) is Boltzmann's constant, and \({ T }\) is the temperature.

Understanding Partition Function

The partition function, denoted as \( Z \), is a key concept in statistical mechanics. It encapsulates all the possible states of a system and helps in computing various thermodynamic quantities. For a particle that can either be bonded or non-bonded, the partition function sums the probabilities of these states:
\[ Z = e^{0} + e^{-\frac{\text{\text{ε}}_0}{kT}} = 1 + e^{-\frac{\text{\text{ε}}_0}{kT}} \] Here, \(\text{\text{ε}}_0 \) is the energy difference between the non-bonded and bonded states, \({ k }\) is Boltzmann's constant, and \({ T }\) is the temperature. The partition function \( Z \) assists in finding average energy and other thermodynamic properties by considering all accessible states.

Average Energy Calculation

The average energy \( < U > \) in a system with multiple states can be calculated using the energies of the states and their probabilities, which are determined by the partition function.
For a particle with two states, the average energy is given by:
\[ = \frac{\text{\text{ε}}_0 \times e^{-\frac{\text{\text{ε}}_0}{kT}}}{1 + e^{-\frac{\text{\text{ε}}_0}{kT}}} \] Here, \(\text{\text{ε}}_0 \) is the energy difference between the non-bonded and bonded states, \({ k }\) is Boltzmann's constant, and \({ T }\) is the temperature. This formula considers the proportion of particles in each state, weighted by their energies, and normalized by the partition function to sum up to 1.