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Enthalpy and Calorimetry

Applications (Work, Enthalpy, Heat Capacity)

This section delves into specific thermodynamic quantities and their applications, particularly focusing on energy transfer and the properties of substances related to heat.

Work

Definition: In thermodynamics, work ($w$) is a form of energy transfer that occurs when a force acts over a distance. It is energy transferred into or out of a system by a means other than a temperature difference (which is heat).

Mechanical Work (Expansion/Compression Work): The most common type of work discussed in chemical thermodynamics is pressure-volume ($PV$) work, also known as expansion or compression work. This occurs when the volume of a system changes against an external pressure.

Formula: For an infinitesimal change in volume ($dV$) against a constant external pressure ($P_{ext}$):

$$dw = -P_{ext} dV$$

The negative sign indicates that if the system expands ($dV > 0$), work is done BY the system on the surroundings ($dw < 0$). If the system is compressed ($dV < 0$), work is done ON the system ($dw > 0$).

Work Done in a Reversible Process: For a reversible process, the external pressure is infinitesimally close to the system's internal pressure ($P_{sys}$), so $P_{ext} \approx P_{sys}$. The work done is then:

$$w = -\int_{V_1}^{V_2} P_{sys} dV$$

For an ideal gas at constant temperature (isothermal process), $P_{sys} = \frac{nRT}{V}$:

$$w_{rev, isothermal} = -\int_{V_1}^{V_2} \frac{nRT}{V} dV = -nRT \ln\left(\frac{V_2}{V_1}\right) = -nRT \ln\left(\frac{P_1}{P_2}\right)$$

Work at Constant Pressure (Isobaric Process): If the process occurs at constant external pressure ($P_{ext}$), the work done is simply:

$$w = -P_{ext}(V_2 - V_1) = -P_{ext}\Delta V$$

Significance: Work is a crucial component of the First Law of Thermodynamics ($\Delta U = q + w$). Understanding the conditions under which work is done (e.g., expansion, compression) is vital for analyzing energy changes in chemical and physical processes.

Enthalpy, H

Definition: Enthalpy ($H$) is a thermodynamic property defined as the sum of the internal energy ($U$) of a system and the product of its pressure ($P$) and volume ($V$):

$$H = U + PV$$

Enthalpy Change ($\Delta H$): Enthalpy itself is a state function, but its absolute value is difficult to determine. We are usually interested in the change in enthalpy ($\Delta H$) during a process.

Relationship to Heat at Constant Pressure: At constant pressure, the change in enthalpy ($\Delta H$) is equal to the heat ($q_p$) transferred into or out of the system.

Derivation:

Consider the First Law of Thermodynamics: $dU = dq + dw$.

For a process at constant pressure, $dw = -P dV$. So, $dU = dq - P dV$.

Now, let's look at the change in enthalpy, $dH = dU + d(PV)$.

Using the product rule for $d(PV)$: $d(PV) = P dV + V dP$.

So, $dH = dU + P dV + V dP$.

Substitute $dU = dq - P dV$ (for constant pressure process where $dw = -PdV$):

$dH = (dq - P dV) + P dV + V dP$

$dH = dq + V dP$

For a process at constant pressure, $dP = 0$. Therefore:

$$dH = dq_p$$

And for a finite process at constant pressure:

$$\Delta H = q_p$$

Significance:

Enthalpy is a very useful concept for describing the heat changes in chemical reactions, especially in processes that occur at atmospheric pressure.

Heat Capacity

Definition: Heat capacity ($C$) is a measure of the amount of heat required to raise the temperature of a substance by one degree Celsius (or one Kelvin). It quantifies how much heat energy a substance can absorb or release for a given temperature change.

Formula:

$$C = \frac{q}{\Delta T}$$

Where:

Specific Heat Capacity ($c$ or $s$): This is the heat capacity per unit mass of a substance. It's an intrinsic property of the substance.

$$c = \frac{q}{m\Delta T}$$

Units: J/(g·K) or J/(kg·K).

Molar Heat Capacity ($C_m$ or $C_{mol}$): This is the heat capacity per mole of a substance.

$$C_m = \frac{q}{n\Delta T}$$

Units: J/(mol·K).

Heat Capacity at Constant Volume ($C_v$) and Constant Pressure ($C_p$):

For most substances, especially gases, $C_p$ is greater than $C_v$ because at constant pressure, the system must also do work to expand as its temperature increases. This additional energy required for expansion means more heat must be supplied to achieve the same temperature increase as in a constant volume process.

The Relationship Between $C_p$ And $C_v$ For An Ideal Gas

Derivation:

We know that for an ideal gas:

  1. From the First Law of Thermodynamics: $dU = dq + dw$.
  2. For a constant volume process: $dw = 0$, so $dU = dq_v$.
  3. The molar heat capacity at constant volume is defined as $C_v = \frac{1}{n}\left(\frac{dq_v}{dT}\right)_V$. Thus, $dq_v = n C_v dT$, and $dU = n C_v dT$.
  4. From the definition of enthalpy: $H = U + PV$. For an ideal gas, $PV = nRT$. So, $H = U + nRT$.
  5. Taking the differential of H: $dH = dU + nR dT$.
  6. The molar heat capacity at constant pressure is defined as $C_p = \frac{1}{n}\left(\frac{dq_p}{dT}\right)_P$.
  7. From the relationship $\Delta H = q_p$, we have $dH = dq_p$. So, $dH = n C_p dT$.

Now, substitute the expressions for $dH$ and $dU$ into the equation $dH = dU + nR dT$:

$$n C_p dT = n C_v dT + nR dT$$

Divide by $n dT$:

$$C_p = C_v + R$$

This fundamental relationship shows that the molar heat capacity at constant pressure is greater than the molar heat capacity at constant volume by the ideal gas constant ($R$). This difference accounts for the work done by the gas during expansion at constant pressure.


Measurement Of $\Delta U$ And $\Delta H$: Calorimetry

Calorimetry: Calorimetry is the experimental technique used to measure the heat changes associated with chemical reactions or physical processes. A device used for measuring heat changes is called a calorimeter.

$\Delta U$ Measurements

Constant Volume Calorimetry (Bomb Calorimetry):

Apparatus: Typically uses a bomb calorimeter. The system (the reaction) is sealed in a strong, rigid container called a "bomb," which is immersed in a known amount of water. This setup ensures that the volume remains constant throughout the process.

Principle: Since the volume is kept constant, no $PV$ work is done ($w=0$). According to the First Law of Thermodynamics, $\Delta U = q + w$. With $w=0$, the change in internal energy is equal to the heat absorbed or released by the system:

$$\Delta U = q_v$$

Measurement:

  1. A known amount of reactant is placed inside the bomb.
  2. The bomb is sealed and placed in a calorimeter containing a known mass of water.
  3. The temperature of the water is measured.
  4. The reaction is initiated (e.g., by electrical ignition).
  5. The heat released or absorbed by the reaction causes a change in the temperature of the water and the calorimeter. This temperature change is measured accurately.
  6. The heat absorbed by the calorimeter ($q_{cal}$) and the water ($q_{water}$) is calculated using their heat capacities: $q_{calorimeter} = C_{cal} \Delta T$ and $q_{water} = m_{water} c_{water} \Delta T$. The total heat capacity of the calorimeter ($C_{cal}$) is determined beforehand through calibration.
  7. The heat released by the reaction ($q_{rxn}$) is equal in magnitude but opposite in sign to the heat absorbed by the calorimeter and water: $q_{rxn} = -(q_{water} + q_{calorimeter})$.
  8. Therefore, $\Delta U = q_{rxn}$.

Applications: Bomb calorimetry is ideal for measuring the heat of combustion reactions, where a substance burns in excess oxygen at constant volume.

$\Delta H$ Measurements

Constant Pressure Calorimetry (Coffee-Cup Calorimetry):

Apparatus: Typically uses a simple calorimeter, often made of insulated containers (like Styrofoam cups), with a lid and a thermometer. This setup approximates constant pressure conditions (atmospheric pressure).

Principle: For a process occurring at constant pressure, the change in enthalpy is equal to the heat transferred:

$$\Delta H = q_p$$

Measurement:

  1. Known amounts of reactants are placed in the calorimeter, often dissolved in water.
  2. The initial temperature of the mixture is measured.
  3. The reaction or process occurs, leading to a change in temperature.
  4. The final temperature is measured.
  5. The heat absorbed or released by the reaction ($q_{rxn}$) is calculated from the temperature change of the solution and the calorimeter, using their heat capacities: $q_{rxn} = -(m_{solution} c_{solution} \Delta T + C_{cal} \Delta T)$. For simple setups, the heat capacity of the calorimeter itself might be negligible or incorporated into the specific heat of the solution.
  6. Therefore, $\Delta H = q_{rxn}$.

Applications: Constant pressure calorimetry is suitable for measuring the heat of dissolution, neutralization reactions, and other reactions that occur readily at atmospheric pressure.

Conversion from $\Delta U$ to $\Delta H$ (if needed):

If $\Delta U$ is measured (e.g., by bomb calorimetry) and the reaction involves a change in the number of moles of gas ($\Delta n_g$), $\Delta H$ can be calculated using the relationship:

$$\Delta H = \Delta U + \Delta (PV)$$

For an ideal gas, $PV = nRT$. So, $\Delta (PV) = \Delta (nRT) = (\Delta n_g)RT$ (assuming constant temperature).

$$\Delta H = \Delta U + (\Delta n_g)RT$$

Where $\Delta n_g$ is the change in the number of moles of gas ($n_{g,products} - n_{g,reactants}$).


Lattice Enthalpy (from Chemical Bonding)

Lattice Enthalpy ($\Delta H_{lattice}$):

Definition: Lattice enthalpy is defined as the enthalpy change required to convert one mole of an ionic compound completely into its gaseous constituent ions. It is a measure of the strength of the ionic bonds in a crystal lattice.

Two Definitions (Important Distinction):

  1. Endothermic Definition (More Common): The energy required to break one mole of the ionic solid into gaseous ions.

    $$MX(s) \rightarrow M^+(g) + X^-(g) \quad \Delta H_{lattice} > 0$$

    This definition is usually positive, representing the energy input needed to break the lattice.

  2. Exothermic Definition (Born-Haber Cycle): The energy released when one mole of an ionic compound is formed from its gaseous constituent ions.

    $$M^+(g) + X^-(g) \rightarrow MX(s) \quad \Delta H_{lattice} < 0$$

    This definition is usually negative, representing the energy output during lattice formation. This is the definition commonly used in Born-Haber cycles.

Factors Affecting Lattice Enthalpy:

Lattice enthalpy is primarily governed by two factors, as described by the Kapustinskii equation or derived from Coulomb's Law:

  1. Ionic Charge: Lattice enthalpy is directly proportional to the product of the charges on the ions. Higher charges lead to stronger electrostatic attraction and thus higher lattice enthalpy.

    2. For example, $MgO$ (Mg$^{2+}$, O$^{2-}$) has a much higher lattice enthalpy than $NaCl$ (Na$^+$, Cl$^-$) because the product of charges ($+2 \times -2 = -4$) is greater than ($+1 \times -1 = -1$).

  2. Ionic Size (Interionic Distance): Lattice enthalpy is inversely proportional to the distance between the centers of the ions (sum of their ionic radii). Smaller ions are closer together, leading to stronger electrostatic attraction and higher lattice enthalpy.

    3. For example, $LiF$ has a higher lattice enthalpy than $CsI$ because both $Li^+$ and $F^-$ are smaller than $Cs^+$ and $I^-$.

Born-Haber Cycle:

Lattice enthalpy cannot be directly measured. It is indirectly determined using a thermodynamic cycle called the Born-Haber cycle. This cycle relates the lattice enthalpy to other measurable enthalpy changes:

For a compound like $MX$ (where M is a metal and X is a non-metal):

$$MX(s) \xrightarrow{\Delta H_{lattice}} M^+(g) + X^-(g)$$

The Born-Haber cycle considers the following steps to form the ionic solid $MX(s)$ from its elements in their standard states:

  1. Atomization of the metal ($M$): $M(s) \rightarrow M(g)$ ($\Delta H_{atom, M}$, enthalpy of atomization)
  2. Ionization of the metal atom: $M(g) \rightarrow M^+(g) + e^-$ ($IE_1$, first ionization energy)
  3. Atomization of the non-metal ($X_2$): $\frac{1}{2}X_2(g) \rightarrow X(g)$ ($\frac{1}{2}\Delta H_{atom, X}$)
  4. Electron Affinity of the non-metal atom: $X(g) + e^- \rightarrow X^-(g)$ ($EA$, electron affinity)
  5. Formation of the ionic solid from gaseous ions: $M^+(g) + X^-(g) \rightarrow MX(s)$ ($\Delta H_{lattice}$, lattice enthalpy, usually defined as negative for formation)
  6. Enthalpy of formation of the compound from elements: $M(s) + \frac{1}{2}X_2(g) \rightarrow MX(s)$ ($\Delta H_f^\circ$, enthalpy of formation)

According to Hess's Law, the enthalpy of formation ($\Delta H_f^\circ$) is the sum of the enthalpy changes of all steps in the cycle:

$$\Delta H_f^\circ = \Delta H_{atom, M} + IE_1 + \frac{1}{2}\Delta H_{atom, X} + EA + \Delta H_{lattice}$$

By rearranging this equation, the lattice enthalpy can be calculated:

$$\Delta H_{lattice} = \Delta H_f^\circ - \left( \Delta H_{atom, M} + IE_1 + \frac{1}{2}\Delta H_{atom, X} + EA \right)$$

Significance: Lattice enthalpy is a key factor determining the stability and properties of ionic compounds, such as their melting points and solubility. A higher (more positive or less negative, depending on definition) lattice enthalpy indicates stronger ionic bonding.