Thermodynamics Lecture Notes

Thermodynamics

Energy Change with Work and Heat

• The total change in a system's internal energy is expressed as:
  • $\Delta E = q + w$
  • Where:
  • $\Delta E$ = total change in internal energy
  • $q$ = energy transferred as heat
  • $w$ = energy transferred as work
• The values of $q$ and $w$ can either be positive or negative.
• Sign Convention:
  • The sign of the energy change is determined from the system's perspective:
  • Energy transferred into the system is positive, indicating an increase in energy.
  • Energy transferred out from the system is negative, indicating a decrease in energy.

Energy Transferred as Heat Only

• If a system transfers energy only as heat and does not do any work (where $w = 0$), the equation simplifies to:
  • $\Delta E = q$
  • This means:
  • If heat flows out of the system, $q$ is negative, indicating that the system is losing heat.
  • If heat flows into the system, $q$ is positive, indicating that the system is gaining heat.
• Analogy: Think of money being transferred in and out of a bank account, reflecting heat flow into and out of a system.

Enthalpy

• Enthalpy (H) is a thermodynamic variable relevant for reactions at constant pressure. It simplifies certain calculations, eliminating the need to measure PV work.
• The change in enthalpy ($\Delta H$) can be defined as:
  • $\Delta H = \Delta E + P \Delta V$
  • Where:
  • $\Delta E$ = change in internal energy
  • $P$ = pressure (assumed constant)
  • $\Delta V$ = change in volume
• In cases where only heat is transferred, we find:
  • $\Delta H = q_p$
  • Meaning the change in enthalpy is equal to heat added or lost at constant pressure.

Calorimetry

• Specific Heat Capacity (c):
  • The specific heat capacity of a substance is defined as the amount of heat required to raise the temperature of 1 gram of that substance by 1 Kelvin (or 1 °C). The formula for heat transfer is:
  • $q = m \times c \times \Delta T$
    • Where:
    • $q$ = heat gained or lost (in Joules)
    • $m$ = mass of the substance (in grams)
    • $c$ = specific heat capacity (in $J/g \cdot K$)
    • $\Delta T$ = change in temperature (in K or °C)

Coffee-Cup Calorimeter

• This instrument is designed to measure heat transfer at constant pressure with the following significant assumptions:
  • No heat escapes from the calorimeter (idealized condition).
  • All heat lost from a hot object is gained by the water in the calorimeter until thermal equilibrium is achieved.
• The heat exchange is modeled as:
  • $-q<em>{sample} = q</em>{water}$
• The expression for heat transfer in terms of mass, specific heat, and temperature change can be further broken down:
  • $c<em>s = - \frac{m</em>w \times c<em>w \times \Delta T</em>w}{m<em>s \times \Delta T</em>s}$
    • Where:
    • $c_s$ = specific heat of the sample
    • $m_w$ = mass of water
    • $c_w$ = specific heat of water (assumed as constant)
    • $\Delta T_w$ = change in water temperature
    • $m_s$ = mass of sample
    • $\Delta T_s$ = change in sample temperature

Reaction in the Calorimeter

• In a chemical reaction within a calorimeter, the heat released by the reaction ($q_{rxn}$) must be absorbed by the calorimeter system:
  • $-q<em>{rxn} = +q</em>{soln}$
• After the chemical reaction completes, the resultant solution is the only entity capable of absorbing this energy.
• Thus:
  • The heat generated from the reaction goes into the solution:
  • $q<em>{rxn, system} = - q</em>{soln, surroundings}$
• To calculate $q_{soln}$, we utilize the known values of mass, temperature change, and specific heat capacity regarding the solution.

Enthalpy Change of an Aqueous Reaction Example

1. Example Setup:
   • 50.0 mL of 0.500 M NaOH is mixed with 25.0 mL of 0.500 M HCl in a coffee-cup calorimeter.
   • Both solutions initially at 25.00°C, and final temperature after mixing is 27.21°C.
   • For calculations, assume:
     • Volumes are additive,
     • Density of solution is 1.00 g/mL,
     • Specific heat capacity of solution is 4.184 J/g·K.
2. Calculation of $q_{soln}$ (in J):
   • $m_{soln} = (50.0 ext{ mL} + 25.0 ext{ mL}) \times 1.00 \text{ g/mL} = 75.0 ext{ g}$
   • Temperature change, $\Delta T_{soln} = 27.21°C - 25.00°C = 2.21°C = 2.21 K$
   • Then:
   • $q<em>{soln} = m</em>{soln} \times c<em>{soln} \times \Delta T</em>{soln}$
     • So,
     • $q_{soln} = 75.0 ext{ g} \times 4.184 ext{ J/g·K} \times 2.21 ext{ K} = 693 ext{ J}$
3. Calculating $\Delta H_{rxn}$ (in kJ/mol of H2O formed):
   • Given $q_{soln} = 693 ext{ J}$,
   • Since this is a coffee-cup calorimeter, by principle, it follows that:
     • $q<em>{soln} = -q</em>{rxn}$
   • Thus, $\Delta H$ can be computed using moles of water formed.
4. Balanced Chemical Equation for Reaction:
   • The reaction occurring between NaOH and HCl is:
     • $HCl(aq) + NaOH(aq) → NaCl(aq) + H_2O(l)$
   • Molar ratios from the balanced equation become crucial for determining the limiting reagent, which in turn defines how many moles of $H_2O$ are produced.
5. Calculations of Moles:
   • For 50.0 mL NaOH:
     • $50.0 ext{ mL} \times \frac{1 ext{ L}}{1000 ext{ mL}} \times 0.500 \text{ mol/L} \times \frac{1 ext{ mol } H<em>2O}{1 ext{ mol NaOH}} = 0.0250 ext{ mol } H</em>2O$
   • For 25.0 mL HCl:
     • $25.0 ext{ mL} \times \frac{1 ext{ L}}{1000 ext{ mL}} \times 0.500 \text{ mol/L} \times \frac{1 ext{ mol } H<em>2O}{1 ext{ mol HCl}} = 0.0125 ext{ mol } H</em>2O$
6. Identifying Limiting Reagent:
   • HCl is the limiting reagent, thus only 0.0125 moles of $H_2O$ can be formed.
7. Calculating $\Delta H$ (in kJ/mol):
   • Using the previous findings, we have:
   • If $q<em>{rxn} = -693$ J (as $q</em>{soln}$ was positive), then:
   • $\Delta H \left( kJ/mol \right) = \frac{q<em>{rxn}}{mol \text{ H}</em>2O} \times \frac{1 ext{ kJ}}{1000 ext{ J}}$
   • Consequently:
     • $\Delta H = \frac{-693 ext{ J}}{0.0125 ext{ mol H}<em>2O} \times \frac{1 ext{ kJ}}{1000 ext{ J}} = -55.4 ext{ kJ/mol H}</em>2O$

Heat Capacity vs Specific Heat Capacity

• Heat Capacity: Defined as the amount of energy required to increase the temperature of a substance by one degree (K or °C) without regard to mass; thus, it does not depend on mass.
• Difference from Specific Heat Capacity: Specific heat capacity is defined per unit mass of a substance. Heat capacity may also refer to molar heat capacity, which is the energy required to raise the temperature of one mole of substance by one degree (K or °C).
• Formula for Heat Capacity:
  • $q_{cal} = \text{heat capacity} \times \Delta T$

Limitations of Coffee-Cup Calorimeter

• Despite its usefulness for rapid estimates, the coffee-cup calorimeter is not perfect.
  • It does not account for all potential heat losses, as it is based on the assumption that it completely insulates the contents.
• Correcting factor for heat transfer associated with the calorimeter is given by:
  • $-q<em>{rxn} = q</em>{soln} + q_{cal}$

Standard Enthalpies of Formation at 25°C

• For an element in its standard state:
  • $\Delta H_f^\circ = 0$
• Standard states to note:
  • Metals are in solid state (e.g., $Ca$)
  • Molecular elements are in their molecular form (e.g., $Cl_2$)
  • For elements existing as allotropes, only one form is designated as the standard state (e.g., $C(graphite)$).

Hess's Law

• Principle: The total enthalpy change for a reaction is equal to the sum of the enthalpy changes for the individual steps.
  • $\Delta H<em>{overall} = \Delta H</em>1 + \Delta H<em>2 + \ldots + \Delta H</em>n$
• Hess's law can be applied to calculate $\Delta H_{rxn}$ using standard enthalpy of formation values:
  • $\Delta H<em>{rxn}^\circ = \Sigma m \Delta H</em>f^\circ(products) - \Sigma n \Delta H_f^\circ(reactants)$

Experimental Procedures for Coffee-Cup Calorimeter

General Setup

1. Constraining Structure: Stack two coffee cups and secure a lid on the top. Poke a small hole for a thermometer and avoid contact with the cup.
2. Stirring Method: Use a glass stirring rod through the lid; remove the rod after stirring. After filling, ensure to keep the lid secured.

Procedure 1: Preparation for Heating Metal

1. Hot-Water Bath Preparation:
   • Fill a 400-mL beaker with ~300 mL of water and use a magnetic stir bar.
   • Heat to a gentle boil while observing to prevent excessive boiling.
2. Metal Sample: Weigh the metal sample before submerging in the water bath.
3. Suspension with Stir Rod: Loop a string around the metal and hang it in the water bath.

Procedure 2: Heat of Solution

1. Initial Set-Up: Measure 50.0 mL of DI water in a graduated cylinder. Add to the calorimeter.
2. Temperature Stabilization: Secure the lid and wait for the temperature to stabilize.
3. Adding Substance: Weigh ~3.2 g of ammonium nitrate (NH₄NO₃) and introduce it quickly to the calorimeter, immediately sealing it.
4. Final Temperature Recording: Stir gently and record the equilibrated final temperature.
5. Rinsing: Rinse the calorimeter and dry it after draining.

Procedure 3: Heat of Neutralization for Strong Acid & Strong Base

1. HCl Preparation: Measure 50.0 mL of 3.0 M hydrochloric acid (HCl) and allow temperature to equilibrate; record $T_{i,HCl}$.
2. NaOH Preparation: Measure 50.0 mL of 3.0 M sodium hydroxide (NaOH) and equilibrate; record $T_{i,NaOH}$.
3. Mixing: Quickly add HCl to the NaOH in the calorimeter, seal immediately, and stir.
4. Final Temperature Measurement: After equilibrium, record final temperature and drain.

Procedure 4: Heat of Neutralization for Weak Acid & Strong Base

1. Acetic Acid Preparation: Measure 50.0 mL of 3.0 M acetic acid (HC₂H₃O₂) and equilibrate.
2. Repeat NaOH Measurement: Do the same with the NaOH and then mix them rapidly.
3. Final Temperatures: Record final temperature post-reaction and drain solution appropriately.

Procedure 5: Specific Heat of a Metal

1. Initial Water Setup: Measure 100.0 mL of DI water in the calorimeter, equilibrate, and measure its initial temperature $(T<em>i, H</em>2O)$.
2. Metal Preparation: Right before retrieval, record the boiling water temperature as $T_i, metal$.
3. Metal Introduction: Insert the hot metal into the calorimeter, ensure no contact with the thermometer, seal it immediately.
4. Final Temperature Measurement: Allow to equilibrate, and record final temperature. Upon completion, drain and return all materials to their proper locations.