10.2

General Heat Capacity

• General heat capacity is expressed as $C<em>G$ or $C</em>{cal}$, where "cal" refers to calorimeter.

• Measurement Units: Joules per degree Celsius (J/°C).

• This type of capacity is the summation of the heat capacities of all components of the calorimeter.

• It is used to quantify heats involved in reaction processes using apparatus called a calorimeter.

Calorimeter

• A calorimeter measures the flow of heat using its various components.

• Components:

  • Thermometer: Measures temperature change ((\Delta T)).

  • Ignition wire: Used to initiate a reaction, often in the case of combustion.

  • Stirrer: Ensures thermal mixing and eliminates convection currents.

  • Water (H2O): Fills the calorimeter and acts as a medium to absorb heat.

Bomb Calorimeter

• A type of calorimeter that maintains constant volume conditions.

• Setup:

  • A sealed chamber (the bomb) where a reaction occurs under high pressure.

  • Contains sample material, oxygen for combustion, and is filled with water.

• Key Elements of a Bomb Calorimeter:

  • It conducts combustion reactions in a sealed container, maintaining constant volume.

  • Heat generated from the reaction is transferred to the water, increasing its temperature.

• Heat Calculations:

  • $Q<em>{reaction} = -Q</em>{calorimeter}$

  • Here, (Q{reaction}) is equal to heat absorbed by the calorimeter ((Q{calorimeter})).

• Heat of the calorimeter can be expressed as:

  • $Q<em>{calorimeter} = C</em>G \Delta T + m \cdot C_{water} \Delta T$,
    where:

  • (C_G) is the general heat capacity,

  • (m) is the mass of water,

  • (C_{water}) is the specific heat capacity of water,

  • (\Delta T) is the change in temperature.

• The equation depends on the given data:

  • If the mass of water is provided, use both terms.

  • If not, use only the general capacity term.

Heat Flow and Energy Change Calculations

• During a bomb calorimeter experiment:

  1. Measure the heat absorbed by the calorimeter through temperature change.

  2. Convert total heat of calorimeter to that of the reaction using the first law of thermodynamics.

  3. Calculate change in energy in Kilojoules per mole:

  • $\Delta E = \frac{Q<em>{reaction}}{moles}$ where (Q{reaction} = -Q_{calorimeter}).

Example Problem

• Given:

  • Sample: 1.5 grams of rocket fuel (dimethyl hydrazine).

  • Temperature change: from 24.33 °C to 26.25 °C.

  • Calorimeter heat capacity: 1.84 kJ/°C.

  • Water mass: 5 kg.

• Steps to calculate (\Delta E):

  1. Calculate (\Delta T):(\Delta T = 26.25 - 24.33 = 1.92 °C).

  2. Calculate (Q_{calorimeter}):

  • (Q{calorimeter} = CG \Delta T + m \cdot C_{water} \Delta T)

  • Consider (C_{water} = 4.18 \, \text{J/g °C} = 4.18 \, \text{kJ/kg °C}).

  • Plugging in numbers:

    • (Q_{calorimeter} = 1.84 \, \text{kJ/°C} * 1.92 \, °C + 5 \, \text{kg} * 4.18 \, \text{kJ/kg °C} * 1.92 \, °C )

    • This gives (Q_{calorimeter} = 43.7 \, \text{kJ}).

  1. Calculate (Q_{reaction}):

  • (Q{reaction} = -Q{calorimeter} = -43.7 \, \text{kJ}).

  1. Convert grams to moles:

  • Molar mass of dimethyl hydrazine: 60 g/mol.

  • Moles of sample = (\frac{1.5 \, g}{60 \, g/mol} = 0.0250 \, moles).

  1. Calculate (\Delta E):

  • (\Delta E = \frac{Q_{reaction}}{moles} = \frac{-43.7 \, kJ}{0.0250 \, moles} = -1748 \, kJ/mol).

Coffee Cup Calorimeter

• Different from bomb calorimeter as it operates under constant pressure.

• Setup:

  • Typically a Styrofoam cup used to minimize heat exchange with the environment.

  • Similar measurement setup with thermometer.

• Measurements:

  • Uses similar equation to calculate heat flow with adaptations:

  • $Q<em>{calorimeter} = m \cdot C</em>{water} \Delta T + C<em>{cup} \Delta T$ if (C{cup}) provided.

  • If not, only measure (Q{calorimeter} = m \cdot C{water} \Delta T).

• Key Features:

  • Because of no work being done (due to lack of gas phase), it relates to enthalpy changes:

  • $\Delta H = \frac{Q_{system}}{moles}$ and is referred to as the enthalpy of reaction.

Summary of Calorimetry Steps

1. For both calorimeters, the equation for heat transfer ((Q_{calorimeter})) begins with specific capacities.

   • For bomb: (Q{calorimeter} = CG \Delta T + m \cdot C_{water} \Delta T).

   • For coffee cup: (Q{calorimeter} = m \cdot C{water} \Delta T + C_{cup} \Delta T).

2. Relate (Q{calorimeter}) to the reaction: (Q{reaction} = -Q_{calorimeter}).

3. Calculate energy per mole for bomb calorimetry as (\Delta E = \frac{Q{reaction}}{moles}) and for coffee cup calorimetry as (\Delta H = \frac{Q{reaction}}{moles}).