Calorimetry

The principle of calorimetry states that the total heat given by hot objects equals the total heat received by cold objects (assuming no heat exchange with surroundings).

Mathematically: $Q_{lost} = Q_{gained}$

For multiple objects:

• $\sum_{hot} m_i c_i (T_{initial,i} - T_{final}) = \sum_{cold} m_j c_j (T_{final} - T_{initial,j})$

Example: When a hot metal piece at 90°C is placed in cold water at 20°C in an insulated calorimeter, heat flows until equilibrium is reached. The heat lost by the metal equals the heat gained by the water and calorimeter.

Note: This principle assumes perfect thermal isolation from the environment.

Critical measurements:

1. Mass of the solid ($m_1$)
2. Mass of calorimeter and stirrer ($m_2$)
3. Mass of water ($m_3$)
4. Initial temperature of solid ($θ_1$)
5. Initial temperature of calorimeter, stirrer and water ($θ_2$)
6. Final equilibrium temperature of mixture ($θ$)

Calculation formula: $s_1 = \frac{(m_2s_2 + m_3s_3)(θ - θ_2)}{m_1(θ_1 - θ)}$

Where:

• $s_1$ = specific heat capacity of solid (unknown)
• $s_2$ = specific heat capacity of calorimeter material
• $s_3$ = specific heat capacity of water

Process requires:

• Heating solid in steam chamber until temperature stabilizes
• Transferring quickly to calorimeter
• Measuring temperature change accurately
• Minimizing heat loss to environment

Searle's Cone Method determines J by converting mechanical work to heat:

Setup:

• Inner conical vessel filled with water
• Outer vessel rotates, causing friction
• Friction torque balanced by suspended weights
• Temperature rise measured

Calculation: $J = \frac{2\pi n Mgr}{(m_1s_1 + m_2s_2)(θ_2 - θ_1)}$

Where:

• $n$ = number of rotations
• $M$ = mass on hanging pan
• $g$ = acceleration due to gravity
• $r$ = radius of disc
• $m_1$ = mass of water
• $m_2$ = mass of vessels
• $s_1, s_2$ = specific heat capacities
• $θ_1, θ_2$ = initial and final temperatures

Physical principle: This demonstrates energy conservation - mechanical work (rotation against friction) converts completely to thermal energy (temperature increase), establishing the equivalence between work and heat.

Experimental value: $J = 4.186$ joules/calorie

Experimental measurement:

1. Setup components:

   • Steam generator/boiler
   • Steam trap to ensure dry steam
   • Calorimeter with water at known temperature
   • Accurate thermometers and weighing balance

2. Procedure:

   • Pass steam into calorimeter containing water
   • Measure initial and final temperatures
   • Determine mass of condensed steam

3. Calculation formula: $L = \frac{(m_1s_1 + m_2s_2)(θ_3 - θ_2)}{m_3} - s_2(θ_1 - θ_3)$

   Where:

   • $m_1$ = mass of calorimeter
   • $m_2$ = mass of water in calorimeter
   • $m_3$ = mass of condensed steam
   • $θ_1$ = temperature of steam
   • $θ_2, θ_3$ = initial and final temperatures of calorimeter

4. Necessary corrections:

   • Subtract heat released by cooling condensed water from steam temperature to final temperature
   • Account for heat lost to surroundings
   • Correct for heat capacity of calorimeter and stirrer
   • Ensure steam is dry (no water droplets)

Example: If 1.5g of steam at 100°C condenses in a calorimeter, raising water temperature from 25°C to 30°C, both the heat of condensation and the heat released during cooling from 100°C to 30°C must be accounted for.

Temperature plateaus occur because all added heat goes into phase change rather than temperature increase:

Characteristics:

• During phase transitions (melting, boiling), temperature remains constant
• Plateau duration proportional to mass and inversely proportional to heating rate
• Slope changes occur at phase transition temperatures

Information extractable:

• Melting/boiling points (plateau temperature)
• Latent heat (from plateau duration): $L = \frac{Q \times t_{plateau}}{m}$ where Q is heat input rate
• Relative quantities of substances (from plateau lengths)
• Purity of substances (pure substances have sharp, flat plateaus)

Example: A temperature-time graph for ice heated at constant rate shows:

1. Linear increase from -20°C to 0°C (solid phase)
2. Plateau at 0°C (melting)
3. Linear increase from 0°C to 100°C (liquid phase)
4. Plateau at 100°C (vaporization)
5. Linear increase above 100°C (gas phase)

Specific Heat Capacity (s):

• Definition: Heat required to raise temperature of 1 kg of substance by 1 K
• Units: J·kg⁻¹·K⁻¹ or cal·g⁻¹·°C⁻¹
• Formula: $Q = ms\Delta T$
• Depends on mass of substance

Molar Heat Capacity (C):

• Definition: Heat required to raise temperature of 1 mole of substance by 1 K
• Units: J·mol⁻¹·K⁻¹
• Formula: $Q = nC\Delta T$
• Depends on number of moles

Relationship:

• $C = sM$ where M is molar mass
• $C = s \times \frac{\text{kg}}{\text{mol}}$

Applications:

• Specific heat: Used in calorimetry calculations with known masses
• Molar heat: Used in thermochemistry and molecular-level analysis

Example: Water has s = 4186 J·kg⁻¹·K⁻¹ and C = 75.4 J·mol⁻¹·K⁻¹ because one mole of water (18g) requires 75.4J to increase by 1K.

Regnault's apparatus determines specific heat capacity through controlled heat transfer between bodies at different temperatures:

• A solid sample is heated to a steady high temperature (typically 100°C) in a steam chamber • The heated solid is rapidly transferred to a calorimeter containing water at a known lower temperature • The final equilibrium temperature is measured after heat exchange occurs • Using the equation: m₁s₁(θ₁ - θ) = m₂s₂(θ - θ₂) + m₃s₃(θ - θ₂)

Where:

• m₁, s₁, θ₁: mass, specific heat capacity, initial temperature of solid
• m₂, s₂, θ₂: mass, specific heat capacity, initial temperature of calorimeter
• m₃, s₃, θ₂: mass, specific heat capacity, initial temperature of water
• θ: final equilibrium temperature

Example: A 50g copper sample heated to 100°C is placed in 200g of water at 20°C in a calorimeter. The equilibrium temperature reaches 25°C, allowing calculation of copper's specific heat capacity.

To calculate the specific heat capacity of a solid using Regnault's apparatus data:

1. Rearrange the heat exchange equation to solve for s₁: s₁ = [(m₂s₂ + m₃s₃)(θ - θ₂)]/[m₁(θ₁ - θ)]

Where:

• s₁: specific heat capacity of the solid (unknown)
• m₁: mass of the solid
• θ₁: initial temperature of the solid (typically 100°C)
• m₂, s₂: mass and specific heat capacity of the calorimeter
• m₃, s₃: mass and specific heat capacity of water (4186 J/kg·K)
• θ₂: initial temperature of calorimeter and water
• θ: final equilibrium temperature

2. Substitute all known values into the equation and solve for s₁

Example: If a 75g aluminum sample cooled from 100°C to 28°C while 150g of water warmed from 20°C to 28°C in a 50g copper calorimeter (s = 389 J/kg·K), the calculated specific heat capacity of aluminum would be approximately 900 J/kg·K.

Note: For accurate results, minimize heat loss to surroundings by using a well-insulated calorimeter and conducting the transfer quickly.

Potential sources of error in Regnault's method and their minimization:

1. Heat loss to surroundings: • Use double-walled calorimeters with vacuum or insulating materials • Conduct experiment quickly after transferring the heated solid • Apply correction factors based on Newton's law of cooling

2. Temperature measurement errors: • Calibrate thermometers before use • Ensure proper thermal contact between thermometer and media • Use precision thermometers with 0.1°C or better resolution

3. Incomplete thermal equilibrium: • Stir water continuously during the experiment • Wait until temperature stabilizes before recording final reading • Ensure sample is fully submerged in calorimeter water

4. Heat transfer during sample transfer: • Minimize transfer time from steam chamber to calorimeter • Use a mechanism that allows rapid transfer without manual handling • Consider correction for estimated heat loss during transfer

5. Water evaporation from calorimeter: • Use a lid on the calorimeter with minimal openings • Account for mass loss in calculations if significant • Maintain laboratory humidity at moderate levels

Practical improvement: Modern implementations often replace the wooden partition with automated mechanisms to reduce transfer time from approximately 2-3 seconds to under 1 second, significantly improving accuracy.

To determine the mass of condensed steam:

1. Weigh the empty calorimeter and stirrer initially (m₁)
2. Add water and weigh again to find water mass (m₂ - m₁)
3. Record initial temperature of calorimeter and water (θ₂)
4. Pass steam into the calorimeter for a controlled period
5. Record the final temperature (θ₃)
6. Weigh the entire system (calorimeter + water + condensed steam)
7. Calculate mass of condensed steam: m₃ = (final system mass) - (m₁ + m₂)

The accuracy of this measurement is critical because:

• It directly affects the calculated value of latent heat
• Small errors in mass measurement cause significant errors in the final result
• The mass should be sufficient to cause a measurable temperature change (typically 5-10°C)
• Condensation on the exterior of the apparatus must be prevented to avoid measurement errors

Example: If initial calorimeter+water mass is 250g, and final mass is 255g, the mass of condensed steam is 5g.