Calorimetry

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

Analysis process:

1. Determine possible heat transfers in sequence:

   • Heat to warm ice to 0°C: $Q_1 = m_{ice} \times c_{ice} \times (0°C - T_{ice})$
   • Heat to melt ice at 0°C: $Q_2 = m_{ice} \times L_f$
   • Heat to warm melted water: $Q_3 = m_{ice} \times c_{water} \times (T_{final} - 0°C)$
   • Heat lost by warm water: $Q_4 = m_{water} \times c_{water} \times (T_{water} - T_{final})$

2. Compare available heat with required heat:

   • If $Q_4 < Q_1$: Final temperature below 0°C, no melting
   • If $Q_1 < Q_4 < (Q_1 + Q_2)$: Final temperature = 0°C, partial melting
   • If $Q_4 > (Q_1 + Q_2)$: Complete melting, final temperature above 0°C

3. For case of partial melting, calculate mass melted: $m_{melted} = \frac{Q_4 - Q_1}{L_f}$

Example: For 5g ice at -20°C added to 10g water at 30°C, first calculate heat available from water cooling to 0°C (1260J) and compare to heat needed to warm ice to 0°C (210J) plus heat to melt all ice (1680J).

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.

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.

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.

The principle involves transferring heat from condensing steam to cold water and measuring the resulting temperature change:

1. Steam at 100°C condenses to water at 100°C, releasing latent heat (L)
2. The condensed water cools from 100°C to the final mixture temperature, releasing sensible heat
3. The calorimeter and cold water absorb this heat, increasing in temperature
4. Using conservation of energy: heat lost by steam = heat gained by cold system

This is mathematically expressed as: m₃L + m₃s₂(θ₁ - θ₃) = m₁s₁(θ₃ - θ₂) + m₂s₂(θ₃ - θ₂)

Where:

• m₃ = mass of condensed steam
• L = specific latent heat of vaporization
• θ₁ = steam temperature (100°C)
• θ₃ = final temperature
• θ₂ = initial cold water temperature
• s₁, s₂ = specific heat capacities of calorimeter material and water

Solving for L gives the specific latent heat of vaporization.

In Searle's Cone apparatus, torque is balanced through two opposing forces:

1. Frictional torque (Γ) between the conical vessels acts in the direction of rotation of the outer vessel, attempting to drag the inner vessel with it.

2. Counter-torque (Mgr) is created by hanging weights on a pan connected to a cord wrapped around a grooved wooden disc attached to the inner vessel.

This balance is crucial because:

• When perfectly balanced (Γ = Mgr), the inner vessel remains stationary
• This equilibrium ensures all mechanical work from rotation converts precisely to heat energy
• Without this balance, some energy would go into rotational motion of the inner vessel
• The balanced state allows for the direct measurement of work-to-heat conversion
• The mathematical relationship: work done = Mgr·2πn (where n is number of rotations)

This precise energy accounting enables accurate calculation of J (mechanical equivalent of heat) by comparing mechanical work input to thermal energy output.

Energy transfer pathway in Searle's Cone Method:

1. Electrical/manual energy → rotation of outer vessel
2. Rotational energy → frictional work between conical vessels
3. Frictional work → heat energy
4. Heat energy → temperature increase of water and calorimeter

Potential sources of error:

• Heat loss to surrounding environment (should be minimized through insulation)
• Incomplete immersion of thermometer or uneven temperature distribution
• Inaccurate measurement of rotation count or speed variations
• Friction in the pulley system affecting the balancing weight
• Non-uniform friction between conical surfaces
• Water evaporation during longer experiments
• Heat capacity of stirrer and thermometer not accounted for
• Insufficient time for thermal equilibrium to be reached
• Variation in friction coefficient with temperature

Corrections:

• Use water equivalent of calorimeter in calculations
• Apply cooling correction factors for heat loss
• Ensure consistent rotation speed
• Verify torque balance throughout experiment
• Perform multiple trials and average results

When properly executed with corrections, this method yields J ≈ 4.186 J/cal, confirming heat as a form of energy.