Thermal and Chemical Effects of Electric Current

To experimentally verify Joule's first law (H ∝ i²):

1. Setup the apparatus:

   • Fill calorimeter with measured quantity of K-oil
   • Insert resistor, thermometer, and stirrer
   • Connect to circuit with battery, rheostat, and ammeter

2. Conduct the experiment:

   • Record initial temperature (θ₁) of K-oil
   • Close the key and adjust rheostat for current i₁
   • Maintain constant current for fixed time t
   • Stir continuously and record final temperature (θ₂)
   • Calculate temperature rise Δθ₁ = θ₂ - θ₁

3. Repeat the process:

   • Let system cool to room temperature
   • Adjust rheostat to get different current i₂
   • Pass this current for same time t
   • Measure temperature rise Δθ₂

4. Analyze results:

   • Calculate ratio Δθ₁/Δθ₂
   • Compare with ratio i₁²/i₂²
   • If Δθ₁/Δθ₂ = i₁²/i₂², then H ∝ i² is verified

The heat produced is proportional to temperature rise in the calorimeter.

The neutral temperature (θn) in a thermocouple:

• Definition: The temperature at which the thermo-emf reaches its maximum value • Mathematical significance: At θn, the derivative of thermo-emf with respect to temperature equals zero (dE/dθ = 0) • Formula: θn = -aAB/bAB (where aAB and bAB are thermocouple constants) • Relationship: Always follows θn = (θi + θc)/2, where θi is inversion temperature and θc is cold junction temperature

Experimental determination:

1. Maintain the cold junction at a known reference temperature (θc)
2. Gradually increase the hot junction temperature while measuring the thermo-emf
3. Plot thermo-emf vs. temperature and identify the peak of the curve
4. Alternatively, if inversion temperature (θi) is known, calculate using θn = (θi + θc)/2

Practical importance:

• Determines the optimal operating range for the thermocouple
• Helps select appropriate thermocouples for specific temperature ranges
• Used in calibration of thermoelectric devices
• Essential for understanding the behavior of thermocouples in precision measurements

Note: The neutral temperature depends on the specific metals used in the thermocouple and can be affected by factors like purity and heat treatment of the metals.

The Seebeck EMF (EAB) in a thermocouple is expressed as:

EAB = (ΠAB)T - (ΠAB)T₀ + (T - T₀)(σA - σB)

Where:

• (ΠAB)T represents the Peltier EMF at the hot junction at temperature T
• (ΠAB)T₀ represents the Peltier EMF at the cold junction at temperature T₀
• σA and σB are the Thomson coefficients of metals A and B
• (T - T₀) is the temperature difference between the junctions

This equation demonstrates that the Seebeck effect is a combination of two Peltier EMFs and two Thomson EMFs.

The relationship between junction temperatures, neutral temperature, and inversion temperature in a thermocouple circuit is:

Ti - Tn = Tn - T₀

Which can be rearranged as: Tn = (Ti + T₀)/2

Where:

• T₀ = Cold junction temperature
• Tn = Neutral temperature (where thermo-EMF reaches maximum value)
• Ti = Inversion temperature (where thermo-EMF becomes zero and then reverses)

Characteristics:

• Neutral temperature always lies halfway between cold junction temperature and inversion temperature
• If cold junction temperature increases, inversion temperature decreases by the same amount
• At the neutral temperature, the thermoelectric power (dE/dT) equals zero
• For a general thermocouple with EMF equation E = aθ + ½bθ², the neutral temperature equals -a/b
• The inversion temperature equals -2a/b

Example: For a copper-nickel thermocouple with cold junction at 10°C and inversion temperature at 530°C, the neutral temperature would be 270°C.

• Primary cells: Can only be discharged; used only once until their chemical energy is depleted
• Secondary cells: Can be both discharged and charged; the chemical reactions are reversible
• In both cases, during discharge, current leaves the cell at the positive terminal and enters at the negative terminal
• For secondary cells during charging, current flows in the opposite direction (enters at positive terminal, leaves at negative)

1. At zinc electrode (negative):

   • SO₄²⁻ ions move to zinc electrode
   • Form ZnSO₄, releasing electrons to zinc
   • Zinc becomes negatively charged

2. At copper electrode (positive):

   • Cu²⁺ ions move toward copper electrode
   • Accept electrons and deposit as copper metal
   • Copper becomes positively charged

3. In electrolyte:

   • H⁺ ions move through porous cup from zinc side
   • Combine with SO₄²⁻ ions in CuSO₄ solution to form H₂SO₄

4. Net result: EMF of approximately 1.09V is generated between electrodes

At negative electrode:

• PbSO₄ + 2H⁺ + 2e⁻ → Pb + H₂SO₄
• Lead sulfate converts back to pure lead

At positive electrode:

• PbSO₄ + SO₄²⁻ + 2H₂O → PbO₂ + 2H₂SO₄
• Lead sulfate converts back to lead dioxide

Overall effects:

• PbSO₄ deposits are dissolved from both electrodes
• Original electrode materials (Pb and PbO₂) are restored
• H₂SO₄ concentration increases (specific gravity rises)
• Cell's EMF returns to fully charged value of approximately 2.05V

Purpose of amalgamation:

• Zinc is treated with mercury to form a zinc-mercury amalgam on the surface

Benefits:

• Prevents local action (unwanted chemical reactions at impurity sites)
• Reduces zinc consumption when the cell is not in use
• Creates more uniform electrode surface
• Extends cell life

Without amalgamation:

• Impurities in zinc form tiny galvanic cells with the zinc itself
• These micro-cells cause zinc to dissolve even when the cell is not in use
• Hydrogen bubbles form on impurity sites
• Cell's capacity is reduced through wasteful zinc consumption
• Shorter overall cell lifespan

At Zinc (Negative) Electrode:

• SO₄²⁻ ions combine with zinc to form ZnSO₄
• Negative charge transfers to zinc electrode
• Reaction: Zn + SO₄²⁻ → ZnSO₄ + 2e⁻

At Copper (Positive) Electrode:

• Cu²⁺ ions from CuSO₄ solution receive electrons
• Copper deposits on copper electrode
• Reaction: Cu²⁺ + 2e⁻ → Cu

Polarization Prevention:

• The H⁺ ions move through porous barrier from H₂SO₄ solution
• They combine with SO₄²⁻ in CuSO₄ solution to form H₂SO₄
• This arrangement prevents hydrogen gas accumulation at the anode
• Without this mechanism, hydrogen buildup would create an opposing potential, reducing cell effectiveness (polarization)
• The continuous chemical cycle maintains stable electrode potentials

Fundamental Comparison

| Property | Joule Heating | Peltier Effect | |----------|-------------------|---------------------| | Location | Throughout entire conductor | Only at junctions between different materials | | Direction | Always produces heat | Can heat OR cool depending on current direction | | Mathematical relationship | $H_{Joule} = i^2Rt$ (proportional to $i^2$) | $H_{Peltier} = \Pi_{AB} \cdot i \cdot t$ (proportional to $i$) | | Physical cause | Collisions between charge carriers and lattice atoms | Energy transfer when carriers cross material boundary | | Reversibility | Irreversible (energy degradation) | Reversible (can be undone) | | Current direction dependence | Independent of current direction | Reverses effect when current direction reverses |

Experimental Identification Methods

1. Current Direction Test:

   • Pass current through a junction of two different materials
   • Measure temperature change
   • Reverse current direction
   • If heating switches to cooling (or vice versa): Peltier effect
   • If heating occurs in both directions: Joule heating

2. Current Magnitude Test:

   • Vary current magnitude and measure temperature change
   • Plot temperature vs. current
   • Linear relationship (∝ $i$): Primarily Peltier effect
   • Quadratic relationship (∝ $i^2$): Primarily Joule heating

Note: Both effects occur simultaneously in real circuits, but their different dependencies on current allow them to be distinguished experimentally.