Thermal and Chemical Effects of Electric Current

When two resistors of different resistances (R₁ and R₂) carry identical current:

• The temperature rise (ΔT) in each resistor is directly proportional to its resistance
• The ratio of temperature increases follows: ΔT₁/ΔT₂ = R₁/R₂
• Heat developed per unit time is given by H = i²R, so with current (i) being equal:
  • A resistor with twice the resistance will develop twice the heat
  • A resistor with half the resistance will develop half the heat

Example: If two copper calorimeters with equal amounts of oil contain resistors of 3Ω and 6Ω respectively, and the same current passes through both for 5 minutes, the temperature rise in the 6Ω calorimeter will be exactly twice that of the 3Ω calorimeter.

Note: This assumes identical thermal environments and heat capacities for both systems.

A Joule's calorimeter is an apparatus used to verify Joule's laws of heating, consisting of:

• A copper calorimeter filled with K-oil
• A non-conducting lid to minimize heat loss
• An immersed resistor connected to an external circuit
• A thermometer to measure temperature changes
• A stirrer to ensure uniform temperature distribution

The external circuit includes:

• A battery (power source)
• A rheostat (for current control)
• An ammeter (to measure current)
• A plug key (to complete/break the circuit)

To verify laws, experimenters:

1. Pass different currents for fixed time intervals (for H ∝ i²)
2. Pass constant current for different time intervals (for H ∝ t)
3. Use different resistors with same current and time (for H ∝ R)

The temperature rise is proportional to heat produced, allowing quantitative verification.

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.

Thomson coefficients contribute to thermoelectric EMF through:

1. Differential contribution: The term (σA - σB)(T - T₀) in the Seebeck EMF equation represents the net Thomson effect contribution

2. Physical mechanism: When current flows through regions with temperature gradients, Thomson heat is absorbed or evolved depending on the metal's Thomson coefficient

3. Direction-dependent effect:

   • For metals with positive σ (like copper): Heat is absorbed when current flows from cold to hot
   • For metals with negative σ (like iron): Heat is absorbed when current flows from hot to cold

4. Magnitude influence: The greater the difference between σA and σB, the larger the Thomson effect contribution to the total EMF

5. Temperature dependence: The contribution scales linearly with the temperature difference (T - T₀)

Note: The Thomson effect occurs throughout the conductors rather than just at junctions, making it a volumetric rather than interfacial phenomenon.

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.

The quantitative relationship in electrolysis is:

• Mass deposited (m) = (Chemical equivalent × Charge) ÷ Faraday constant
• m = (E × Q) ÷ F = (E × I × t) ÷ F

To calculate silver mass when copper mass is known:

1. Both cells receive identical charge (Q) in series connection
2. For copper: m₁ = (E₁ × Q) ÷ F
3. For silver: m₂ = (E₂ × Q) ÷ F
4. Dividing these equations: m₂/m₁ = E₂/E₁
5. Rearranging: m₂ = m₁ × (E₂/E₁)

Example calculation:

• Given 0.100g copper deposited
• Chemical equivalent of Cu = 63.5/2 = 31.75
• Chemical equivalent of Ag = 108/1 = 108
• Mass of silver = 0.100g × (108/31.75) = 0.340g

Note: This works regardless of current magnitude or electrolysis duration as long as the cells are in series.

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

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

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.