🔥 Calorimetry: The Art of Heat Accounting

The Big Idea (In One Sentence)

Calorimetry is like keeping a heat bank account—what one thing loses, another thing gains. The total always stays the same!

🌡️ What is Calorimetry?

Imagine you have a piggy bank. If you take coins out and give them to your friend, you have less, but your friend has more. The total coins stay the same!

Heat works the same way.

When hot things and cold things meet:

• The hot thing loses heat (like giving away coins)
• The cold thing gains heat (like receiving coins)
• The total heat stays the same (nothing is lost!)

This is called Calorimetry—the science of measuring heat transfer.

⚖️ Principle of Calorimetry

The Golden Rule

Heat Lost = Heat Gained

When two objects at different temperatures touch:

• The hot object cools down (loses heat)
• The cold object warms up (gains heat)
• They keep exchanging until they reach the same temperature

Real Life Example 🍵

You pour hot milk into cold coffee:

• Hot milk gives away heat → gets cooler
• Cold coffee takes heat → gets warmer
• Both end up at the same temperature in the middle!

graph TD
    A[🔥 Hot Object] -->|Loses Heat| C[🤝 Meet]
    B[❄️ Cold Object] -->|Gains Heat| C
    C --> D[🌡️ Same Temperature!]

The Math Behind It

Heat lost by hot object = Heat gained by cold object

Q_lost = Q_gained

m₁ × c₁ × (T₁ - Tᶠ) = m₂ × c₂ × (Tᶠ - T₂)

Where:

• m = mass (how much stuff)
• c = specific heat (how easily it heats up)
• T = temperature
• Tᶠ = final temperature (where they meet)

🧪 Method of Mixtures

What Is It?

The Method of Mixtures is a clever trick scientists use to find out the specific heat of a substance.

Think of it like this: If you know how much heat water gains, you can figure out how much heat the mystery object lost!

The Recipe 📝

Step 1: Heat up a metal block (like copper or iron)

Step 2: Drop it into cold water in a container called a calorimeter

Step 3: Wait until everything reaches the same temperature

Step 4: Use the heat equation to find what you need!

Example: Finding Specific Heat of a Metal

Setup:

• Metal block: 200g at 100°C
• Water: 500g at 20°C
• Final temperature: 25°C

Solution:

Heat lost by metal = Heat gained by water

m_metal × c_metal × ΔT_metal
= m_water × c_water × ΔT_water

200 × c_metal × (100-25)
= 500 × 4.2 × (25-20)

200 × c_metal × 75 = 500 × 4.2 × 5

c_metal = 10500 ÷ 15000
c_metal = 0.7 J/g°C

The metal’s specific heat is 0.7 J/g°C (that’s like iron!)

Why a Calorimeter? 🥤

A calorimeter is like a super insulated coffee cup:

• It traps all the heat inside
• No heat escapes to the room
• Makes our measurements accurate

🌊 Mixing of Substances

Mixing Two Liquids at Different Temperatures

When you mix two amounts of the same liquid at different temperatures, finding the final temperature is easy!

Formula:

Tᶠ = (m₁T₁ + m₂T₂) ÷ (m₁ + m₂)

Example: Mixing Water 💧

You mix:

• 200g of water at 80°C (hot)
• 300g of water at 20°C (cold)

What’s the final temperature?

Tᶠ = (200 × 80 + 300 × 20) ÷ (200 + 300)
Tᶠ = (16000 + 6000) ÷ 500
Tᶠ = 22000 ÷ 500
Tᶠ = 44°C

The mixture settles at 44°C!

Notice: It’s closer to 20°C than 80°C because there’s more cold water.

Mixing Different Substances

When mixing different substances (like metal in water), we use the full equation:

m₁c₁(T₁ - Tᶠ) = m₂c₂(Tᶠ - T₂)

Example: Hot Metal in Cold Oil 🛢️

• Iron piece: 100g at 200°C (c = 0.5 J/g°C)
• Oil: 400g at 30°C (c = 2.0 J/g°C)

Find the final temperature:

Heat lost by iron = Heat gained by oil

100 × 0.5 × (200 - Tᶠ) = 400 × 2.0 × (Tᶠ - 30)

50 × (200 - Tᶠ) = 800 × (Tᶠ - 30)

10000 - 50Tᶠ = 800Tᶠ - 24000

10000 + 24000 = 800Tᶠ + 50Tᶠ

34000 = 850Tᶠ

Tᶠ = 40°C

The final temperature is 40°C!

🎯 Quick Mental Model

Think of temperature like a seesaw:

graph TD
    A[🔥 Hot Side<br>Goes DOWN] --- B[⚖️ Balance Point<br>Final Temperature]
    B --- C[❄️ Cold Side<br>Goes UP]

• More mass = more influence
• Higher specific heat = more stubborn (harder to change)
• They always meet somewhere in the middle!

🧠 Key Takeaways

🌟 The Big Picture

Calorimetry is everywhere:

• 🍳 Cooking (heat from stove to food)
• 🧊 Cooling drinks (ice absorbs heat)
• 🏠 Heating homes (radiators share heat)
• 🌡️ Your body (maintaining 37°C)

You’re now a Heat Accountant! You understand that energy is never lost or created—it just moves around, and we can track every bit of it!

📊 Formula Summary

┌─────────────────────────────────────┐
│  Heat Energy: Q = m × c × ΔT       │
├─────────────────────────────────────┤
│  Principle: Q_lost = Q_gained      │
├─────────────────────────────────────┤
│  Same substance mixing:            │
│  Tᶠ = (m₁T₁ + m₂T₂) ÷ (m₁ + m₂)   │
├─────────────────────────────────────┤
│  Different substances:             │
│  m₁c₁(T₁-Tᶠ) = m₂c₂(Tᶠ-T₂)        │
└─────────────────────────────────────┘

You’ve got this! 🚀