Heat — the Energy That Flows From Hot to Cold 🔥

Heat — the Energy That Flows From Hot to Cold 🔥

A subtle question 🤔: are “heat” and “temperature” the same thing? Most people say yes — but they aren’t. A cup of tea at 90°C is very hot, but its total energy is ~40 kJ. A 25°C swimming pool feels cool, but its total energy is around 5.2 gigajoules — 100,000× more, while being colder 🏊. Understanding that distinction between temperature (how hot) and heat (how much energy is exchanged) is the whole point of this section.

The core idea in one paragraph 📌

Heat (\( Q \)) is energy in transit between two bodies due to a temperature difference — not a property of a body. A body has a temperature and an internal energy; the moment that energy crosses from a hotter body to a colder one, we call it heat. The direction is always hot-to-cold (second law of thermodynamics) until they reach thermal equilibrium (same temperature). Heat required to change a substance’s temperature: \( Q = m c \Delta T \), where \( c \) is the specific heat capacity — the energy needed to raise 1 kg by 1 K. SI unit: joule (J); food-industry unit: calorie (\( 1\,\text{cal} \approx 4.184\,\text{J} \)).

The key formula 📐

\[ \boxed{Q = m \, c \, \Delta T} \]

Total heat capacity of an object: \( C = mc \), units J/K.

Specific heat capacity of common materials 📊

Material \( c \) (J/(kg·K)) Note
Liquid water 4186 Highest of common substances — key to temperate climate 🌊
Ice (0°C) 2100 Roughly half of water
Water vapour 2010
Ethanol 2440
Human body (avg) 3500 Because ~60% water
Wood 1700
Air (constant pressure) 1005
Concrete 880
Glass 840
Aluminium 900
Iron/steel 450 1/9 of water — pans heat up fast 🍳
Copper 385
Silver 235
Gold 129
Lead 128 ~3% of water

The deep takeaway: water is exceptionally high-capacity; that’s why it heats slowly, cools slowly, and acts as the planet’s thermal regulator.

Calorimetry law: exchange in a closed system 🧪

If two bodies are placed in thermal contact with no loss, heat gained = heat lost:

\[ \boxed{Q_\text{hot} + Q_\text{cold} = 0 \quad\Rightarrow\quad m_1 c_1 (T_f – T_1) + m_2 c_2 (T_f – T_2) = 0} \]

Solving for the final temperature \( T_f \) in the two-body case:

\[ T_f = \frac{m_1 c_1 T_1 + m_2 c_2 T_2}{m_1 c_1 + m_2 c_2} \]

Example 1: Tea + cold water ☕

Mix \( m_1 = 200\,\text{g} \) of tea at \( 90\,°\text{C} \) with \( m_2 = 100\,\text{g} \) of water at \( 20\,°\text{C} \). Final temperature? (Both have water’s \( c \).)

\[ T_f = \frac{0.2(90) + 0.1(20)}{0.2 + 0.1} = \frac{18 + 2}{0.3} = 66.7\,°\text{C} \]

Result: adding 50% cold water only cooled the tea by 23°, not to the naive average — because the tea has more mass.

Example 2: Why does hot metal cool fast in water? 🔥

Drop \( 500\,\text{g} \) of steel at \( 200\,°\text{C} \) into \( 500\,\text{g} \) of water at \( 20\,°\text{C} \):

\[ T_f = \frac{m_s c_s T_s + m_w c_w T_w}{m_s c_s + m_w c_w} = \frac{0.5(450)(200) + 0.5(4186)(20)}{0.5(450) + 0.5(4186)} \]

\[ = \frac{45{,}000 + 41{,}860}{225 + 2093} = \frac{86{,}860}{2318} \approx 37.5\,°\text{C} \]

The lesson: even though the steel starts at \( 200\,°\text{C} \), the final temperature is only ~37° — because water’s \( c \) is 9× that of steel.

Example 3: How much energy for a morning shower? 🚿

Heat \( 50\,\text{L} \) of water from \( 15\,°\text{C} \) to \( 40\,°\text{C} \):

\[ Q = m c \Delta T = 50 \times 4186 \times 25 = 5.23 \times 10^{6}\ \text{J} = 5.23\ \text{MJ} \]

\[ = \frac{5.23\times 10^6}{3.6\times 10^6} \approx 1.45\ \text{kWh} \]

At \( 0.15 per kWh that’s about \)0.22 per shower. Gas heaters are far cheaper because gas has a much higher energy density than electricity.

Calorie vs joule — why does food use calories? 🍫

A 200-kcal chocolate bar is ~840 kJ — enough, in principle, to warm 200 L of water by 1°C. The “diet calorie” is exactly the physics kilocalorie, just labelled misleadingly by the food industry.

Python analysis 🐍

1) Generic calorimeter — mixing two bodies

def final_temp(m1, c1, T1, m2, c2, T2):
    """Equilibrium temperature of two bodies in loss-free thermal contact."""
    return (m1*c1*T1 + m2*c2*T2) / (m1*c1 + m2*c2)

cases = [
    ("Tea + cold water",  0.2, 4186, 90, 0.1, 4186, 20),
    ("Hot steel in water", 0.5, 450, 200, 0.5, 4186, 20),
    ("Aluminium + copper", 0.3, 900, 150, 0.3, 385, 20),
    ("Hot gold in mercury", 0.05, 129, 300, 0.5, 140, 20),
]
for name, m1, c1, T1, m2, c2, T2 in cases:
    Tf = final_temp(m1, c1, T1, m2, c2, T2)
    print(f"{name:22s} → T_f = {Tf:.2f}°C")

2) Compare heat-up time of 5 materials on the same stove

import numpy as np
import matplotlib.pyplot as plt

# 1500-W stove, 1 kg sample, starting at 20°C
P = 1500          # W
m = 1.0           # kg
T0 = 20
materials = {"Water": 4186, "Oil": 2000, "Alcohol": 2440,
             "Aluminium": 900, "Iron": 450}
t = np.linspace(0, 300, 300)   # 5 minutes

for name, c in materials.items():
    # Q = m c ΔT  and  Q = P·t  →  ΔT = P·t / (m·c)
    T = T0 + P * t / (m * c)
    plt.plot(t, T, label=name)

plt.xlabel("Time (s)"); plt.ylabel("Temperature (°C)")
plt.title("Heating 1 kg on a 1500 W stove")
plt.legend(); plt.grid(alpha=0.3); plt.axhline(100, color="r", ls=":", label="Water boil")
plt.show()

3) Sea vs land diurnal temperature swing

import numpy as np, matplotlib.pyplot as plt

hours = np.arange(0, 24, 0.25)
# Sinusoidal solar radiation, peak at noon
solar = np.maximum(0, np.sin((hours - 6)/12 * np.pi)) * 800  # W/m²

# Simple thermal model: dT/dt ∝ Q_absorbed / (ρ·c·h)
def diurnal(c, rho, h_m, absorb=0.6, cool=0.15):
    T = np.zeros_like(hours)
    T[0] = 15
    for i in range(1, len(hours)):
        dt = (hours[i] - hours[i-1]) * 3600
        heating = absorb * solar[i] / (rho * c * h_m)
        cooling = cool * (T[i-1] - 15) / (rho * c * h_m) * 1e5
        T[i] = T[i-1] + (heating - cooling) * dt
    return T

T_land = diurnal(c=800,  rho=1600, h_m=0.5)   # sand
T_sea  = diurnal(c=4186, rho=1000, h_m=2.0)   # ocean (mixed layer)

plt.plot(hours, T_land, label="Land (sand)", color="orange")
plt.plot(hours, T_sea,  label="Sea",         color="steelblue")
plt.xlabel("Hour of day"); plt.ylabel("Temperature (°C)")
plt.title("Why coasts are milder than inland")
plt.legend(); plt.grid(alpha=0.3); plt.show()

Take-home summary 🎁

Heat ≠ temperature. Temperature = kinetic energy of particles; heat = energy in thermal transit. Direction is always hot-to-cold; when transit stops, we have thermal equilibrium. \( Q=mc\Delta T \) is the workhorse; in a closed system \( Q_\text{hot}+Q_\text{cold}=0 \) gives the calorimetry formula. Water’s specific heat (~4200) is exceptional — steel is 1/9 of water, which is why a pan warms instantly but the food inside doesn’t. Food-label calories are actually kilocalories ~ 4.2 kJ 🎯.

“Nice to know” box: The word “calorie” comes from a dead theory ⚗️

Before the 19th century, scientists believed heat was an invisible fluid called “caloric” that flowed from hot to cold. They even tried to weigh it! The word “calorie” is a leftover of that theory. Then in 1843, James Prescott Joule did his famous paddle-wheel experiment (a falling weight stirred water; the water got warmer by the exact amount predicted from mechanical work) and showed heat is just another form of energy — not a fluid. The SI unit of energy, “joule,” is named after the very person who killed caloric 180 years ago ✨. Yet food packaging still uses the word “calorie” from the dead theory!

Test yourself 📝

References and further exploration 📚

Articles and reference

Videos (YouTube)

External simulators

On this site 🔗

Next up we answer a fun question: why does water’s temperature stay locked at 100°C while it boils, no matter how much you crank up the flame? 🧊💧💨 See you there! 👋

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