Newton’s law of cooling experiment with a live graph (Pico W + 2 DS18B20 probes)

Parts list

Part	Qty	Approx.	Buy
Raspberry Pi Pico W (MicroPython installed)	1	₹620	Buy Raspberry Pi Pico W (MicroPython installed)
DS18B20 waterproof probe (1 m cable)	2	₹240	Buy DS18B20 waterproof probe (1 m cable)
4.7 kΩ resistor	1	₹1	Buy 4.7 kΩ resistor
Breadboard + jumper wires	1	₹90	Buy Breadboard + jumper wires
Steel tumbler or paper cup + thermocol lid (for a second run)	1	₹30	Buy Steel tumbler or paper cup + thermocol lid (for a second run)
Micro-USB cable + 5 V adapter or power bank	1	₹150	Buy Micro-USB cable + 5 V adapter or power bank
Total		₹1,131

Affiliate disclosure: “Buy” links may earn FlashReady a small commission at no extra cost to you. Prices are indicative.

Wiring

Every connection
Raspberry Pi Pico W	Part · pin	Wire	Notes
GP22	DS18B20 (1-Wire) · DQ	yellow	Hot-water probe (yellow or white wire) — physical pin 29
3V3	DS18B20 (1-Wire) · DQ	red	Via 4.7 kΩ — One pull-up for the whole 1-Wire bus
3V3	DS18B20 (1-Wire) · VDD	red
GND	DS18B20 (1-Wire) · GND	black
GP22	DS18B20 (1-Wire) #2 · DQ	yellow	Room-air probe on the same bus
3V3	DS18B20 (1-Wire) #2 · VDD	red
GND	DS18B20 (1-Wire) #2 · GND	black

Board quirks: Use hot water from a kettle or geyser, around 70–80 °C — not boiling, and an adult pours it. The DS18B20 probe’s steel tube is fine in water up to about 100 °C, but the cable’s PVC sleeve can soften above that. The probe that is warmer at power-up is taken as the hot-water probe. Put the hot probe in the cup before switching on (or open /reset after it is in). One 4.7 kΩ pull-up from DQ to 3V3 serves both probes — they share the 1-Wire bus and are told apart by their unique 64-bit ROM codes. Fanless still air gives the cleanest curve. A ceiling fan speeds cooling (a bigger k) — which makes a good second experiment. Without Wi-Fi settings the Pico still logs: the CSV lines appear in Thonny’s shell and can be copied into a spreadsheet.

Code

main.pyMicroPythonOpen in Playground Flash

# Newton's law of cooling experiment - Raspberry Pi Pico W (MicroPython) + 2 waterproof DS18B20 probes.
# One probe sits in a cup of hot water, the other in the room air. Every 5 s the Pico W logs both
# temperatures. It fits T(t) = Ta + (T0 - Ta) * e^(-k t) by a straight-line fit of ln(T - Ta) against t,
# and serves a live graph at http://<pico-ip>/ (CSV at /data.csv, new run at /reset).
# Without Wi-Fi it still logs: the CSV lines appear in Thonny's shell.
# Both probes: DQ -> GP22 (one 4.7 kOhm pull-up to 3V3 for the bus), VDD -> 3V3, GND -> GND.
from machine import Pin
import onewire
import ds18x20
import network
import socket
import time
import math
import json

SAMPLE_S = 5
MAX_POINTS = 720  # 1 hour at 5 s
MIN_DIFF_C = 2.0  # points closer than this to room temperature are left out of the fit (noise)

WIFI_SSID = "your-wifi"
WIFI_PASS = "your-password"

# The FlashReady flasher saves your settings to /settings.json; they win over the lines above.
try:
    with open("/settings.json") as f:
        _s = json.load(f)
except (OSError, ValueError):
    _s = {}
WIFI_SSID = _s.get("wifi_ssid") or WIFI_SSID
WIFI_PASS = _s.get("wifi_pass") or WIFI_PASS

led = Pin("LED", Pin.OUT)
bus = ds18x20.DS18X20(onewire.OneWire(Pin(22)))
roms = bus.scan()
print("DS18B20 probes found:", len(roms))
hot_rom = room_rom = None

t0 = time.ticks_ms()
times, hot, room = [], [], []


def read_both():
    """Returns (hot, room) in deg C; room is None with only one probe."""
    global hot_rom, room_rom
    bus.convert_temp()
    time.sleep_ms(750)
    vals = [bus.read_temp(r) for r in roms]
    if hot_rom is None and vals:
        # The warmer probe at the start is the one in the hot water.
        i = vals.index(max(vals))
        hot_rom = roms[i]
        room_rom = roms[1 - i] if len(roms) > 1 else None
        print("Hot-water probe:", hot_rom.hex(), " room probe:", room_rom.hex() if room_rom else "none")
    h = vals[roms.index(hot_rom)]
    r = vals[roms.index(room_rom)] if room_rom else None
    return h, r


def fit():
    """(k per minute, T0 fitted, Ta used, points used) or None."""
    if not hot:
        return None
    ta = sum(room) / len(room) if room and room[0] is not None else min(hot) - 1
    xs, ys = [], []
    for t, h in zip(times, hot):
        if h - ta >= MIN_DIFF_C:
            xs.append(t / 60)
            ys.append(math.log(h - ta))
    n = len(xs)
    if n < 5:
        return None
    sx, sy = sum(xs), sum(ys)
    sxx = sum(x * x for x in xs)
    sxy = sum(x * y for x, y in zip(xs, ys))
    den = n * sxx - sx * sx
    if den == 0:
        return None
    slope = (n * sxy - sx * sy) / den
    icpt = (sy - slope * sx) / n
    return -slope, ta + math.exp(icpt), ta, n


def csv():
    yield "t_s,hot_c,room_c\n"
    for t, h, r in zip(times, hot, room):
        yield "%d,%.2f,%s\n" % (t, h, "" if r is None else "%.2f" % r)


PAGE = """<!doctype html><meta name=viewport content="width=device-width"><title>Cooling curve</title>
<style>body{font-family:sans-serif;margin:16px}canvas{width:100%;max-width:720px;border:1px solid #ccc}</style>
<h2>Newton's law of cooling</h2><p id=s>Loading...</p><canvas id=c width=720 height=360></canvas>
<p><a href=/data.csv>Download CSV</a> &middot; <a href=/reset>Start a new run</a></p>
<script>
async function u(){const d=await (await fetch('/data.json')).json();const c=document.getElementById('c'),g=c.getContext('2d');
g.clearRect(0,0,720,360);if(!d.t.length){return}const all=d.hot.concat(d.room.filter(x=>x!==null));
const lo=Math.floor(Math.min(...all))-2,hi=Math.ceil(Math.max(...all))+2,tm=Math.max(60,d.t[d.t.length-1]);
const X=t=>40+t/tm*670,Y=v=>340-(v-lo)/(hi-lo)*330;g.fillStyle='#888';g.fillText(hi+' C',2,14);g.fillText(lo+' C',2,340);
g.fillText((tm/60).toFixed(1)+' min',660,355);
const line=(a,col)=>{g.strokeStyle=col;g.beginPath();a.forEach((v,i)=>{if(v===null)return;i?g.lineTo(X(d.t[i]),Y(v)):g.moveTo(X(d.t[i]),Y(v))});g.stroke()};
line(d.hot,'#e11d48');line(d.room,'#2563eb');
if(d.k){g.strokeStyle='#16a34a';g.setLineDash([5,4]);g.beginPath();for(let t=0;t<=tm;t+=tm/60){const v=d.ta+(d.t0-d.ta)*Math.exp(-d.k*t/60);t?g.lineTo(X(t),Y(v)):g.moveTo(X(t),Y(v))}g.stroke();g.setLineDash([])}
document.getElementById('s').textContent='Hot '+d.hot[d.hot.length-1].toFixed(1)+' C, room '+(d.room[d.room.length-1]??'-')+' C, '+d.t.length+' points'+(d.k?', k = '+d.k.toFixed(4)+' per min (fit on '+d.n+' points)':'');}
u();setInterval(u,5000);
</script>"""


def data_json():
    f = fit()
    d = {"t": times, "hot": hot, "room": room}
    if f:
        d.update(k=f[0], t0=f[1], ta=f[2], n=f[3])
    return json.dumps(d)


def serve(server):
    global t0, full
    try:
        conn, _ = server.accept()
    except OSError:
        return
    try:
        conn.settimeout(2)
        req = conn.recv(512).decode()
        path = req.split(" ")[1] if " " in req else "/"
        if path.startswith("/data.json"):
            conn.send("HTTP/1.0 200 OK\r\nContent-Type: application/json\r\n\r\n")
            conn.send(data_json())
        elif path.startswith("/data.csv"):
            conn.send("HTTP/1.0 200 OK\r\nContent-Type: text/csv\r\nContent-Disposition: attachment; filename=cooling.csv\r\n\r\n")
            for line in csv():
                conn.send(line)
        elif path.startswith("/reset"):
            times.clear()
            hot.clear()
            room.clear()
            t0 = time.ticks_ms()
            full = False
            print("New run started")
            conn.send("HTTP/1.0 303 See Other\r\nLocation: /\r\n\r\n")
        else:
            conn.send("HTTP/1.0 200 OK\r\nContent-Type: text/html\r\n\r\n")
            conn.send(PAGE)
    except OSError:
        pass
    finally:
        conn.close()


server = None
if WIFI_SSID == "your-wifi":
    print("Web page off - no Wi-Fi name set. Flash from flashready.in or type it into main.py. Logging to the shell only.")
else:
    wlan = network.WLAN(network.STA_IF)
    wlan.active(True)
    wlan.connect(WIFI_SSID, WIFI_PASS)
    for _ in range(40):
        if wlan.isconnected():
            break
        time.sleep(0.5)
    if wlan.isconnected():
        print("Open http://%s/ on a phone or laptop on the same Wi-Fi" % wlan.ifconfig()[0])
        server = socket.socket()
        server.setsockopt(socket.SOL_SOCKET, socket.SO_REUSEADDR, 1)
        server.bind(socket.getaddrinfo("0.0.0.0", 80)[0][-1])
        server.listen(2)
        server.settimeout(0.2)
    else:
        print("Wi-Fi not found - logging to the shell only")

if not roms:
    print("No DS18B20 found - check DQ on GP22, the 4.7 kOhm pull-up to 3V3, VDD and GND")
    while True:
        led.toggle()
        time.sleep(0.2)

print("t_s,hot_c,room_c")
next_sample = time.ticks_ms()
full = False
while True:
    if server:
        serve(server)
    if time.ticks_diff(time.ticks_ms(), next_sample) < 0:
        continue
    next_sample = time.ticks_add(next_sample, SAMPLE_S * 1000)
    led.on()
    h, r = read_both()
    led.off()
    if len(times) < MAX_POINTS:
        times.append(time.ticks_diff(time.ticks_ms(), t0) // 1000)
        hot.append(round(h, 2))
        room.append(None if r is None else round(r, 2))
        print("%d,%.2f,%s" % (times[-1], h, "" if r is None else "%.2f" % r))
        f = fit()
        if f and len(times) % 12 == 0:  # once a minute
            print("fit: k = %.4f /min, T0 = %.1f C, Ta = %.1f C, %d points" % f)
    elif not full:
        full = True
        print("Run full (1 hour) - open /reset to start again")

# Newton's law of cooling experiment - Raspberry Pi Pico W (MicroPython) + 2 waterproof DS18B20 probes.
# One probe sits in a cup of hot water, the other in the room air. Every 5 s the Pico W logs both
# temperatures. It fits T(t) = Ta + (T0 - Ta) * e^(-k t) by a straight-line fit of ln(T - Ta) against t,
# and serves a live graph at http://<pico-ip>/ (CSV at /data.csv, new run at /reset).
# Without Wi-Fi it still logs: the CSV lines appear in Thonny's shell.
# Both probes: DQ -> GP22 (one 4.7 kOhm pull-up to 3V3 for the bus), VDD -> 3V3, GND -> GND.
from machine import Pin
import onewire
import ds18x20
import network
import socket
import time
import math
import json

SAMPLE_S = 5
MAX_POINTS = 720  # 1 hour at 5 s
MIN_DIFF_C = 2.0  # points closer than this to room temperature are left out of the fit (noise)

WIFI_SSID = "your-wifi"
WIFI_PASS = "your-password"

# The FlashReady flasher saves your settings to /settings.json; they win over the lines above.
try:
    with open("/settings.json") as f:
        _s = json.load(f)
except (OSError, ValueError):
    _s = {}
WIFI_SSID = _s.get("wifi_ssid") or WIFI_SSID
WIFI_PASS = _s.get("wifi_pass") or WIFI_PASS

led = Pin("LED", Pin.OUT)
bus = ds18x20.DS18X20(onewire.OneWire(Pin(22)))
roms = bus.scan()
print("DS18B20 probes found:", len(roms))
hot_rom = room_rom = None

t0 = time.ticks_ms()
times, hot, room = [], [], []


def read_both():
    """Returns (hot, room) in deg C; room is None with only one probe."""
    global hot_rom, room_rom
    bus.convert_temp()
    time.sleep_ms(750)
    vals = [bus.read_temp(r) for r in roms]
    if hot_rom is None and vals:
        # The warmer probe at the start is the one in the hot water.
        i = vals.index(max(vals))
        hot_rom = roms[i]
        room_rom = roms[1 - i] if len(roms) > 1 else None
        print("Hot-water probe:", hot_rom.hex(), " room probe:", room_rom.hex() if room_rom else "none")
    h = vals[roms.index(hot_rom)]
    r = vals[roms.index(room_rom)] if room_rom else None
    return h, r


def fit():
    """(k per minute, T0 fitted, Ta used, points used) or None."""
    if not hot:
        return None
    ta = sum(room) / len(room) if room and room[0] is not None else min(hot) - 1
    xs, ys = [], []
    for t, h in zip(times, hot):
        if h - ta >= MIN_DIFF_C:
            xs.append(t / 60)
            ys.append(math.log(h - ta))
    n = len(xs)
    if n < 5:
        return None
    sx, sy = sum(xs), sum(ys)
    sxx = sum(x * x for x in xs)
    sxy = sum(x * y for x, y in zip(xs, ys))
    den = n * sxx - sx * sx
    if den == 0:
        return None
    slope = (n * sxy - sx * sy) / den
    icpt = (sy - slope * sx) / n
    return -slope, ta + math.exp(icpt), ta, n


def csv():
    yield "t_s,hot_c,room_c\n"
    for t, h, r in zip(times, hot, room):
        yield "%d,%.2f,%s\n" % (t, h, "" if r is None else "%.2f" % r)


PAGE = """<!doctype html><meta name=viewport content="width=device-width"><title>Cooling curve</title>
<style>body{font-family:sans-serif;margin:16px}canvas{width:100%;max-width:720px;border:1px solid #ccc}</style>
<h2>Newton's law of cooling</h2><p id=s>Loading...</p><canvas id=c width=720 height=360></canvas>
<p><a href=/data.csv>Download CSV</a> &middot; <a href=/reset>Start a new run</a></p>
<script>
async function u(){const d=await (await fetch('/data.json')).json();const c=document.getElementById('c'),g=c.getContext('2d');
g.clearRect(0,0,720,360);if(!d.t.length){return}const all=d.hot.concat(d.room.filter(x=>x!==null));
const lo=Math.floor(Math.min(...all))-2,hi=Math.ceil(Math.max(...all))+2,tm=Math.max(60,d.t[d.t.length-1]);
const X=t=>40+t/tm*670,Y=v=>340-(v-lo)/(hi-lo)*330;g.fillStyle='#888';g.fillText(hi+' C',2,14);g.fillText(lo+' C',2,340);
g.fillText((tm/60).toFixed(1)+' min',660,355);
const line=(a,col)=>{g.strokeStyle=col;g.beginPath();a.forEach((v,i)=>{if(v===null)return;i?g.lineTo(X(d.t[i]),Y(v)):g.moveTo(X(d.t[i]),Y(v))});g.stroke()};
line(d.hot,'#e11d48');line(d.room,'#2563eb');
if(d.k){g.strokeStyle='#16a34a';g.setLineDash([5,4]);g.beginPath();for(let t=0;t<=tm;t+=tm/60){const v=d.ta+(d.t0-d.ta)*Math.exp(-d.k*t/60);t?g.lineTo(X(t),Y(v)):g.moveTo(X(t),Y(v))}g.stroke();g.setLineDash([])}
document.getElementById('s').textContent='Hot '+d.hot[d.hot.length-1].toFixed(1)+' C, room '+(d.room[d.room.length-1]??'-')+' C, '+d.t.length+' points'+(d.k?', k = '+d.k.toFixed(4)+' per min (fit on '+d.n+' points)':'');}
u();setInterval(u,5000);
</script>"""


def data_json():
    f = fit()
    d = {"t": times, "hot": hot, "room": room}
    if f:
        d.update(k=f[0], t0=f[1], ta=f[2], n=f[3])
    return json.dumps(d)


def serve(server):
    global t0, full
    try:
        conn, _ = server.accept()
    except OSError:
        return
    try:
        conn.settimeout(2)
        req = conn.recv(512).decode()
        path = req.split(" ")[1] if " " in req else "/"
        if path.startswith("/data.json"):
            conn.send("HTTP/1.0 200 OK\r\nContent-Type: application/json\r\n\r\n")
            conn.send(data_json())
        elif path.startswith("/data.csv"):
            conn.send("HTTP/1.0 200 OK\r\nContent-Type: text/csv\r\nContent-Disposition: attachment; filename=cooling.csv\r\n\r\n")
            for line in csv():
                conn.send(line)
        elif path.startswith("/reset"):
            times.clear()
            hot.clear()
            room.clear()
            t0 = time.ticks_ms()
            full = False
            print("New run started")
            conn.send("HTTP/1.0 303 See Other\r\nLocation: /\r\n\r\n")
        else:
            conn.send("HTTP/1.0 200 OK\r\nContent-Type: text/html\r\n\r\n")
            conn.send(PAGE)
    except OSError:
        pass
    finally:
        conn.close()


server = None
if WIFI_SSID == "your-wifi":
    print("Web page off - no Wi-Fi name set. Flash from flashready.in or type it into main.py. Logging to the shell only.")
else:
    wlan = network.WLAN(network.STA_IF)
    wlan.active(True)
    wlan.connect(WIFI_SSID, WIFI_PASS)
    for _ in range(40):
        if wlan.isconnected():
            break
        time.sleep(0.5)
    if wlan.isconnected():
        print("Open http://%s/ on a phone or laptop on the same Wi-Fi" % wlan.ifconfig()[0])
        server = socket.socket()
        server.setsockopt(socket.SOL_SOCKET, socket.SO_REUSEADDR, 1)
        server.bind(socket.getaddrinfo("0.0.0.0", 80)[0][-1])
        server.listen(2)
        server.settimeout(0.2)
    else:
        print("Wi-Fi not found - logging to the shell only")

if not roms:
    print("No DS18B20 found - check DQ on GP22, the 4.7 kOhm pull-up to 3V3, VDD and GND")
    while True:
        led.toggle()
        time.sleep(0.2)

print("t_s,hot_c,room_c")
next_sample = time.ticks_ms()
full = False
while True:
    if server:
        serve(server)
    if time.ticks_diff(time.ticks_ms(), next_sample) < 0:
        continue
    next_sample = time.ticks_add(next_sample, SAMPLE_S * 1000)
    led.on()
    h, r = read_both()
    led.off()
    if len(times) < MAX_POINTS:
        times.append(time.ticks_diff(time.ticks_ms(), t0) // 1000)
        hot.append(round(h, 2))
        room.append(None if r is None else round(r, 2))
        print("%d,%.2f,%s" % (times[-1], h, "" if r is None else "%.2f" % r))
        f = fit()
        if f and len(times) % 12 == 0:  # once a minute
            print("fit: k = %.4f /min, T0 = %.1f C, Ta = %.1f C, %d points" % f)
    elif not full:
        full = True
        print("Run full (1 hour) - open /reset to start again")

Compiled for picow on micropython (mpy-cross v1.29.0) · 3 Nov 2026

How it works

Newton’s law of cooling says that a hot object cools at a rate proportional to how much hotter it is than its surroundings. That gives an exponential curve, T(t) = Tₐ + (T₀ − Tₐ)·e^(−kt). In the school lab it is usually a thermometer, a stopwatch and a lot of writing; here two digital probes record every 5 seconds and the Pico W draws the curve live on any phone on the same Wi-Fi.

How it works

Two DS18B20 probes share one data wire (the 1-Wire bus on GP22). Each DS18B20 has a unique 64-bit address, so the Pico W can read them separately. The one that is warmer at power-up is taken as the hot-water probe; the other measures the room air Tₐ.

Every 5 seconds the Pico W reads both probes and stores the values (up to one hour). It serves a small web page that redraws the graph every 5 seconds: the red line is the water, the blue line the room, and the dashed green line the fitted law.

The fit. Take the log of the law: ln(T − Tₐ) = ln(T₀ − Tₐ) − k·t. Plotting ln(T − Tₐ) against time gives a straight line with slope −k. The Pico W fits that line by least squares, leaving out points less than 2 °C above room temperature (there, sensor noise dominates). It shows k per minute; the shell prints it once a minute too.

/data.csv downloads every point for your lab file; /reset starts a new run.

What to show at the exhibition

Run two cups side by side on two Pico Ws (or one after the other): one open, one with a thermocol lid. Visitors watch the curves separate within a few minutes. Show the fitted k for each and ask: “Which cup keeps tea hot longer, and by how much?”

Questions the examiner may ask

Why is the curve exponential? The rate of heat loss is proportional to the temperature difference. As the water cools, the difference shrinks, so it cools more and more slowly.
Why take the logarithm? It turns the exponential into a straight line, and the slope of a straight line is easy to fit and to understand.
Why measure room temperature too? Tₐ is in the law. A wrong Tₐ bends the log plot and changes k.
Does the law hold exactly? Roughly, for small temperature differences. At high temperatures, evaporation and radiation make the early cooling faster than the simple law predicts — you can often see the first few minutes stray from the fitted line.

Cost in India

About ₹1,100. Waterproof DS18B20 probes cost ₹100–150 each; the Pico W ₹600–650. Without Wi-Fi, the same build works as a USB data logger with Thonny.

Troubleshooting

“No DS18B20 found”

Check the pull-up between DQ and 3V3, and the probe colours: usually red = VDD, black = GND, yellow (or white) = DQ. Some cheap probes use other colours — check the seller’s sheet.

A probe reads 85.0 °C at the start

85 °C is the DS18B20’s power-on value, returned if a reading is taken before a conversion finishes. The sketch waits 750 ms per conversion; if you still see it, the supply is weak — use short wires.

The page doesn’t open

The phone or laptop must be on the same Wi-Fi as the Pico W. Use the address printed in Thonny’s shell, with http:// (not https://).

k changes a lot between runs

It depends on the cup, the amount of water, a lid, draughts and stirring. Keep everything the same and change one thing per run — that is the experiment.

Upgrades

Compare cups: steel, glass, paper and a flask — which has the smallest k?
Add a third probe and cool two cups at once (one with a lid).
Save each run to the Pico’s flash as a CSV so you can download it later.
Show the live temperature on an SSD1306 OLED for use without a phone.

Parts list

Wiring

Code

How it works

How it works

What to show at the exhibition

Questions the examiner may ask

Cost in India

Troubleshooting

Upgrades

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