Temperature Scales & Absolute Zero

The Ultimate Cold: Absolute Zero

In our last lesson, we established that a higher temperature means more molecular motion. So, what happens if you keep cooling something down? The molecules move slower... and slower... and slower.

Logically, there must be a point where you've removed as much of this motion energy as you possibly can. A point where the molecules are nearly perfectly still. This theoretical point of minimum possible energy is called absolute zero.

It's the floor, the rock bottom of temperature. You can't get any colder.

As the article mentions, in the strange world of quantum mechanics, particles can never be perfectly still, but for our purposes, you can think of absolute zero as the state of minimum possible thermal energy. It's a fundamental limit set by the laws of physics.

This concept is so important that two of our temperature scales are built directly on top of it.

The Four Rulers: Relative vs. Absolute Scales

We use four main scales to measure temperature. It's useful to group them into two types: relative and absolute.

1. Relative Scales (Celsius & Fahrenheit): These scales set their zero points at convenient, but physically arbitrary, locations.

   • Celsius (°C): Used by most of the world and in science. It sets 0°C at the freezing point of water and 100°C at the boiling point of water (at standard atmospheric pressure).
   • Fahrenheit (°F): Used primarily in the United States. It sets the freezing point of water at 32°F and the boiling point at 212°F. Its zero point was originally based on a freezing brine mixture.

2. Absolute Scales (Kelvin & Rankine): These scales anchor their zero points at the true floor of temperature: absolute zero. This makes them essential for scientific laws.

   • Kelvin (K): This is the official SI unit for temperature. It's the go-to scale for scientists worldwide. Its zero point, 0 K, is absolute zero. You'll notice there is no "degree" symbol used with Kelvin.
   • Rankine (°R or R): This is the absolute scale that corresponds to Fahrenheit, used in some engineering fields, especially in the US. Its zero point, 0 °R, is also absolute zero.

To get a better feel for these scales and their history, let's watch a short video.

The Art of Conversion: Translating Temperatures

Since different scales are used in different contexts, being able to convert between them is a fundamental skill. We'll build the conversion formulas from intuition.

The Simple "Shift": Celsius ↔ Kelvin and Fahrenheit ↔ Rankine

Let's start with the easiest conversions. The "size" of one degree Celsius is exactly the same as the "size" of one Kelvin. The only difference is their starting point.

• Water freezes at 0°C, which is 273.15 K.
• This means the Kelvin scale is simply the Celsius scale shifted up by 273.15.

To convert from Celsius to Kelvin:
$T_K=T_{°C}+273.15$

To convert from Kelvin to Celsius:
$T_{°C}=T_K-273.15$

The exact same logic applies to Fahrenheit and Rankine. The "size" of one degree Fahrenheit is the same as one degree Rankine. The only difference is the starting point.

• Absolute zero is 0 R, which is -459.67 °F.
• This means the Rankine scale is simply the Fahrenheit scale shifted up by 459.67.

To convert from Fahrenheit to Rankine:
$T_R=T_{°F}+459.67$

To convert from Rankine to Fahrenheit:
$T_{°F}=T_R-459.67$

These are just simple shifts, like changing the floor number in an elevator without changing the height of each floor.

The "Shift and Scale": Celsius ↔ Fahrenheit

This conversion is a bit more involved because the size of the degrees is different, and the zero points are different.

Let's figure out the relationship from scratch. Look at the interval between the freezing and boiling points of water:

• In Celsius, this range is from 0°C to 100°C, which is 100 units.
• In Fahrenheit, this range is from 32°F to 212°F, which is 180 units.

The ratio of Fahrenheit units to Celsius units is $\frac{180}{100}=\frac{9}{5}=1.8$.
This means that for every 1-degree increase in Celsius, the temperature increases by 1.8 degrees in Fahrenheit. This gives us our scaling factor: 9/5.

Now we account for the different zero points. The freezing point is 0 on the Celsius scale but 32 on the Fahrenheit scale. So, we need to add a 32-degree offset.

Combining these gives us the conversion formulas:

To convert from Celsius to Fahrenheit:
$T_{°F}=\frac{9}{5}{T}_{°C}+32$

To convert from Fahrenheit to Celsius (by rearranging the formula):
$T_{°C}=\frac{5}{9}(T_{°F}-32)$

Let's watch a video that walks through this logic and gives several clear examples.

The Absolute Connection: Kelvin ↔ Rankine

Since both Kelvin and Rankine start at absolute zero, converting between them is purely a scaling operation—no shift required!

We already know that a change of 1 K is the same as a change of 1°C, and a change of 1 R is the same as a change of 1°F. We also know that a 1°C change is equal to a 9/5 °F change.

Therefore, a 1 K change must be equal to a 9/5 R change.

To convert from Kelvin to Rankine:
$T_R=\frac{9}{5}{T}_K$

An Important Note on Temperature Changes (ΔT)

This is a critical point that will save you a lot of headaches later on. When you're dealing with a change in temperature (ΔT, pronounced "delta T"), things get simpler.

• Since the size of a Celsius degree and a Kelvin are the same, a change of 10°C is exactly the same as a change of 10 K.
  $\Delta T(°C)=\Delta T(K)$
• Similarly, since the size of a Fahrenheit degree and a Rankine degree are the same, a change of 10°F is exactly the same as a change of 10 R.
  $\Delta T(°F)=\Delta T(R)$

The video below works through a simple example that makes this crystal clear.

Quick Recap

Alright, let's pause our walk and review.

• We have four main temperature scales: Celsius, Fahrenheit, Kelvin, and Rankine.
• Celsius and Fahrenheit are relative scales based on the properties of water.
• Kelvin and Rankine are absolute scales based on absolute zero, the coldest possible temperature.
• We have a set of formulas to convert between them, based on shifting their zero points and scaling the size of their degrees.
• A temperature change in Celsius is equal to a change in Kelvin. A change in Fahrenheit is equal to a change in Rankine.

Now, let's put this knowledge into practice.

Let's Practice: Problems and Questions

Time to get our hands dirty with some numbers. Follow along step-by-step.

Worked Problems

Problem 1: Normal Body Temperature

A healthy person has a body temperature of approximately 37°C. What is this temperature in Fahrenheit, Kelvin, and Rankine?

Solution:

1. To Fahrenheit:

   • Formula: $T_{°F}=\frac{9}{5}{T}_{°C}+32$
   • Calculation: $T_{°F}=\frac{9}{5}(37)+32=66.6+32=98.6°F$

2. To Kelvin:

   • Formula: $T_K=T_{°C}+273.15$
   • Calculation: $T_K=37+273.15=310.15K$

3. To Rankine: We can convert from Fahrenheit.

   • Formula: $T_R=T_{°F}+459.67$
   • Calculation: $T_R=98.6+459.67=558.27R$

Problem 2: A Cold Winter Day

The temperature in Anchorage, Alaska is -10°F. What is this temperature in Celsius?

Solution:

1. To Celsius:
   • Formula: $T_{°C}=\frac{5}{9}(T_{°F}-32)$
   • Calculation: $T_{°C}=\frac{5}{9}(-10-32)=\frac{5}{9}(-42)\approx -23.3°C$

Problem 3: Cooking a Turkey

A recipe says to roast a turkey until its internal temperature reaches 165°F. Your fancy new digital thermometer is set to Celsius. What temperature should you look for?

Solution:

1. To Celsius:
   • Formula: $T_{°C}=\frac{5}{9}(T_{°F}-32)$
   • Calculation: $T_{°C}=\frac{5}{9}(165-32)=\frac{5}{9}(133)\approx 73.9°C$

Problem 4: A Multi-Step Conversion

The surface temperature of the sun is approximately 5800 K. What is this in degrees Fahrenheit?

Solution:

There is no direct formula from K to °F, so we have to go through Celsius first.

1. Step 1: Kelvin to Celsius

   • Formula: $T_{°C}=T_K-273.15$
   • Calculation: $T_{°C}=5800-273.15=5526.85°C$

2. Step 2: Celsius to Fahrenheit

   • Formula: $T_{°F}=\frac{9}{5}{T}_{°C}+32$
   • Calculation: $T_{°F}=\frac{9}{5}(5526.85)+32=9948.33+32=9980.33°F$
   • So, the sun's surface is nearly 10,000°F!

Conceptual Questions

Lesson Summary

Excellent work today. We've mastered one of the most practical and fundamental skills in thermodynamics: navigating the language of temperature.

Here are the key takeaways:

• Absolute Zero is the theoretical minimum temperature, where particles have the least possible thermal energy.
• We use four main scales: Celsius and Fahrenheit (relative scales useful for everyday life) and Kelvin and Rankine (absolute scales essential for science and engineering).
• We have a set of simple formulas to convert between these scales.
• For temperature changes (ΔT), a change in Celsius is the same size as a change in Kelvin, and a change in Fahrenheit is the same size as a change in Rankine.

Preview of the Next Lesson

We now have a solid grip on the concepts of pressure (the force of molecular collisions) and temperature (the average energy of those molecules). We're finally ready to put them together.

In our next lesson, we will explore the beautiful relationship between a gas's pressure, volume, and temperature. This will lead us to one of the most powerful and widely used equations in all of thermodynamics: the Ideal Gas Law. This is where things really start to get interesting! See you then.