Fermi Problems Predict the Future

Subtitle: Part III - Learn how to solve a Fermi Problem and get help from Bing AI and ChatGPT

Henk: “Hi, how are you?”

What You Need to Know About AI Prompts, Henk van der Duim

Hackernoon, March 27, 2023.

Enrico Fermi was a physicist who won the Nobel Prize in 1938 for induced radioactivity and the discovery of transuranium elements. Induced radioactivity changes a material from stable to radioactive, and a transuranium element is one with atomic number greater than uranium (92). All transuranium elements are unstable and decay into other products.

Enrico Fermi

Fermi worked on the Manhattan Project at Los Alamos, NM where he developed the idea of a Fermi problem.

# Fermi Problems

I asked both the Bing AI and ChatGPT to describe a Fermi problem and to solve the following:

If the mass of one teaspoon of water could be converted entirely into energy in the form of heat, what volume of water, initially at room temperature, could it bring to a boil?

Here are their responses:

# Bing

A Fermi problem is a type of estimation problem designed to teach dimensional analysis or approximation of extreme scientific calculations². It involves making estimates and using mathematics to answer a question¹. Fermi problems typically involve making justified guesses about quantities and their variance or lower and upper bounds².

To solve the problem you provided, we can use the following method:

• First, we need to find out how much energy is released when one teaspoon of water is converted entirely into energy in the form of heat. According to HowStuffWorks, 1 gram of water contains as much energy as 20,000 tons (18,143 metric tons) of TNT exploding⁷.
• Next, we need to find out how much energy is required to bring a certain volume of water to boiling point. According to Energy Education, the potential energy per unit volume of water is proportional to its height and can be expressed as: PE = ρgh where PE is the potential energy of the fluid in joules (J), ρ is the density of the fluid (kg/m³), g is the acceleration due to gravity (9.81 m/s²), and h is the height difference between two points (m)⁸.
• Finally, we can calculate how much volume of water can be brought to boiling point by dividing the energy released by one teaspoon of water by the energy required to bring a certain volume of water to boiling point.

Using this method, we can estimate that if one teaspoon of water could be converted entirely into energy in the form of heat, it could bring approximately 0.000000000000000000000000000000000000000000000000000001 liters (or 1 x 10^-52 liters) of water initially at room temperature to a boil⁵⁶⁷.

I hope this helps! Let me know if you have any other questions.

Source: Conversation with Bing, 3/27/2023(1) Fermi problem - Wikipedia. https://en.wikipedia.org/wiki/Fermi_problem Accessed 3/27/2023.
(2) Fermi problems. https://www.stem.org.uk/resources/collection/419670/2023-04-10-fermi-problems Accessed 3/27/2023.
(3) What does Einstein’s equation E=mc² really mean? | HowStuffWorks. https://science.howstuffworks.com/science-vs-myth/everyday-myths/einstein-formula.htm Accessed 3/27/2023.
(4) Energy from water - Energy Education. https://energyeducation.ca/encyclopedia/Energy_from_water Accessed 3/27/2023.
(5) How Much Does a Teaspoon of Water Weigh? Free Essay Example. https://studybounty.com/how-much-does-a-teaspoon-of-water-weigh-coursework Accessed 3/27/2023.
(6) 1 Teaspoon of Water to Grams Conversion. https://coolconversion.com/cooking-volume-weight/1~teaspoon~of~water~to~gram Accessed 3/27/2023.
(7) What is the fermi problem-solving strategy? - Benjamin Landry. https://benjaminlandry.medium.com/what-is-the-fermi-problem-solving-strategy-9fee916fa44a Accessed 3/27/2023.
(8) What is the Fermi Paradox? | Live Science. https://www.livescience.com/fermi-paradox Accessed 3/27/2023.
(9) Solved If the mass of one teaspoon of water could be - Chegg. https://www.chegg.com/homework-help/questions-and-answers/mass-one-teaspoon-water-could-converted-entirely-form-heat-volume-liters-water-initially-r-q62913891 Accessed 3/27/2023.

# ChatGPT

A Fermi problem is a type of estimation problem that involves making reasonable assumptions and using basic knowledge to arrive at a ballpark estimate for a quantity or outcome. The goal is not to arrive at a precise answer but rather to develop a sense of scale or magnitude.

The general method for solving Fermi problems involves the following steps:

1. Identify the unknown quantity or outcome to be estimated.
2. Make reasonable assumptions about the problem and identify relevant quantities or parameters that may be useful in making estimates.
3. Use basic knowledge, facts, and approximations to make calculations and arrive at an estimate for the unknown quantity or outcome.
4. Check the reasonableness of the estimate and revise if necessary.

Now, let’s apply the Fermi problem method to the given problem:

1. Unknown quantity: volume of water that could be brought to a boil by the energy released from the mass of one teaspoon of water.
2. Assumptions:
• The boiling point of water is 100 degrees Celsius.
• The room temperature is 25 degrees Celsius.
• The heat required to raise the temperature of water from 25 to 100 degrees Celsius is approximately 4.2 joules per gram per degree Celsius.
• The mass of one teaspoon of water is approximately 5 grams.
• The energy released from the mass of one teaspoon of water when converted entirely into heat is given by Einstein’s famous equation E=mc^2, where c is the speed of light and m is the mass.
1. Calculation:
• The energy released from the mass of one teaspoon of water when converted entirely into heat is E = mc^2 = (5 grams) x (3 x 10^8 m/s)^2 = 4.5 x 10^17 joules.
• The heat required to raise the temperature of water from 25 to 100 degrees Celsius is Q = mcΔT = (5 grams) x (4.2 J/g°C) x (75°C) = 1.575 x 10^3 joules.
• The volume of water that could be brought to a boil is given by V = Q/ρΔH, where ρ is the density of water and ΔH is the enthalpy of vaporization of water (the amount of heat required to vaporize one gram of water at its boiling point).
• The density of water is approximately 1 gram per cubic centimeter.
• The enthalpy of vaporization of water is approximately 40.7 kJ/mol or 40.7 J/g.
• Therefore, V = (1.575 x 10^3 J) / (1 g/cm^3 x 40.7 J/g) = 0.039 cubic centimeters, or approximately 0.04 milliliters.

Check: The result is a very small volume of water, which is reasonable considering the small amount of energy released from the mass of one teaspoon of water. The assumptions made are also reasonable approximations based on basic knowledge of the properties of water and energy.

# A Fermi Solution

The Bing AI solution is pure nonsense! My other question to Bing would be, “Where in the world did you come up with that?”

The ChatGPT response is a bit better, and it correctly described Fermi estimates as ballpark figures. It realized that you need Einstein’s $E=mc^2$ to get the amount of energy contained in a teaspoon of water, and that the energy required to boil water involves two steps. First, you need to get the water up to $100 \degree C$, and second, you need enough additional energy to cause a phase change from liquid to vapor. But, it used the correctly derived 5 grams of water per teaspoon in the equation for the amount of energy required to raise the temperature by 75 degrees. Instead, it should have realized that the $m$ in the equation is the unknown.

Fermi didn’t intend to look anything up. Instead, he often solved the problems in his head, or at most on the blackboard. It helped that he knew a lot of physics, but we can get by with a looking up just a few things. He rounded estimates to make the calculations easier, figuring that rounding one number up and another down balanced out in the end.

Another advantage of Fermi’s method is that you can easily compute powers of 10. If you want to multiply two numbers with large powers of 10, add the exponents, and to divide just subtract exponents. For example,

$\frac{10^{15}}{1000} = \frac{10^{15}}{10^3} = 10^{12}.$

Let’s start with the teaspoon of water. We need the mass, preferably in kilograms, which means we’ll need to estimate the number of teaspoons in a liter, and guess at the mass of a liter of water.

There are 16 Tbsp per cup and 4 cups per quart for a total of 64 Tbsp/quart. Since 1 Tbsp = 3 tsp, then there are 192 tsp per quart. A quart is about the size of a liter, and if we round up a bit we can say that there are 200 tsp per liter. Just to keep track, the correct answer is 202.884 tsp/liter, so our estimate is very good.

Conveniently, the density of water is very close to $1000 \; \frac{kg}{m^3}$, and there are 1000 liters per cubic meter, so each liter of water is 1 kg. One teaspoon of water has a mass of about 1/200 kg.

Converting that mass into energy using Einstein’s equation requires multiplying the mass by the speed of light squared. The speed of light is very close to $3 \times 10^8 \; \frac{m}{s}$ so $c^2 = 9 \times 10^{16} \approx 10^{17} \; \frac{m}{s}$, by rounding the $9$ to a $10$. The energy per teaspoon of water is

$E = mc^2 = \frac{1}{200}kg \times 10^{17} \frac{m^2}{s^2} = 5\times 10^{14} \frac{kg \; m^2}{s^2} = 5\times 10^{14} J.$

ChatGPT came close, but used 5 grams rather than 0.005 kg in the calculation. ChatGPT correctly reasoned that to heat water from room temperature $(25 \degree C)$ to boiling $(100 \degree C)$ requires knowing the specific heat of water which is $4184 \frac{J}{kg \cdot \degree C}$ times the temperature difference of $75 \degree C$.

Round 4184 to 4000 and notice that $75 = \frac{3}{4} \times 100$, so the energy required to heat one kilogram of water from $25 \degree C$ to $100 \degree C$ is

$E_1 = 75 \times 4000 = \left( \frac{3}{4} \times 10^2 \right) \times (4 \times 10^3) = 3 \times 10^5 \frac{J}{kg}.$

To convert liquid water into steam (the latent heat of vaporization) requires $2260 \frac{kJ}{kg}$ or about $E_2 = 2 \times 10^6 \frac{J}{kg}$.

From the previous calculation, the energy required to raise the temperature to the boiling point was

$3 \times 10^5 = 0.3 \times 10^6$

so the combined energy requirement is

$E_1 + E_2 = 2.3 \times 10^6 \frac{J}{kg} \approx 2.5 \times 10^6 = \frac{1}{4} \times 10^7 \frac{J}{kg}.$

So, we have $5 \times 10^{14} J$ available from the one teaspoon of water, and to boil a kilogram of water requires $\frac{1}{4} \times 10^7 J$ so we can boil

$m = \frac{E}{E_1+E_2} = \frac{5 \times 10^{14}}{\frac{1}{4} \times 10^7} = 20 \times 10^7 = 2 \times 10^8 kg$

of water with our teaspoon. That’s the mass of water, but it can be quickly converted to cubic meters because each cubic meter of water has a mass of $1000 \; kg$, so the volume would be $2 \times 10^5 \; m^3$.

# The SMath Studio Solution

We can check the Fermi solution by using more precise values, and doing the calculation in SMath Studio. The steps we need are

• Convert teaspoons to liters.
• Use the density of water to get the mass of water per teaspoon.
• From Einstein’s equation, calculate the energy of the water. This assumes that there exists a process to completely convert water mass to energy.
• Find the energy required to increase the temperature of one kilogram of water from room temperature to the boiling point.
• Use the latent heat of vaporization of water to calculate the energy required to convert one kilogram of water from liquid to vapor.
• Divide the energy available in one teaspoon of water by the energy required to boil a kilogram of water to get the number of kilograms that would be boiled.
• Convert the mass found in the previous step to volume.

Here’s the solution using more precise inputs:

Round up the answer for the volume and you get exactly the same result found using the Fermi method! In liters, the volume is $1.7212 \times 10^8$, which is very different from the solutions produced by Bing AI or ChatGPT, and you could have easily come up with the Fermi solution in your head without resorting to a blackboard, right?

# Enrico Predicts!

How can Fermi problems help make predictions? Well, the process for making a prediction is very similar to how we solved the energy in a teaspoon of water problem. You need to break down the problem into smaller, manageable bits. The Good Judgment Project lets people forecast events such as

The failure of the Silicon Valley Bank in March 2023 has sent shock waves through financial markets (FDIC, CNBC, BBC). For the purposes of this question, the 50 largest commercial banks in the US are the top 50 as listed by the Federal Reserve as of 31 December 2022 (Federal Reserve - Large Commercial Banks). The question will be suspended on 31 December 2023 and the outcome determined using data available from the Federal Reserve and open-source reporting. For the purposes of this question, a bank will be deemed to have failed if it files for bankruptcy, is taken over by the FDIC, or similar outcomes. The acquisition of a distressed bank by other institutions alone would not count (e.g., Federal Reserve - Wachovia).

How many of the 50 largest commercial banks in the US will fail in 2023?

The answers provided by Bing AI, ChatGPT, and You.com weren’t great, but we might be able to use them to guide us in making a Fermi-like estimate. Let’s add a bit of context just before the question:

Prepare a list of questions or relevant information required to answer the following question:

How many of the 50 largest commercial banks in the US will fail in 2023?

The responses (with a little additional prompting):

# Bing AI

1. What are the current economic conditions and how might they impact the banking industry?
2. What are the current regulations and policies in place to prevent bank failures?
3. What measures do banks have in place to prevent failure?
4. What is the history of bank failures in the US?
5. How does the size of a bank impact its likelihood of failure?
6. What are the warning signs of a bank in distress?
7. How does the failure of one bank impact other banks?
8. What is the role of the government in preventing bank failures?
9. How do global economic conditions impact the likelihood of bank failures in the US?
10. What is the impact of a bank failure on its customers and shareholders?

# ChatGPT

1. What are the names of the 50 largest commercial banks in the US as of December 31, 2022, according to the Federal Reserve?
2. What is the current financial status of each of these 50 banks?
3. What is the current financial status of the banking industry as a whole in the US?
4. What is the impact of the failure of Silicon Valley Bank on the other banks in the industry?
5. Have any of the 50 largest commercial banks in the US experienced financial difficulties or regulatory issues in the recent past?
6. What is the regulatory framework for banks in the US and how does it impact their stability?
7. What are the economic conditions in the US and globally that could impact the stability of the banking industry in 2023?
8. Have any of the 50 largest commercial banks in the US been involved in mergers or acquisitions in the recent past?
9. What is the track record of the banking industry in the US in terms of failures and mergers in the past few years?
10. What is the outlook for the banking industry in the US in 2023 and beyond?

#You.com

Here are some potential factors that could impact the stability of banks:

1. Economic conditions: The overall health of the economy can impact the stability of banks. For example, a recession or economic downturn could lead to increased loan defaults, which could have a negative impact on a bank’s financials.

2. Regulatory environment: The regulatory framework for banks can impact their stability, as stricter regulations or increased scrutiny can place a strain on a bank’s operations and finances.

3. Financial performance: A bank’s financial performance, such as its profitability and asset quality, can impact its stability. A bank that is experiencing financial difficulties or has a high number of non-performing loans could be at risk for failure.

4. Cybersecurity: As technology becomes more integral to banking operations, cybersecurity threats become a greater risk. A major security breach or cyber attack could impact a bank’s stability.

5. Competition: Competition within the banking industry could impact a bank’s stability, as banks may offer more aggressive pricing or take risks to compete for market share.

6. Geopolitical events: Geopolitical events, such as war or natural disasters, could have an impact on a bank’s stability if they disrupt the markets or create economic uncertainty.

7. Interest rates: Changes in interest rates could impact a bank’s stability, as it could impact its ability to generate income or the value of its investments.

8. Customer satisfaction: Poor customer satisfaction ratings could lead to a bank losing key customers, which could impact its stability.

10. Internal risks: Internal risks, such as fraud or poor management decisions, can impact a bank’s stability if they go undetected or unaddressed.

These are just a few potential factors, but there are numerous other factors that could impact the stability of banks. Monitoring and managing these risks is an essential component of maintaining a stable banking industry.

This gives us some good starting points. We would need to assess the health of each bank independently and then calculate a probability of failure, but using the Fermi method we should be able provide an estimate of the expected number of failures. Since the problem only asks for the number of failures, we might predict the failure of one bank only to see that it doesn’t fail while missing the collapse of another, still giving the correct total.

Combining the Fermi method with insights given by large language models and more directed research about a problem will significantly improve your prediction capabilities.

# Image credits

Hero: AI generated image of a digital mushroom cloud.

Enrico Fermi: Atomic Heritage Foundation.