Problem 90 - Introductory Chemistry Atoms First

Question

What is the coldest temperature possible in Celsius and Fahrenheit? Give your answers to an uncertainty of plus or minus one-hundredth of a degree.

Step-by-Step Solution

Answer

The coldest temperature possible is \(-273.15 \pm 0.01\) °C in Celsius and \(-459.67 \pm 0.01\) °F in Fahrenheit.

Step 1: Finding Absolute Zero in the Celsius Scale

Absolute zero in the Celsius scale is defined as the temperature at which particles have minimum possible kinetic energy. This temperature is -273.15 degrees Celsius. Given the uncertainty condition, the coldest temperature possible in Celsius will be: \(-273.15 \pm 0.01\) °C

Step 2: Converting Absolute Zero from Celsius to Fahrenheit

Now, we need to convert the absolute zero temperature from the Celsius scale to the Fahrenheit scale. The conversion formula is given by: \(F = \frac{9}{5}C + 32\) where F is the temperature in Fahrenheit and C is the temperature in Celsius. Plugging the value of absolute zero in Celsius into the formula, we get: \(F = \frac{9}{5}(-273.15) + 32\)

Step 3: Calculating the Coldest Temperature in Fahrenheit

Now we will compute the coldest temperature in Fahrenheit: \(F = (-273.15 \times \frac{9}{5}) + 32 \approx -459.67\) Given the uncertainty condition, the coldest temperature possible in Fahrenheit will be: \(-459.67 \pm 0.01\) °F

Step 4: Reporting the Results

The coldest temperature possible in both Celsius and Fahrenheit, with an uncertainty of plus or minus one-hundredth of a degree, are: - In Celsius: \(-273.15 \pm 0.01\) °C - In Fahrenheit: \(-459.67 \pm 0.01\) °F

Key Concepts

Temperature Conversion

Understanding temperature conversion is crucial for various scientific calculations, particularly when working between temperature scales. An important baseline for these conversions is absolute zero, which is defined as the temperature where all kinetic energy in atoms ceases, and it's universally accepted that absolute zero is -273.15 degrees Celsius.

To convert a temperature from Celsius to Fahrenheit, one uses the formula:
\[ F = \frac{9}{5}C + 32 \]
where F denotes Fahrenheit and C denotes Celsius. In the context of absolute zero (-273.15 °C), when we substitute this value into the formula, we can determine the equivalent in Fahrenheit. This conversion is essential for scientists and students working across regions that use different temperature scales.

Kinetic Energy

Kinetic energy is the energy that an object possesses due to its motion. It is a key concept in physics, directly related to temperature. At a microscopic level, the temperature of an object is a measure of the average kinetic energy of its particles. The lower the temperature, the less kinetic energy and therefore the less motion in the particles. As we approach absolute zero, theoretical physics suggests particles would have minimal motion, effectively reaching the lowest possible kinetic energy.

This concept is elegantly illustrated by the definition of absolute zero itself—no kinetic energy, no particle movement. In our daily lives, we don't experience temperatures anywhere close to this extreme, but understanding the relationship between temperature and kinetic energy is fundamental in areas like thermodynamics and quantum mechanics.

Uncertainty in Measurements

Measurements in science are never exact due to the limitations of measuring instruments and the inherent variability of the systems being measured. This is why measurements are often expressed with an associated uncertainty. As per the exercise, the coldest temperatures in both Celsius and Fahrenheit were expressed with an uncertainty of plus or minus one-hundredth of a degree.

This tiny range reflects a high degree of precision, which is particularly impressive considering we are discussing measurements of nearly the lowest possible temperatures in the universe. Uncertainty is a fundamental concept in scientific measurements, emphasizing that no measurement is complete without an acknowledgment of its precision. This honesty in reporting brings integrity to data and allows for meaningful comparison between different measurements and experiments.