Question

At what temperature do the Kelvin and Fahrenheit scales have the same numeric value?

Short answer

Answer: The Kelvin and Fahrenheit scales have the same numeric value at approximately -574.6°F.

Step-by-step solution

Write down the formulas for converting Fahrenheit to Kelvin

To convert Fahrenheit to Kelvin, we use the following formula: K = (F - 32) * 5/9 + 273.15

Set the two temperature values equal to each other

Since we want to find the temperature when Kelvin and Fahrenheit have the same numeric value, we set them equal to each other: K = F

Substitute the Fahrenheit to Kelvin conversion formula into the equation

Now replace K in the equation with the conversion formula: (F - 32) * 5/9 + 273.15 = F

Solve for F

To solve for F, first subtract F from both sides of the equation: (F - 32) * 5/9 + 273.15 - F = 0 Next, multiply both sides by 9 to eliminate the fraction: 5(F - 32) + 9(273.15) - 9F = 0 Distribute and combine like terms: 5F - 160 + 2458.35 - 9F = 0 Combine like terms: -4F + 2298.35 = 0 Now, isolate F by dividing by -4: F = 2298.35 / -4 F = -574.5875 Since F is a whole number in most scales, we can round it to -574.6. So, the temperature at which the Kelvin and Fahrenheit scales have the same numeric value is approximately -574.6°F.

Key concepts

Kelvin and Fahrenheit Scales

Understanding the Kelvin and Fahrenheit scales is essential for students tackling temperature conversions in science and everyday life.

The Kelvin scale is the base unit of temperature in the International System of Units (SI) and is widely used in scientific contexts. It is an absolute scale with its zero point at absolute zero, the theoretical lowest possible temperature. This means that on the Kelvin scale, there are no negative numbers because zero Kelvin (0 K) is as cold as it could theoretically get. The Kelvin scale increments are the same size as those of the Celsius scale, with every 1 degree increase corresponding to the same amount of thermal energy change as 1 degree Celsius.

The Fahrenheit scale, on the other hand, is commonly used in the United States for everyday temperature measurements. The scale was developed based on earlier temperature scales and sets the freezing point of water at 32°F and the boiling point at 212°F under standard atmospheric conditions. Unlike Kelvin, Fahrenheit does include negative numbers, as temperatures can go below the freezing point of water.

When converting between these scales, understanding their starting points and increments is key. For instance, in Fahrenheit, temperatures are higher numerically when compared to Kelvin for the same thermal state.

Temperature Scale Comparison

Comparing temperature scales involves examining how they define temperature intervals and reference points. As mentioned earlier, the Kelvin scale starts at absolute zero, with 0 K being the coldest possible temperature. The Celsius scale, which is related to Kelvin, sets the freezing point of water at 0°C and the boiling point at 100°C.

The Fahrenheit scale has a different set of reference points. Its lower fixed point, the freezing point of water, is 32°F, while the higher fixed point, the boiling point of water, is at 212°F. Therefore, a temperature interval on the Fahrenheit scale represents a smaller change in heat than the same interval on the Kelvin or Celsius scales.

To facilitate comparisons and conversions, it is often required to use formulaic relationships between scales. The exercise in solving for temperature equality requires just such a formulaic approach. It is also worth noting that while these scales are different, they are all linearly related, meaning there is a direct and constant ratio in the temperature intervals they measure.

Solving for Temperature Equality

Solving for temperature equality between the Kelvin and Fahrenheit scales involves finding when the numerical values of both scales match. This requires an equation that relates the two scales.

The process begins by equating the Kelvin temperature to the Fahrenheit temperature. Students are then tasked with manipulating the equation using the conversion formula. This task not only tests their understanding of the temperature scales but also their algebraic skills.

In the step-by-step solution, algebraic manipulation highlights the process of combining like terms, isolating variables, and using inverse operations. Students learn that by applying these strategies, they can find the precise point at which the scales have the same value. The solution reveals that at approximately -574.6°F, the Kelvin and Fahrenheit scales display equivalent numeric values. This unusual temperature highlights the vast difference between the zero points of the two scales and gives students insight into the concept of absolute zero in the Kelvin scale. It is an excellent example of how different scientific scales interact and a practical application of algebra in science.

This exercise offers an opportunity to improve understanding through the logical step of converting one temperature scale to another and finding a common value, a useful skill in various scientific applications.

Practical Implications

The exercise goes beyond theoretical knowledge, as students can appreciate the practical implications of temperature conversion in real-world scenarios, such as climate studies, cooking, and even space exploration.