Problem 95 - Chemistry The Molecular Nature of Matter

Question

There exists a single temperature at which the value reported in \({ }^{\circ} \mathrm{F}\) is numerically the same as the value reported in \(^{\circ} \mathrm{C}\). What is this temperature?

Step-by-Step Solution

Answer

-40 degrees is the temperature at which both Celsius and Fahrenheit scales are equal.

Step 1: Write down the conversion formula

The formula to convert Fahrenheit to Celsius is: \(C = \frac{5}{9}(F - 32)\).

Step 2: Set the Celsius and Fahrenheit values equal

Set the temperature in Celsius equal to the temperature in Fahrenheit since they are numerically the same at this point: \(C = F\).

Step 3: Substitute the value of C in the conversion formula

Substitute \(C\) with \(F\) in the conversion formula: \(F = \frac{5}{9}(F - 32)\).

Step 4: Solve for Fahrenheit

Now solve the equation \(F = \frac{5}{9}(F - 32)\) for the value of \(F\). First multiply both sides by 9 to get rid of the fraction: \(9F = 5(F - 32)\).

Step 5: Expand and simplify the equation

Expand the right side of the equation: \(9F = 5F - 160\). Then bring all the \(F\) terms to one side: \(9F - 5F = -160\).

Step 6: Solve for F

Combine the \(F\) terms and divide by the coefficient: \(4F = -160\), then \(F = \frac{-160}{4} = -40\). The temperature at which Celsius and Fahrenheit are the same is \(-40^\circ\).

Key Concepts

Celsius to Fahrenheit Conversion

Understanding how to convert temperatures between the Celsius and Fahrenheit scales is a practical skill that can help us interpret weather reports, cook recipes from different countries, and even tackle scientific problems. The formula to convert a temperature from Fahrenheit (\( F \)) to Celsius (\( C \)) is given by: \[\begin{equation}C = \frac{5}{9}(F - 32)\end{equation}\]This equation arises from the relationship between the two scales. To convert from Celsius to Fahrenheit, we can use the inverse of this formula:\[\begin{equation}F = \frac{9}{5}C + 32\end{equation}\]The reason for the '32' in these formulas is due to the offset between the two scales - the freezing point of water is 0 degrees Celsius, but 32 degrees Fahrenheit.

When working with these conversions, it's important to accurately apply mathematical operations such as multiplication, division, and addition or subtraction of constants. For example, in the exercise, the conversion is used to find a unique temperature where Celsius and Fahrenheit values are the same. It's an interesting intersection point that occurs at -40 degrees, showing a direct numeric equivalence on both scales.

Solving Linear Equations

Linear equations form the backbone of algebra and offer a gateway to understanding a variety of complex mathematical concepts. To solve a linear equation, we aim to isolate the variable (usually represented as 'x' or 'y' - in our case, it's the temperature 'F') on one side of the equation. This involves performing the same mathematical operations on both sides of the equation to maintain equality.

Solving the exercise's linear equation began with substituting one temperature variable for another, followed by multiplication to eliminate fractions, and then using addition or subtraction to collect like terms. Finally, we divide both sides by the coefficient of the variable to solve for 'F'. The steps taken in the exercise demonstrate a systematic approach to simplifying and resolving a linear equation, showcasing a method that can be applied to countless similar problems in algebra.

Temperature Scales

The concept of temperature scales is important for understanding how we measure the warmth or coldness of an object or environment. There are several scales used worldwide, with Celsius (\( ^\r\circ C \)) and Fahrenheit (\( ^\r\circ F \)) being two of the most common. The Celsius scale, based on the freezing and boiling points of water at 0 and 100 degrees respectively, is widely used in science and by most countries around the world for day-to-day temperature reporting.

The Fahrenheit scale, on the other hand, sets the freezing point of water at 32 degrees and the boiling point at 212 degrees. It's primarily used in the United States and some Caribbean countries. There are also other scales like Kelvin, which is used in scientific measurements, where 0 Kelvin is absolute zero - the theoretical lowest possible temperature.

The exercise provided offers insight into the relationship between these temperature scales, specifically the point at which Celsius and Fahrenheit readings converge. It's a fascinating detail that not only reflects on the history and development of thermometry but also on the interconnected nature of different measurement systems.