Problem 73 - Chemistry: Principles and Reactions

Question

At what point is the temperature in \({ }^{\circ} \mathrm{F}\) exactly twice that in \({ }^{\circ} \mathrm{C} ?\)

Step-by-Step Solution

Answer

Answer: At 160°C and 320°F.

Step 1: Write down the formula to convert Celsius to Fahrenheit

The formula to convert Celsius to Fahrenheit is: °F = (9/5) * °C + 32

Step 2: Create an equation relating the Fahrenheit and Celsius temperatures

According to the problem, we want to find the point where °F is twice the value of °C, so we can write the equation: °F = 2 * °C

Step 3: Substitute the conversion formula for Fahrenheit into the equation

Now we can substitute the °F conversion formula into the equation: (9/5) * °C + 32 = 2 * °C

Step 4: Solve the equation for Celsius temperature

Subtract (9/5) * °C from both sides of the equation to isolate the Celsius term and simplify the equation: 32 = 2 * °C - (9/5) * °C 32 = (1/5) * °C To solve for °C, multiply both sides by 5: °C = 160

Step 5: Convert the Celsius temperature back into Fahrenheit

Use the conversion formula to find the Fahrenheit temperature when the Celsius temperature is 160: °F = (9/5) * °C + 32 °F = (9/5) * 160 + 32 °F = 288 + 32 °F = 320 At the point where Fahrenheit is exactly twice the Celsius temperature, the temperature is 160° Celsius and 320° Fahrenheit.

Key Concepts

Celsius and Fahrenheit relationship

Understanding the relationship between Celsius and Fahrenheit is fundamental when dealing with temperature conversions. These are two of the most commonly used temperature scales in scientific, medical, and everyday contexts. Celsius, also known as centigrade, is primarily used in most parts of the world. Fahrenheit is commonly used in the United States.

The conversion between Celsius and Fahrenheit can be conducted using a simple formula, which allows you to express a temperature in Fahrenheit (\( ^{\circ}\mathrm{F} \)) based on a given Celsius temperature (\( ^{\circ}\mathrm{C} \)). The standard conversion formula is:

• \[ ^{\circ} \mathrm{F} = \left( \frac{9}{5} \right) \times ^{\circ} \mathrm{C} + 32 \]

This formula comes from the fact that each degree Celsius represents a larger interval than each degree Fahrenheit. Thus, the conversion factor of \( \frac{9}{5} \) scales the Celsius temperature to the appropriate Fahrenheit level, and then we shift the baseline by 32 to align with the scale's zero points.

Temperature formulas

Temperature formulas play a crucial role in converting values from one temperature scale to another. When working with temperatures and equations, understanding these formulas is essential.

In most chemistry problems, such as the one involving temperature convergence, formulas like the Celsius to Fahrenheit conversion come into play. Here's the specific formula:

• \[ ^{\circ} \mathrm{F} = \left( \frac{9}{5} \right) \times ^{\circ} \mathrm{C} + 32 \]

This formula enables us to translate Celsius values to Fahrenheit, making it easier to comprehend temperature readings in different contexts. When you have a problem where you must find a Fahrenheit temperature that is twice the Celsius temperature, equations leveraging these conversions are indispensable.

For example, in the original exercise, the task was to find when Fahrenheit is double Celsius. Using the conversion, \( \text{°F} = 2 \times \text{°C} \), you create an equation to solve for the critical point where this relationship stands.

Solving equations in chemistry

Solving equations in chemistry often involves the application of basic algebra to understand how different variables relate to each other. Let's take a closer look at applying equations during a temperature conversion problem.

When determining the point at which one temperature scale equals twice another, like in the case of Celsius and Fahrenheit, you start by setting up an equation. For the problem at hand, we have \( \text{°F} = 2 \times \text{°C} \).

Next, substitute the Celsius to Fahrenheit formula into this equation:

• \[ \left( \frac{9}{5} \right) \times ^{\circ} \mathrm{C} + 32 = 2 \times ^{\circ} \mathrm{C} \]

Simplifying and solving this linear equation lets you find the unknown Celsius temperature. As seen in the solution, you rearrange the formula to isolate \( ^{\circ} \mathrm{C} \), then solve. This method highlights the usefulness of algebra in chemistry, especially when dealing with interrelated variables like temperature scales.

Finally, you verify the result by converting the solved Celsius back to Fahrenheit to ensure consistency with the problem statement.