Question

A centigrade and a Fahrenheit thermometer one dipped in boiling water. The water temperature is lowered until the Fahrenheit thermometer registers \(140^{\circ}\). What is the fall in temperatures as registered by the centigrade thermometer? (A) \(30^{\circ}\) (B) \(40^{\circ}\) (C) \(60^{\circ}\) (D) \(80^{\circ}\)

Short answer

The fall in temperatures as registered by the centigrade thermometer is \(\boxed{(B) \: 40^{\circ}}\).

Step-by-step solution

Relationship between Fahrenheit and Celsius

The relationship between Fahrenheit (F) and Celsius (C) temperatures can be expressed as: \[F = \frac{9}{5}C + 32\]

Determine Initial Boiling Point Temperature

The boiling point of water occurs at a temperature of \(100^{\circ}\) Celsius. We can convert this to Fahrenheit using the formula above: \[F = \frac{9}{5}(100) + 32 = 212^{\circ} \text{ Fahrenheit}\]

Calculate Final Temperature in Celsius

The final temperature in Fahrenheit is given as \(140^{\circ}\). To find the corresponding Celsius temperature, we can use the above formula and re-arrange it to solve for C: \[C = \frac{5}{9}(F - 32)\] Then, plug in \(F = 140^{\circ}\): \[C = \frac{5}{9}(140 - 32) = 60^{\circ} \text{ Celsius}\]

Find the Fall in Temperatures

Now we know both the initial Celsius temperature (\(100^{\circ}\)) and the final Celsius temperature (\(60^{\circ}\)). Let's find the difference to have the fall in temperature: \[\text{Fall in temperature} = 100^{\circ}C - 60^{\circ}C = 40^{\circ}C\] Therefore, the fall in temperatures as registered by the centigrade thermometer is: \[\boxed{(B) \: 40^{\circ}}\]

Key concepts

Celsius to Fahrenheit conversion

Temperature scales are essential for measuring heat and cold. Two of the most commonly used scales are Celsius (°C) and Fahrenheit (°F). Understanding the conversion between these two scales is crucial for various scientific and daily applications. The conversion formula allows us to translate temperatures from one scale to the other with ease.
To convert a Celsius temperature to Fahrenheit, use the formula:

• \[ F = \frac{9}{5}C + 32 \]

This formula shows how many degrees to adjust the Celsius reading to change it into Fahrenheit. A conversion, for example, will show that a typical room temperature of 25°C is equivalent to 77°F.
Visa-versa, converting Fahrenheit to Celsius involves rearranging the equation:

• \[ C = \frac{5}{9}(F - 32) \]

So the temperature conversion process is straightforward if you just follow the formulas. By substituting the given temperature into these equations, you can easily switch between the Celsius and the Fahrenheit scales.

temperature change

Temperature change refers to the difference in temperature measured over time or between two different states. It’s crucial in many scientific disciplines, including chemistry and physics, and it's often important in cooking and weather forecasting as well.
In this particular exercise, we calculated how much the temperature drops when water cools from its boiling point. You start by knowing the initial and final temperatures.
The formula for finding the change in temperature is simple:

• \[ \text{Change} = \text{Initial Temperature} - \text{Final Temperature} \]

By finding this difference, you can measure how much the temperature has increased or decreased. Remember, a positive result indicates a temperature drop, meaning it has cooled down. Conversely, a negative result would mean a temperature rise, showing heating.
This is a beneficial concept to understand as it clarifies how energy is transferred in various processes.

boiling point of water

The boiling point of water is a fundamental concept in chemistry. It is the temperature at which water changes from a liquid to a gas (steam). This transformation occurs at a specific and predictable temperature under typical atmospheric conditions. For water, this point is precisely at 100°C (Celsius). In Fahrenheit, the boiling point is 212°F.
Understanding this concept helps us in numerous real-world situations. For instance, cooking or when calculating energy requirements for heating water in scientific experiments. It’s crucial while learning about phase changes in states of matter too.
At higher altitudes, the air pressure is lower, which can lower the boiling point of water. So altitude must be considered in processes that rely on precise temperatures.
Knowing the boiling point is not just about numbers; it's understanding the behavior of substances at different energy levels. This is important in various fields such as meteorology, food science, and environmental science.