Question

Liquid nitrogen has a temperature of \(-208^{\circ} \mathrm{C}\). An orange is placed in the liquid nitrogen, which causes the liquid to boil at \(-196^{\circ} \mathrm{C}\). What is the amount of the increase in temperature?

Short answer

Answer: The increase in temperature is 12°C.

Step-by-step solution

Identify the initial and final temperatures

The initial temperature of the liquid nitrogen is -208°C, and the final temperature when the liquid starts to boil is -196°C.

Find the temperature difference

To find the temperature difference, subtract the initial temperature from the final temperature: \[-196^{\circ} \mathrm{C} - (-208^{\circ} \mathrm{C}).\]

Calculate the increase in temperature

By calculating the difference in temperatures:\[ -196^{\circ} \mathrm{C} + 208^{\circ} \mathrm{C} = 12^{\circ} \mathrm{C}.\]Therefore, the increase in temperature is 12°C.

Key concepts

Finding Temperature Difference

Understanding how to find the temperature difference is a fundamental skill in solving many physics and chemistry problems, such as calculating heat transfer or reactions' rates. Simply put, the temperature difference is the amount of thermal change between two points. This can often be visualized as the distance between two values on a thermometer.

To find the temperature difference, you'll need to subtract one temperature from another. In the case of the liquid nitrogen and orange example, we subtract the initial temperature of the nitrogen (-208^{ circ} C) from its final temperature when it starts to boil (-196^{ circ} C). You should always pay attention to the signs (positive or negative) of each temperature, as they will directly affect the result of your subtraction.

In some cases, when dealing with temperature changes, you may encounter an increase or decrease in the values depending on the context of the problem. An increase in temperature is indicated by a positive difference, while a decrease reveals a negative difference.

Negative Number Subtraction

Subtracting negative numbers can be a tricky concept to grasp, but it's a foundational component of prealgebra and important for temperature calculations. Think of it like owing money: if you have a debt (-208\() and it decreases by another debt (-196\)), you actually owe less, hence you add the second debt to the first.

In temperature terms, to subtract a negative number, you actually need to add the absolute value of that number. The subtraction of negative numbers involves changing the sign of the number you are subtracting and then adding it to the first number. The formula for this is: if you have two numbers a and b, and you want to subtract b from a, where b is negative, the operation will look like a - (-b) = a + b.

For our example, the subtraction problem -196^{ circ} C - (-208^{ circ} C) becomes -196^{ circ} C + 208^{ circ} C. This operation simplifies to 12^{ circ} C, which represents the increase in temperature as the liquid nitrogen begins to boil after the orange is added.

Prealgebra Temperature Problems

Dealing with temperature problems in prealgebra requires an understanding of the basic principles of arithmetic applied to real-world contexts, like the behavior of substances under changing temperatures. When solving these types of problems, it's helpful to imagine a number line where temperatures can move up and down, becoming warmer or cooler.

To solve prealgebra temperature problems, follow these steps: identify the given temperatures, understand whether you're looking at an increase or decrease (based on the context), and then apply the correct arithmetic operation. Always double-check if you're dealing with negative numbers and remember that increasing a negative number makes it less negative, reflecting a rise in temperature.

Incorporate these guidelines when you're solving textbook exercises, and remember: practice is key. Work through problems systematically, check your understanding of negative numbers, and always consider the real-world implications of the calculations you're performing. By doing so, you'll improve your accuracy and confidence in solving prealgebra temperature problems.