Problem 110 - Chemistry

Question

A \(125 \mathrm{~mL}\) sample of water at \(293.2 \mathrm{~K}\) was heated for \(8 \mathrm{~min}, 25 \mathrm{~s}\) so as to give a constant temperature increase of \(3.0{ }^{\circ} \mathrm{F} / \mathrm{min}\). What is the final temperature of the water in degrees Celsius?

Step-by-Step Solution

Answer

The final temperature is approximately \(34.03^{\circ}\mathrm{C}\).

Step 1: Calculate the total heating time

First, convert the total heating time from minutes and seconds into seconds. We have 8 minutes and 25 seconds, which is equivalent to:\[\text{Total time} = 8 \times 60 + 25 = 505 \text{ seconds}\]

Step 2: Find the total temperature increase

The problem states that the temperature increases by \(3.0^{\circ} \mathrm{F}/\text{min}\). We need to convert the rate from minutes to seconds because the time was converted to seconds:\[\text{Rate in seconds} = \frac{3.0^{\circ} \mathrm{F}}{60 \text{ seconds}} = 0.05^{\circ}\mathrm{F}/\text{sec}\]Thus, the total temperature increase in Fahrenheit is:\[\text{Total increase} = 0.05^{\circ}\mathrm{F}/\text{sec} \times 505 \text{ seconds} = 25.25^{\circ}\mathrm{F}\]

Step 3: Calculate the initial temperature in Fahrenheit

Use the formula for converting temperature from Kelvin to Fahrenheit:\[T(^{\circ}\mathrm{F}) = \frac{9}{5}T(\mathrm{K}) - 459.67\]Substitute the initial temperature\(T = 293.2\mathrm{~K}\):\[T_{\text{initial}}= \frac{9}{5} \times 293.2 - 459.67\approx 68.0^{\circ}\mathrm{F}\]

Step 4: Calculate the final temperature in Fahrenheit

Add the total temperature increase to the initial temperature in Fahrenheit:\[T_{\text{final}}= 68.0^{\circ}\mathrm{F} + 25.25^{\circ}\mathrm{F} = 93.25^{\circ}\mathrm{F}\]

Step 5: Convert the final temperature to Celsius

Use the formula for converting temperature from Fahrenheit to Celsius:\[T(^{\circ}\mathrm{C}) = \frac{5}{9}(T(^{\circ}\mathrm{F}) - 32)\]Substitute the final temperature:\[T_{\text{final}}= \frac{5}{9}(93.25 - 32) \approx 34.03^{\circ}\mathrm{C}\]

Key Concepts

Understanding Temperature Increase Calculation

When looking at changes in temperature, it's important to know how to calculate the total temperature increase over time. For this, you need to know the rate of temperature change. In our problem, the water's temperature increased by \(3.0^{\circ}\mathrm{F}/\mathrm{min}\). To find the total temperature increase, the rate must correlate with the time, which is often measured in a different unit like seconds.
To do this conversion, calculate the rate per second, which is \(0.05^{\circ}\mathrm{F}/\mathrm{sec}\). This is done by dividing the per-minute rate by 60 seconds. Then, multiply this rate with the total time in seconds (505 seconds here).
This way, you find that the temperature increase totals \(25.25^{\circ}\mathrm{F}\). So, understanding how to align your units is key to mastering temperature increase calculations.

Converting Kelvin to Fahrenheit

Kelvin is the base unit of temperature in the International System of Units (SI), while Fahrenheit is used predominantly in the United States. Transitioning between these two systems can be crucial, especially in scientific analyses.
The conversion formula from Kelvin to Fahrenheit is:\[ T(^{\circ}\mathrm{F}) = \frac{9}{5}T(\mathrm{K}) - 459.67 \]This formula accounts for the difference between absolute zero in both scales. An example from the exercise shows us transforming \(293.2\mathrm{~K}\) to \(68.0^{\circ}\mathrm{F}\).
Knowing this conversion can help you understand data presented in different temperature units, equipping you with the ability to fluently work with varying temperature scales.

Converting Fahrenheit to Celsius

Celsius is another commonly used temperature scale, especially in scientific contexts. If you understand how to convert from Fahrenheit, you can communicate findings across different nations and disciplines.
The conversion from Fahrenheit to Celsius is designated by the formula:\[ T(^{\circ}\mathrm{C}) = \frac{5}{9}(T(^{\circ}\mathrm{F}) - 32) \]In our context, the final temperature of \(93.25^{\circ}\mathrm{F}\) converts to approximately \(34.03^{\circ}\mathrm{C}\).
This correspondence is important for standardized understanding and documentation of temperature changes worldwide.

• Always remember to subtract 32 from the Fahrenheit temperature first.
• Then multiply by the fraction \(\frac{5}{9}\) to find the Celsius value.

These methods ensure accuracy in scientific temperature measurements.