Problem 102 - Chemistry Principles and Practice

Question

A scientific oven is programmed to change temperature from \(80.0{ }^{\circ} \mathrm{F}\) to \(215.0{ }^{\circ} \mathrm{F}\) in 1 minute. Express the rate of change in degrees Celsius per second, and use the convention of significant digits to express the uncertainty in the rate.

Step-by-Step Solution

Answer

The rate of change is 1.25 °C/second with three significant figures.

Step 1: Convert Temperature from Fahrenheit to Celsius

To find the rate of change in Celsius, convert the temperatures from Fahrenheit to Celsius using the formula: \[ C = \frac{5}{9} (F - 32) \].For 80.0 °F: \[ C_1 = \frac{5}{9} (80.0 - 32) = \frac{5}{9} \times 48.0 = 26.67{ }^{ullet}C \].For 215.0 °F: \[ C_2 = \frac{5}{9} (215.0 - 32) = \frac{5}{9} \times 183 = 101.67{ }^{ullet}C \].

Step 2: Calculate the Change in Temperature

Subtract the initial temperature in Celsius from the final temperature to find the change.\[ \Delta C = C_2 - C_1 = 101.67 - 26.67 = 75.0{ }^{ullet}C \].

Step 3: Determine the Rate of Change per Minute

Since the temperature changes over the course of 1 minute, the rate of change per minute in Celsius is simply the change in temperature calculated:\[ Rate_{min} = 75.0{ }^{ullet}C/{minute} \].

Step 4: Convert Rate to Celsius per Second

To convert from per minute to per second, divide the rate by 60 (since there are 60 seconds in a minute):\[ Rate_{sec} = \frac{75.0{ }^{ullet}C}{60 ext{ seconds}} = 1.25{ }^{ullet}C/{second} \].

Step 5: Apply Significant Figures

The final step is to ensure the rate is expressed with the correct number of significant figures, based on the initial data (80.0 and 215.0 have three significant figures). Thus, the rate of change should also have three significant figures:\[ 1.25{ }^{ullet}C/{second} \].

Key Concepts

Significant Figures

When performing scientific calculations, it is critical to use the concept of significant figures correctly.
This ensures that the precision of your results is consistent with the precision of the input data.

• Significant figures are the digits in a number that contribute to its accuracy. This includes all digits except leading and trailing zeros which are merely placeholders.
• For measurements, usually, all non-zero digits are significant, zeroes between significant digits, and trailing zeroes in a decimal number.
• In our exercise, both temperatures (80.0°F and 215.0°F) are given with three significant figures. Therefore, every result obtained using these numbers must also be expressed with three significant figures.

By ensuring that calculations, like the rate of change, maintain these figures (in this case, 1.25°C/second), the precision of the measured data is strictly adhered to, providing a reliable result.

Rate of Change

Understanding how quickly something changes over time involves calculating the rate of change.
This is a foundational concept in both mathematics and science.

• The rate of change is often expressed as a "per unit" measure, such as per second, per minute, or per hour.
• In this exercise, we need to find how fast the oven's temperature changes in degrees Celsius per second.

After converting temperatures from Fahrenheit to Celsius, we see the temperature changes by 75.0°C over one minute.
To express this as per second, divide the change by the total number of seconds in a minute (60 seconds), giving an answer of 1.25°C/second.
This value tells us precisely how much the temperature increases each second which is crucial for monitoring and controlling scientific processes.

Celsius to Fahrenheit Conversion

Converting temperatures is essential for many scientific and everyday applications.
The conversion between Fahrenheit and Celsius is governed by a simple formula:

• To convert Fahrenheit (F) to Celsius (C), use the formula: \[ C = \frac{5}{9} (F - 32) \]
• This involves subtracting 32 from the Fahrenheit value, multiplying the result by 5, and then dividing by 9.
• This formula reflects the different starting points and increments in the two temperature scales.

In our example, the temperature is converted from 80.0°F and 215.0°F to 26.67°C and 101.67°C, respectively.
Using these precise conversions allows us to accurately determine how temperature changes in a more universally understood metric—degrees Celsius.