Problem 18 - College Physics

Question

When lightning strikes, it can heat the air around it to more than \(30000 \mathrm{~K}\), five times the surface temperature of the Sun. (a) What is this temperature on the Fahrenheit and Celsius scales? (b) The temperature is sometimes reported to be \(30000^{\circ} \mathrm{C}\). Assuming that \(30000 \mathrm{~K}\) is correct, what is the percentage error of this Celsius value?

Step-by-Step Solution

Answer

(a) 29726.85°C and 53520.33°F. (b) 0.92% error.

Step 1: Convert Kelvin to Celsius

To convert the temperature from Kelvin to Celsius, we use the formula: \[ T_{\text{C}} = T_{\text{K}} - 273.15 \]For this problem, substitute the given temperature: \[ T_{\text{C}} = 30000 - 273.15 = 29726.85 \, ^{\circ}\mathrm{C} \].

Step 2: Convert Celsius to Fahrenheit

To convert from Celsius to Fahrenheit, use the formula: \[ T_{\text{F}} = \frac{9}{5}T_{\text{C}} + 32 \].Now substitute the Celsius temperature obtained from Step 1:\[ T_{\text{F}} = \frac{9}{5} \times 29726.85 + 32 \approx 53520.33 \, ^{\circ}\mathrm{F} \].

Step 3: Calculate Percentage Error in Celsius

The reported temperature was given as \(30000^{\circ} \mathrm{C}\), but the correct value was found to be \(29726.85^{\circ} \mathrm{C}\). The formula for percentage error is:\[ \text{Percentage Error} = \left( \frac{\text{Reported Value} - \text{Correct Value}}{\text{Correct Value}} \right) \times 100\% \].Substitute the given values:\[ \text{Percentage Error} = \left( \frac{30000 - 29726.85}{29726.85} \right) \times 100\% \approx 0.92\% \].

Key Concepts

Kelvin to Celsius Conversion

When it comes to converting temperatures from Kelvin to Celsius, it involves a simple subtraction. This is because the Kelvin and Celsius scales are directly related. Both scales have the same size unit, so you only need to adjust for the zero points. The formula to convert Kelvin to Celsius is: \[T_{\text{C}} = T_{\text{K}} - 273.15\]This equation highlights that you subtract 273.15 from the Kelvin value to get the Celsius temperature. Keep in mind that absolute zero in Kelvin, 0 K, is equivalent to -273.15°C in Celsius. Hence, Kelvin starts counting from this absolute zero while Celsius starts at the freezing point of water.

Celsius to Fahrenheit Conversion

Turning Celsius into Fahrenheit involves a different formula, since the scales differ in both their starting points (i.e., zero points) and intervals. The United States primarily uses the Fahrenheit scale, which can be converted from Celsius with the following equation:\[ T_{\text{F}} = \frac{9}{5}T_{\text{C}} + 32 \]This means you multiply the Celsius temperature by 9/5 and then add 32. This calculation adjusts for the difference in zero points and the size of each unit. For example, 0°C equals 32°F because ice melts at these respective points on each scale. When you convert temperatures, checking your unit conversions ensures accuracy.

Percentage Error Calculation

Percentage error is a valuable tool in measuring the accuracy of reported data against the actual or correct value. It's a crucial concept, especially in scientific measurements and experiments. The formula to determine percentage error is:\[\text{Percentage Error} = \left( \frac{\text{Reported Value} - \text{Correct Value}}{\text{Correct Value}} \right) \times 100\%\]This equation calculates how much the reported measurement deviates from the correct value and expresses this deviation as a percentage. A lower percentage indicates a result closer to the true value, signifying greater precision and reliability. In scientific studies, ensuring low percentage error is vital to maintaining the credibility of results.