Problem 84 - Chemistry: An Atoms-Focused Approach

Question

The temperature of the dry ice (solid carbon dioxide) in ice cream vending carts is \(-78^{\circ} \mathrm{C} .\) What is this temperature on the Fahrenheit and Kelvin scales?

Step-by-Step Solution

Answer

Answer: The temperature of dry ice is -108.4°F in the Fahrenheit scale and 195.15 K in the Kelvin scale.

Step 1: Convert Celsius to Fahrenheit

To convert the temperature from Celsius to Fahrenheit, we use the formula: \(F = \frac{9}{5} C + 32\). In this case, the temperature in Celsius is -78°C, so we substitute this value into the formula: \(F = \frac{9}{5} (-78) + 32\). Now, we simplify the expression: F = \(\frac{-702}{5}+ 32\) F = \( -140.4 + 32\) So, the temperature in Fahrenheit is: F = \(-108.4^{\circ}\mathrm{F}\).

Step 2: Convert Celsius to Kelvin

To convert the temperature from Celsius to Kelvin, we use the formula: \(K = C + 273.15\). In this case, the temperature in Celsius is -78°C, so we substitute this value into the formula: \(K = -78 + 273.15\). Now, we simplify the expression: K = \(195.15\) So, the temperature in Kelvin is: K = \(195.15\,\mathrm{K}\). In conclusion, the temperature of the dry ice in ice cream vending carts, initially given as \(-78^{\circ}\mathrm{C}\), is equivalent to \(-108.4^{\circ}\mathrm{F}\) on the Fahrenheit scale and \(195.15\,\mathrm{K}\) on the Kelvin scale.

Key Concepts

Celsius to Fahrenheit

Temperature conversion from Celsius to Fahrenheit is crucial when dealing with temperature-related problems, especially in science and daily life.
The formula to convert Celsius to Fahrenheit is:\[ F = \frac{9}{5} C + 32 \]This formula stems from the relationship between the two temperature scales, where the freezing point of water is 0°C (32°F) and the boiling point is 100°C (212°F).
To use the formula, simply multiply the Celsius temperature by 9/5, which scales the temperature to match the spread of the Fahrenheit scale, then add 32 to align it with the Fahrenheit zero point.
For example, for dry ice at -78°C:

• First multiply by 9/5: \( \frac{9}{5} \times (-78) = -140.4 \)
• Then add 32: \( -140.4 + 32 = -108.4 \)
  so the temperature is \(-108.4^{\circ}\text{F}\).

This straightforward method allows you to quickly convert any Celsius temperature to Fahrenheit.
Understanding these steps helps in interpreting weather forecasts, scientific experiments, and international travel temperature differences.

Celsius to Kelvin

The Kelvin scale is another way to measure temperature, often used in scientific contexts.
It is an absolute scale based on the theoretical point of absolute zero, the lowest possible temperature where all molecular motion stops.
To convert Celsius to Kelvin, use the formula:\[ K = C + 273.15 \]This conversion is easier than the Celsius to Fahrenheit one because you simply add 273.15 to the Celsius temperature.
This addition adjusts the Celsius temperature to reflect the absolute scale, starting from absolute zero.
For example, the conversion for dry ice:

• Add 273.15 to the Celsius temperature: \( -78 + 273.15 = 195.15 \)
  so, the temperature in Kelvin is \( 195.15 \, \text{K} \).

This method is especially useful in scientific calculations where Kelvin provides a clear, absolute understanding of temperature without negative numbers.
It is mainly used in fields like physics and chemistry to describe thermal dynamics.

Dry Ice Temperature

Dry ice, the solid form of carbon dioxide, is much colder than regular ice, making it useful for preserving temperature-sensitive items.
Its common use in ice cream vending carts demonstrates one of its practical applications.
At \(-78^{\circ}\text{C}\) or \(-108.4^{\circ}\text{F}\), dry ice transitions directly from a solid to a gas in a process called sublimation, without melting into a liquid.
This unique property makes it ideal for cooling without mess or residue.

• Dry ice helps keep items extremely cold over time.
• Its sublimation into carbon dioxide gas is safe under proper ventilation.

Knowing how to convert its temperature to different scales allows for better understanding and handling of dry ice in various industries.
This includes food preservation, shipping pharmaceuticals, and theatrical effects where extreme cold is necessary.