Problem 96 - Chemistry: Atoms First

Question

Convert the temperature of dry ice, \(-77^{\circ} \mathrm{C},\) into degrees Fahrenheit and kelvin.

Step-by-Step Solution

Answer

Dry ice at \(-77^{\text{o}} \text{C}\) is equal to \(-105.4^{\text{o}} \text{F}\) and \(196.15 \text{K}\).

Step 1: Convert Celsius to Fahrenheit

To convert Celsius to Fahrenheit, use the formula: \( F = \frac{9}{5}C + 32 \), where \( C \) is the temperature in Celsius. Plug in the value of \(-77^{\text{o}} \text{C}\) to find Fahrenheit.

Step 2: Calculate Fahrenheit

Calculate the temperature in Fahrenheit: \( F = \frac{9}{5}(-77) + 32 = -137.4 + 32 = -105.4^{\text{o}} \text{F} \).

Step 3: Convert Celsius to Kelvin

To convert Celsius to Kelvin, use the formula: \( K = C + 273.15 \), where \( C \) is the temperature in Celsius. Plug in the value of \(-77^{\text{o}} \text{C}\) to find Kelvin.

Step 4: Calculate Kelvin

Calculate the temperature in Kelvin: \( K = -77 + 273.15 = 196.15 \text{K} \).

Key Concepts

Celsius to Fahrenheit Conversion

Temperature conversion between Celsius and Fahrenheit is a common task in chemistry and daily life. To convert Celsius to Fahrenheit, you use the equation:

\( F = \frac{9}{5}C + 32 \)

where \( F \) represents the temperature in degrees Fahrenheit and \( C \) is the temperature in degrees Celsius. This relationship is derived from the temperature interval scaling where a degree Celsius is larger than that of a degree Fahrenheit. For instance, when converting the temperature of dry ice, with a temperature of \(-77^{\text{o}} \text{C}\), we calculate the corresponding temperature in Fahrenheit by plugging in the values into the formula:

\( F = \frac{9}{5}(-77) + 32 = -137.4 + 32 = -105.4^{\text{o}} \text{F} \).
To enhance understanding, remember that the fraction \(\frac{9}{5}\) reflects the rate at which Celsius temperatures convert to Fahrenheit. The addition of 32 accounts for the offset in the Fahrenheit scale, where the freezing point of water is 32 degrees above the zero of the Fahrenheit scale, as opposed to 0 degrees on the Celsius scale.

Celsius to Kelvin Conversion

The conversion from Celsius to Kelvin is vital in scientific work because the Kelvin scale is the SI base unit for temperature and does not use degrees. The formula to convert Celsius to Kelvin is:

\( K = C + 273.15 \)

The '+' indicates that 273.15 is added to the Celsius temperature to shift the scale from Celsius to Kelvin; this factor converts the freezing point of water from 0 degrees Celsius to 273.15 Kelvin. When working with temperatures like that of dry ice, \(-77^{\text{o}} \text{C}\), the conversion is direct:
\( K = -77 + 273.15 = 196.15 \text{K} \).
Understanding this conversion helps to grasp thermal dynamics' and chemistry calculations that are often standardized in Kelvin for consistency, especially when dealing with absolute temperature measurements, entropy calculations, or equilibrium constants in chemical reactions.

Thermal Properties of Dry Ice

Dry ice, which is solid carbon dioxide (CO2), is known for its extremely low temperature and sublimation—that is, transitioning from a solid directly to a gas at temperatures above \(-78.5^{\text{o}} \text{C}\) (or 194.65 K) under normal atmospheric pressures. The thermal properties of dry ice make it a unique substance suitable for various applications such as keeping foods frozen during transport, stage effects, and scientific research.
A critical feature is its exothermic transition where it absorbs heat as it transforms, making it highly effective for cooling. However, due to its cold nature, it requires careful handling, using insulated gloves to prevent cold burns. In the context of temperature conversion, knowing the exact temperature of dry ice in Celsius, Fahrenheit, and Kelvin is required for safely and effectively using this material in various chemical processes and applications.