Question

Natural gas is mostly methane, a substance that boils at a temperature of \(109 \mathrm{~K}\). What is its boiling point in \({ }^{\circ} \mathrm{C}\) and \({ }^{\circ} \mathrm{F}\) ?

Short answer

The boiling point of methane in Celsius is \( -164.15^\circ C \) and in Fahrenheit is \( -263.47^\circ F \)

Step-by-step solution

Convert to Celsius

To convert the boiling point from Kelvin to Celsius, use the formula \( T(\text{°C}) = T(\text{K}) - 273.15 \) where T(K) is the temperature in Kelvin. Substitute the given temperature in Kelvin to get the Celsius value.

Calculate the Celsius value

Substitute 109 K into the formula to get \( T(\text{°C}) = 109 \text{K} - 273.15 \) which equals \( -164.15 \text{°C} \).

Convert to Fahrenheit

To convert the boiling point from Celsius to Fahrenheit, use the formula \( T(\text{°F}) = T(\text{°C}) \times \frac{9}{5} + 32 \). Use the Celsius value obtained in the previous step for this calculation.

Calculate the Fahrenheit value

Substitute -164.15 °C into the formula to get \( T(\text{°F}) = -164.15 \text{°C} \times \frac{9}{5} + 32 \) which simplifies to \( -263.47 \text{°F} \).

Key concepts

Temperature Conversion Explained

Understanding temperature conversion is crucial for many scientific calculations, including the study of chemistry where reactions are temperature dependent. Temperature conversion involves changing a temperature reading from one scale to another, such as from Celsius to Kelvin, Celsius to Fahrenheit, or Fahrenheit to Kelvin.

Each temperature scale has its own applications and units. For instance, Celsius is commonly used in most countries outside of the U.S. for everyday temperature measurements, while Fahrenheit is primarily used in the United States for the same purpose. On the other hand, Kelvin is the unit of choice in the scientific community, particularly in physics and chemistry, because it is an absolute temperature scale with zero Kelvin representing absolute zero, the theoretical point where particles have minimal thermal motion.

In converting temperatures, we use specific formulas that relate each scale to the others. These formulas account for the different starting points and scaling of each temperature unit. For example, the freezing point of water is 0°C, which is 32°F, and also 273.15K. To transition between these units, we apply conversion formulas that involve either addition/subtraction or multiplication/division, factoring in these scale differences and fixed points.

Kelvin to Celsius Conversion

Converting temperature from the Kelvin (K) scale to the Celsius (°C) scale is straightforward because both scales have the same magnitude of degree. The key difference between them is their starting points: 0K is absolute zero, while 0°C is the freezing point of water.

To convert from Kelvin to Celsius, the formula is: \( T(\text{°C}) = T(\text{K}) - 273.15 \). Using this, it's easy to convert the boiling point of methane from Kelvin to Celsius. Methane boils at 109K, which when converted using the formula translates to \( T(\text{°C}) = 109K - 273.15 = -164.15°C \).

This conversion demonstrates that the boiling point of methane in Celsius is well below the freezing point of water, highlighting the extreme conditions under which methane transitions from a liquid to a gas. Grasping this concept aids scientists and students in understanding thermodynamics and the behavior of gases under various temperatures.

Celsius to Fahrenheit Conversion

The conversion from Celsius to Fahrenheit allows us to translate temperature readings into a scale commonly used in the United States and a few other countries. The formula to convert Celsius into Fahrenheit is slightly more complex because it involves a multiplication and addition to adjust for the different starting points and scaling between the two units.

The formula is: \( T(\text{°F}) = T(\text{°C}) \times \frac{9}{5} + 32 \). When we take the previously calculated boiling point of methane in Celsius, \( -164.15\text{°C} \), and apply it to the formula, we get \( T(\text{°F}) = -164.15\text{°C} \times \frac{9}{5} + 32 = -263.47\text{°F} \).

This conversion process is essential in contexts where Fahrenheit is the preferred temperature scale. Understanding these conversions allows students to communicate their scientific findings accurately with international audiences and ensures that they can follow procedures and experiments that may be documented using different temperature scales.