Problem 47 - Chemistry The Molecular Nature of Matter

Question

The temperature of the core of the sun is estimated to be about 15.7 million \(\mathrm{K}\). What is this temperature in \({ }^{\circ} \mathrm{C}\) and \({ }^{\circ} \mathrm{F}\) ?

Step-by-Step Solution

Answer

The core temperature of the sun is approximately 15,699,726.85\( \degree C\) and 28,259,508.33\( \degree F\).

Step 1: Understand the Temperature Conversion Formulas

To convert Kelvin to Celsius, use the formula: \( \degree C = K - 273.15 \). To convert Celsius to Fahrenheit, use the formula: \( \degree F = \degree C \times \frac{9}{5} + 32 \)

Step 2: Convert Kelvin to Celsius

Substitute the given temperature in Kelvin to the Celsius conversion formula: \( \degree C = 15,700,000\ K - 273.15 \)

Step 3: Calculate the Celsius Temperature

Subtract 273.15 from 15,700,000 to get the temperature in Celsius: \( \degree C = 15,699,726.85 \)

Step 4: Convert Celsius to Fahrenheit

Substitute the Celsius value into the Fahrenheit conversion formula: \( \degree F = 15,699,726.85 \times \frac{9}{5} + 32 \)

Step 5: Calculate the Fahrenheit Temperature

Multiply 15,699,726.85 by \( \frac{9}{5} \) and add 32 to get the temperature in Fahrenheit: \( \degree F = 28,259,508.33 \)

Key Concepts

Kelvin to Celsius Conversion

Understanding temperature conversions is crucial for many scientific calculations, including those in chemistry and physics. Kelvin to Celsius conversion is one of the fundamental concepts in thermodynamics and is especially important when dealing with extreme temperatures such as those found in stellar cores.

The formula to convert Kelvin to Celsius is quite straightforward: you subtract 273.15 from the Kelvin value. Why this particular number? It's because 0 degrees Celsius is equivalent to 273.15 Kelvin; this is because the Celsius scale is set based on the freezing and boiling points of water, which are 0°C and 100°C respectively, while the Kelvin scale is based on absolute zero, the theoretical point of no thermal energy.

For example, if a star's core temperature is 15.7 million Kelvin, we subtract 273.15 to find the Celsius temperature:\begin{align*}\( \degree C &= K - 273.15 \) \ \( \degree C &= 15,700,000 K - 273.15 \) \ \( \degree C &= 15,699,726.85 \) d{align*}This basic conversion is a handy tool not just in astrophysics but across all branches of science where temperature plays a role in experiments and processes.

Celsius to Fahrenheit Conversion

When exploring Celsius to Fahrenheit conversion, we encounter a different aspect of temperature measurement, often used in daily life in some countries like the United States. The formula reflects the scale differences between the two units: to convert from Celsius to Fahrenheit, you multiply the Celsius temperature by \frac{9}{5} and then add 32.

This is because the two scales have different zero points and degrees sizes: the freezing point of water is 32°F and its boiling point is 212°F, establishing a 180-degree interval between the two, which corresponds to 100 degrees on the Celsius scale. Consequently, each degree Celsius is equivalent to 1.8 degrees Fahrenheit.

Let's convert the previously calculated temperature of the sun’s core from Celsius to Fahrenheit: \begin{align*}\( \degree F &= \degree C \times \frac{9}{5} + 32 \) \ \( \degree F &= 15,699,726.85 \times \frac{9}{5} + 32 \) \ \( \degree F &= 28,259,508.33 \) d{align*}This information is not only practical for everyday life but also an essential part of scientific communication, as it allows researchers to present data in units that are familiar to different audiences.

Thermodynamics

Thermodynamics is the study of heat, energy, and the work they can perform. It’s a branch of physical science that explains how thermal energy is converted to and from other forms of energy and how it affects matter. The principles of thermodynamics are based on four fundamental laws that define how temperature, energy, and entropy interact within the universe.

The first law of thermodynamics, also known as the Law of Energy Conservation, states that energy cannot be created or destroyed, only transformed. For example, the extreme heat of the sun’s core originates from nuclear fusion, turning mass into energy.

The second law of thermodynamics states that in any energy transfer or transformation, if no energy enters or leaves the system, the potential energy of the state will always be less than that of the initial state, a concept known as entropy.

The third law of thermodynamics states that the entropy of a system approaches a constant value as the temperature approaches absolute zero. Absolute zero, the theoretical temperature at which particles would have minimal vibrational motion, is 0 Kelvin or -273.15 Celsius.

Finally, the zeroth law of thermodynamics establishes that if two systems are each in thermal equilibrium with a third system, they are in thermal equilibrium with each other.

These principles play a vital role in understanding the vast ranges of temperature in the universe, from the frigid vacuum of space to the fiery core of stars.