Problem 4
Question
\(\cdot\) Inside the earth and the sun. (a) Geophysicists have esti- mated that the temperature at the center of the earth's core is \(5000^{\circ} \mathrm{C}\) (or more), while the temperature of the sun's core is about 15 million \(\mathrm{K}\) . Express both of these temperatures in Fahrenheit degrees.
Step-by-Step Solution
Verified Answer
Earth's core: 9032°F, Sun's core: 26,999,540.33°F.
1Step 1: Convert Earth's Core Temperature to Fahrenheit
The Earth's core temperature is given as \(5000^{\circ} C\). To convert Celsius to Fahrenheit, use the formula: \[ F = \frac{9}{5} \times C + 32 \]Substitute 5000 for \(C\):\[ F = \frac{9}{5} \times 5000 + 32 = 9000 + 32 = 9032^{\circ} F \]
2Step 2: Convert Sun's Core Temperature from Kelvin to Celsius
The Sun's core temperature is given as \(15 \text{ million } K\). First, convert Kelvin to Celsius using the formula:\[ C = K - 273.15 \]Substitute 15 million for \(K\): \[ C = 15,000,000 - 273.15 = 14,999,726.85^{\circ} C \]
3Step 3: Convert Sun's Core Temperature to Fahrenheit
Now, using the Celsius temperature found in Step 2, convert it to Fahrenheit with the formula:\[ F = \frac{9}{5} \times C + 32 \]Substitute 14,999,726.85 for \(C\):\[ F = \frac{9}{5} \times 14,999,726.85 + 32 = 26,999,508.33 + 32 = 26,999,540.33^{\circ} F \]
Key Concepts
Earth's Core TemperatureSun's Core TemperatureCelsius to Fahrenheit ConversionKelvin to Celsius Conversion
Earth's Core Temperature
The Earth's core is incredibly hot and scientists estimate its temperature to be around 5000°C.
Even though we can't directly measure this temperature, these estimates are calculated based on scientific models and simulations.
Understanding this temperature is essential as it gives insights into the geological and thermal dynamics of our planet.
To convert the temperature from Celsius to Fahrenheit, you use the formula: \[ F = \frac{9}{5} \times C + 32 \]This gives approximately 9032°F when you substitute 5000 for \(C\).
This conversion is helpful when dealing with international research where Fahrenheit might be used.
The high temperature influences the Earth's magnetic field and tectonic activities.
This information helps geophysicists study Earth's formation and predict events like earthquakes.
Even though we can't directly measure this temperature, these estimates are calculated based on scientific models and simulations.
Understanding this temperature is essential as it gives insights into the geological and thermal dynamics of our planet.
To convert the temperature from Celsius to Fahrenheit, you use the formula: \[ F = \frac{9}{5} \times C + 32 \]This gives approximately 9032°F when you substitute 5000 for \(C\).
This conversion is helpful when dealing with international research where Fahrenheit might be used.
The high temperature influences the Earth's magnetic field and tectonic activities.
This information helps geophysicists study Earth's formation and predict events like earthquakes.
Sun's Core Temperature
The sun's core is much hotter than Earth’s and is estimated to be around 15 million Kelvin.
These temperatures are crucial in facilitating nuclear fusion, the process that powers the sun.
In fusion, hydrogen atoms combine to form helium, releasing vast amounts of energy.
To understand this temperature in terms that we more commonly use on Earth, Kelvin is often converted to Celsius and then to Fahrenheit.
The conversion from Kelvin to Celsius is done using the formula: \[ C = K - 273.15 \]Once in Celsius, this extreme heat measures approximately 14,999,726.85°C.
Converting it further to Fahrenheit using the formula above results in about 26,999,540.33°F.
Such knowledge is vital in astrophysics to help us understand how stars, including our sun, evolve.
These temperatures are crucial in facilitating nuclear fusion, the process that powers the sun.
In fusion, hydrogen atoms combine to form helium, releasing vast amounts of energy.
To understand this temperature in terms that we more commonly use on Earth, Kelvin is often converted to Celsius and then to Fahrenheit.
The conversion from Kelvin to Celsius is done using the formula: \[ C = K - 273.15 \]Once in Celsius, this extreme heat measures approximately 14,999,726.85°C.
Converting it further to Fahrenheit using the formula above results in about 26,999,540.33°F.
Such knowledge is vital in astrophysics to help us understand how stars, including our sun, evolve.
Celsius to Fahrenheit Conversion
Converting temperatures from Celsius to Fahrenheit is a common task in science and daily life.
It involves a simple mathematical formula: \[ F = \frac{9}{5} \times C + 32 \]This formula allows for the conversion between the two temperature scales, where \(C\) represents the temperature in Celsius and \(F\) the Fahrenheit equivalent.
It's particularly useful in countries like the United States where Fahrenheit is commonly used, whereas Celsius is more prevalent in most of the world.
For example, boiling water at 100°C is equivalent to 212°F.
This conversion helps people across disparate regions understand temperature data globally.
It involves a simple mathematical formula: \[ F = \frac{9}{5} \times C + 32 \]This formula allows for the conversion between the two temperature scales, where \(C\) represents the temperature in Celsius and \(F\) the Fahrenheit equivalent.
It's particularly useful in countries like the United States where Fahrenheit is commonly used, whereas Celsius is more prevalent in most of the world.
For example, boiling water at 100°C is equivalent to 212°F.
This conversion helps people across disparate regions understand temperature data globally.
Kelvin to Celsius Conversion
Kelvin is the base unit of temperature in the International System of Units (SI).
Unlike Celsius, Kelvin does not use degrees, and its scale starts from absolute zero, the point where no more energy can be extracted from a system.
The conversion from Kelvin to Celsius is straightforward: simply subtract 273.15 from the Kelvin temperature.\[ C = K - 273.15 \]This is because the Kelvin and Celsius scales have the same increment size.
For instance, water freezes at 273.15K, which is 0°C.
Kelvin is often used in scientific computations like those of thermodynamics and space sciences to ensure precision.
Unlike Celsius, Kelvin does not use degrees, and its scale starts from absolute zero, the point where no more energy can be extracted from a system.
The conversion from Kelvin to Celsius is straightforward: simply subtract 273.15 from the Kelvin temperature.\[ C = K - 273.15 \]This is because the Kelvin and Celsius scales have the same increment size.
For instance, water freezes at 273.15K, which is 0°C.
Kelvin is often used in scientific computations like those of thermodynamics and space sciences to ensure precision.
Other exercises in this chapter
Problem 2
Temperatures in biomedicine. (a) Normal body temperature. The average normal body temperature measured in the mouth is 310 \(\mathrm{K}\) . What would Celsius a
View solution Problem 3
\(\bullet\) (a) On January \(22,1943,\) the temperature in Spearfish, South Dakota, rose from \(-4.0^{\circ} \mathrm{F}\) to \(45.0^{\circ} \mathrm{F}\) in just
View solution Problem 5
(a) At what temperature do the Fahrenheit and Celsius scales coincide? (b) Is there any temperature at which the Kelvin and Celsius scales coincide?
View solution Problem 6
Convert the following Kelvin temperatures to the Celsius and Fahrenheit scales: (a) the midday temperature at the surface of the moon \((400 \mathrm{K}) ;\) (b)
View solution