Question

The freezing point of silver is \(960.8^{\circ} \mathrm{C}\) and the freezing point of gold is \(1063.0^{\circ} \mathrm{C}\). Convert these two readings to \(\mathrm{Kelvin}\) \(\left({ }^{\circ} \mathrm{K}\right)\), Fahrenheit \(\left({ }^{\circ} \mathrm{F}\right)\), and Rankine \(\left({ }^{\circ} \mathrm{R}\right)\).

Short answer

The freezing points of silver and gold converted to Kelvin, Fahrenheit, and Rankine are as follows: Silver: - Kelvin: \(1233.95^{\circ} \mathrm{K}\) - Fahrenheit: \(1761.44^{\circ} \mathrm{F}\) - Rankine: \(2217.11^{\circ} \mathrm{R}\) Gold: - Kelvin: \(1336.15^{\circ} \mathrm{K}\) - Fahrenheit: \(1945.4^{\circ} \mathrm{F}\) - Rankine: \(2405.07^{\circ} \mathrm{R}\)

Step-by-step solution

Conversion to Kelvin

To convert the temperature from degrees Celsius to Kelvin, we use the following formula: \(T_{K} = T_{C} + 273.15\) For silver: \(T_{K} = 960.8 + 273.15 = 1233.95^{\circ} \mathrm{K}\) For gold: \(T_{K} = 1063.0 + 273.15 = 1336.15^{\circ} \mathrm{K}\)

Conversion to Fahrenheit

To convert the temperature from degrees Celsius to Fahrenheit, we use the following formula: \(T_{F} = \frac{9}{5} T_{C} + 32\) For silver: \(T_{F} = \frac{9}{5}(960.8) + 32 = 1761.44^{\circ} \mathrm{F}\) For gold: \(T_{F} = \frac{9}{5}(1063.0) + 32 = 1945.4^{\circ} \mathrm{F}\)

Conversion to Rankine

To convert the temperature from degrees Celsius to Rankine, we use the following formula: \(T_{R} = \frac{9}{5} T_{C} + 491.67\) For silver: \(T_{R} = \frac{9}{5}(960.8) + 491.67 = 2217.11^{\circ} \mathrm{R}\) For gold: \(T_{R} = \frac{9}{5}(1063.0) + 491.67 = 2405.07^{\circ} \mathrm{R}\) The converted freezing points are as follows: Silver: - Kelvin: \( 1233.95^{\circ} \mathrm{K}\) - Fahrenheit: \( 1761.44^{\circ} \mathrm{F}\) - Rankine: \( 2217.11^{\circ} \mathrm{R}\) Gold: - Kelvin: \(1336.15^{\circ} \mathrm{K}\) - Fahrenheit: \(1945.4^{\circ} \mathrm{F}\) - Rankine: \(2405.07^{\circ} \mathrm{R}\)

Key concepts

Kelvin Scale

The Kelvin scale is an absolute temperature scale used primarily in science and engineering. Unlike the Celsius and Fahrenheit scales, which are relative scales based on the properties of water, the Kelvin scale starts at absolute zero, which is the theoretical point where all molecular motion ceases. The size of one Kelvin unit is the same as one degree Celsius unit, so temperature differences are the same in both scales. To convert from Celsius to Kelvin, you simply add 273.15. For example, the freezing point of silver at 960.8 degrees Celsius is converted to Kelvin by calculating:
\(960.8^\circ \text{C} + 273.15 = 1233.95\text{K}\).
It's important to remember that Kelvin is not referred to in degrees; it's simply 'Kelvin' or 'K'.

Fahrenheit Scale

The Fahrenheit scale is a temperature scale where the freezing point of water is 32 degrees Fahrenheit (\(32^\circ\text{F}\)) and the boiling point is 212 degrees Fahrenheit (\(212^\circ\text{F}\)), at normal atmospheric pressure. This scale is used mainly in the United States. Converting Celsius to Fahrenheit requires multiplying by 9/5 (or 1.8) and then adding 32. Hence, the conversion for the freezing point of silver becomes:
\(\frac{9}{5}(960.8) + 32 = 1761.44^\circ\text{F}\).
This equation ensures that the temperature difference in degrees Fahrenheit is reflective of the Celsius scale's range.

Rankine Scale

The Rankine scale is another absolute temperature scale, similar in concept to the Kelvin scale, but it uses the Fahrenheit degree for its unit size instead. It's mostly used in engineering fields within the United States. Absolute zero on the Rankine scale is also zero, but the freezing and boiling points of water are 491.67\(\text{R}\) and 671.67\(\text{R}\) respectively. To convert from Celsius to Rankine, you multiply by 9/5 and then add 491.67. The resulting conversion for the freezing point of silver is:
\(\frac{9}{5}(960.8) + 491.67 = 2217.11^\circ\text{R}\).
The Rankine scale isn't as commonly used as the Kelvin scale, but it's still important for certain scientific and engineering calculations.

Freezing Point of Metals

The freezing point of a metal is the temperature at which it changes from a liquid to a solid. This physical property is crucial for numerous industrial processes, including metalworking and manufacturing. Each metal has a unique freezing point; for example, silver freezes at 960.8 degrees Celsius and gold at 1063.0 degrees Celsius. Understanding and being able to convert these temperatures into Kelvin, Fahrenheit, and Rankine is essential for accurate thermodynamic calculations in various fields of chemistry and engineering. The conversion formulas we've explored allow scientists and engineers to work within the temperature scale that best suits their needs, whether that's for measuring extreme temperatures in research or manufacturing components that must endure diverse environmental conditions.
To highlight the relevance of these concepts, let's recall that while silver's freezing point in Celsius may seem extreme, its conversions to 1233.95K, 1761.44\(\text{F}\), and 2217.11\(\text{R}\) in the respective scales illustrate the wide range of temperatures that must be managed in metalworking industries.