Problem 88 - Introductory Chemistry Atoms First

Question

Convert: (a) \(22.5^{\circ} \mathrm{C}\) to Fahrenheit and Kelvin (b) \(-3.00{ }^{\circ} \mathrm{F}\) to Celsius and Kelvin (c) \(100.0^{\circ} \mathrm{C}\) to Kelvin and Fahrenheit (d) \(65.1^{\circ} \mathrm{C}\) to Fahrenheit and Kelvin

Step-by-Step Solution

Answer

(a) \(72.5^{\circ}\mathrm{F}\), \(295.65\,\mathrm{K}\) (b) \(-19.44^{\circ}\mathrm{C}\), \(253.71\,\mathrm{K}\) (c) \(212.0^{\circ}\mathrm{F}\), \(373.15\,\mathrm{K}\) (d) \(149.18^{\circ}\mathrm{F}\), \(338.25\,\mathrm{K}\)

Step 1: Celsius to Fahrenheit

Use the formula: \(F = \frac{9}{5} C + 32\). Replace \(C\) with \(22.5\): \(F = \frac{9}{5} (22.5) + 32\) Now, calculate the Fahrenheit value: \(F = 72.5^{\circ} \mathrm{F}\).

Step 2: Celsius to Kelvin

Use the formula: \(K = C + 273.15\). Replace \(C\) with \(22.5\): \(K = 22.5 + 273.15\) Now, calculate the Kelvin value: \(K = 295.65\,\mathrm{K}\). (b) Convert \(-3.00{ }^{\circ} \mathrm{F}\) to Celsius and Kelvin

Step 1: Fahrenheit to Celsius

Use the formula: \(C = \frac{5}{9}(F - 32)\). Replace \(F\) with \(-3.00\): \(C = \frac{5}{9}(-3.00 - 32)\) Now, calculate the Celsius value: \(C = -19.44^{\circ}\mathrm{C}\).

Step 2: Fahrenheit to Kelvin

Use the formula: \(K = \frac{5}{9} (F - 32) + 273.15\). Replace \(F\) with \(-3.00\): \(K = \frac{5}{9}(-3.00 - 32) + 273.15\) Now, calculate the Kelvin value: \(K = 253.71\,\mathrm{K}\). (c) Convert \(100.0^{\circ} \mathrm{C}\) to Kelvin and Fahrenheit

Step 1: Celsius to Fahrenheit

Use the formula: \(F = \frac{9}{5} C + 32\). Replace \(C\) with \(100.0\): \(F = \frac{9}{5}(100.0) + 32\) Now, calculate the Fahrenheit value: \(F = 212.0^{\circ}\mathrm{F}\).

Step 2: Celsius to Kelvin

Use the formula: \(K = C + 273.15\). Replace \(C\) with \(100.0\): \(K = 100.0 + 273.15\) Now, calculate the Kelvin value: \(K = 373.15\,\mathrm{K}\). (d) Convert \(65.1^{\circ}\mathrm{C}\) to Fahrenheit and Kelvin

Step 1: Celsius to Fahrenheit

Use the formula: \(F = \frac{9}{5} C + 32\). Replace \(C\) with \(65.1\): \(F = \frac{9}{5}(65.1) + 32\) Now, calculate the Fahrenheit value: \(F = 149.18^{\circ}\mathrm{F}\).

Step 2: Celsius to Kelvin

Use the formula: \(K = C + 273.15\). Replace \(C\) with \(65.1\): \(K = 65.1 + 273.15\) Now, calculate the Kelvin value: \(K = 338.25\,\mathrm{K}\).

Key Concepts

Celsius to Fahrenheit

Converting temperature from Celsius to Fahrenheit is simple with a specific formula. The formula to use is: \( F = \frac{9}{5} C + 32 \). Here, \( C \) represents the degrees in Celsius. This formula accounts for the difference in temperature scales between Celsius and Fahrenheit, particularly the differing intervals and starting points.

• The scaling factor \( \frac{9}{5} \) adjusts the degree interval from Celsius to Fahrenheit, which is why you multiply it by your Celsius temperature.
• Add 32 because Fahrenheit starts at 32 when Celsius is at zero. This offset helps align the scales at the freezing point of water.

For example, if you want to convert \( 22.5^{\circ}C \) to Fahrenheit, you substitute into the formula to find that it equals \( 72.5^{\circ}F \). Just remember to follow the conversion steps so that you aren't mistakenly converting temperatures.

Fahrenheit to Celsius

To convert temperatures from Fahrenheit to Celsius, you'll use a straightforward formula: \( C = \frac{5}{9}(F - 32) \). In this formula, \( F \) stands for degrees in Fahrenheit. This conversion involves two core adjustments:

• First, you subtract 32 from the Fahrenheit temperature. This adjusts for the baseline difference, since 32°F marks the freezing point of water on this scale.
• The term \( \frac{5}{9} \) is used to scale back the interval differences between the two systems. Celsius degrees are larger than Fahrenheit degrees; hence this fraction reduces the Fahrenheit degree to its corresponding smaller Celsius measure.

As an illustration, when converting \( -3.0^{\circ}F \) to Celsius, this method will yield \( -19.44^{\circ}C \). Keep the equation in mind whenever you need to bridge the gap between these two common temperature scales.

Celsius to Kelvin

The conversion from Celsius to Kelvin is direct and requires minimal calculation. The relation is depicted in the formula: \( K = C + 273.15 \). Here, \( C \) refers to the temperature in Celsius, while \( K \) indicates the temperature in Kelvin.

• The value 273.15 is crucial as it defines the zero point of Kelvin, called absolute zero, precisely offset from Celsius.
• Simply add 273.15 to your Celsius temperature to transition to the Kelvin scale.

For example, converting \( 100.0^{\circ}C \) into Kelvin results in \( 373.15K \). This straightforward conversion reflects how these scales interrelate, with Celsius's zero point (freezing of water) shifting to Kelvin's 273.15.

Fahrenheit to Kelvin

To convert from Fahrenheit to Kelvin, you employ a formula that efficiently transforms the temperature by considering both the conversion to Celsius and the addition necessary to reach Kelvin: \( K = \frac{5}{9} (F - 32) + 273.15 \). This equation functions as a bridge, integrating both Fahrenheit to Celsius, then linking Celsius to Kelvin for a seamless shift.

• First, use \( \frac{5}{9} (F - 32) \) to change Fahrenheit into Celsius, following the prior outlined steps.
• Next, the conversion to Kelvin occurs via adding 273.15, addressing the difference in starting points: 0K equals \(-273.15^{\circ}C\).

For instance, converting \( -3.0^{\circ}F \) gives you \( 253.71K \). Hence, this ensures the entire temperature continuum is accurately represented in scientific contexts.