Finding Temperature When Kelvin and Fahrenheit Scales Align

The question asks for the corresponding reading on a Celsius thermometer when a sample has the same temperature reading on both Kelvin and Fahrenheit thermometers. Let this common reading on the Kelvin and Fahrenheit scales be \( T \). So, \( K = T \) and \( F = T \), where \( K \) is the temperature in Kelvin and \( F \) is the temperature in Fahrenheit.

We need to find the temperature in Celsius, denoted by \( C \), at this condition \( K=F \).

We use the standard conversion formulas between the three temperature scales:

• From Celsius to Kelvin: \( K = C + 273 \)
• From Celsius to Fahrenheit: \( F = \frac{9}{5}C + 32 \)

Since we are given that the Kelvin reading (\( K \)) and the Fahrenheit reading (\( F \)) are the same, we can set the right-hand sides of the conversion formulas (expressed in terms of Celsius) equal to each other:

Substitute \( K = C + 273 \) and \( F = \frac{9}{5}C + 32 \) into the condition \( K = F \):

\[ C + 273 = \frac{9}{5}C + 32 \]

Now, we need to solve this equation for \( C \). Let's rearrange the terms to group the \( C \) terms on one side and the constant terms on the other:

\[ 273 - 32 = \frac{9}{5}C - C \]

\[ 241 = \left(\frac{9}{5} - 1\right)C \]

\[ 241 = \left(\frac{9}{5} - \frac{5}{5}\right)C \]

\[ 241 = \frac{4}{5}C \]

To find \( C \), we multiply both sides by \( \frac{5}{4} \):

\[ C = \frac{241 \times 5}{4} \]

\[ C = \frac{1205}{4} \]

\[ C = 301.25 \]

So, the temperature reading in Celsius is 301.25 °C when the Kelvin and Fahrenheit thermometers show the same reading.

Looking at the given options, the value 301.25 is closest to 301.

Let's verify the readings for \( C = 301.25 \):

• Kelvin: \( K = C + 273 = 301.25 + 273 = 574.25 \) K
• Fahrenheit: \( F = \frac{9}{5}C + 32 = \frac{9}{5}(301.25) + 32 = 1.8 \times 301.25 + 32 = 542.25 + 32 = 574.25 \) °F

Indeed, at 301.25 °C, the Kelvin and Fahrenheit readings are both 574.25. Since 301 is the closest integer option to 301.25, it is the correct choice.

Key Temperature Conversion Formulas

Revision Table: Temperature Scale Conversion

Understanding the relationships between different temperature scales is crucial. The key formulas allow us to convert a temperature value from one scale to another.

• The Celsius scale is widely used and based on the freezing point (0 °C) and boiling point (100 °C) of water at standard atmospheric pressure.
• The Kelvin scale is the base unit of temperature in the International System of Units (SI). It is an absolute scale, meaning 0 K (absolute zero) is the theoretical point where particle motion stops. The steps are the same size as Celsius degrees.
• The Fahrenheit scale is commonly used in the United States. Water freezes at 32 °F and boils at 212 °F.

Additional Information on Temperature Scales

Absolute zero is 0 K, which corresponds to -273.15 °C and -459.67 °F. While conversions often use 273 for simplicity, the more precise value is 273.15.

The size of one degree Celsius is the same as the size of one Kelvin. A change of 1 °C is equal to a change of 1 K.

The size of one degree Fahrenheit is \( \frac{5}{9} \) times the size of one degree Celsius or Kelvin.