A demented scientist creates a new temperature scale, the 'Z scale.' He decides to call the boiling point of nitrogen 0°Z and the melting point of iron 1000°Z. Convert 500°Z to degrees Celsius and to kelvins.

Verified step by step guidance

1. Define the relationship between the Z scale and the Celsius scale. The boiling point of nitrogen is 0°Z, which corresponds to -196°C, and the melting point of iron is 1000°Z, which corresponds to 1538°C. This gives two points to establish a linear relationship between the Z scale and the Celsius scale.
2. Use the formula for a linear relationship: \( T_{Z} = m \cdot T_{C} + b \), where \( m \) is the slope and \( b \) is the intercept. Calculate the slope \( m \) using the two points: \( m = \frac{T_{Z2} - T_{Z1}}{T_{C2} - T_{C1}} = \frac{1000 - 0}{1538 - (-196)} \).
3. Determine the intercept \( b \) by substituting one of the points (e.g., \( T_{Z1} = 0 \) and \( T_{C1} = -196 \)) into the linear equation: \( 0 = m \cdot (-196) + b \). Solve for \( b \).
4. Once the relationship between the Z scale and the Celsius scale is established, convert 500°Z to Celsius using the equation \( T_{C} = \frac{T_{Z} - b}{m} \). Substitute \( T_{Z} = 500 \) into the equation to find \( T_{C} \).
5. To convert from Celsius to kelvins, use the formula \( T_{K} = T_{C} + 273.15 \). Add 273.15 to the Celsius temperature obtained in the previous step to find the temperature in kelvins.

Key Concepts

Temperature Scales

Temperature scales, such as Celsius, Kelvin, and the newly defined Z scale, provide a way to quantify thermal energy. Each scale has specific reference points; for instance, 0°C is the freezing point of water, while 100°C is its boiling point. Understanding how to convert between these scales is essential for solving temperature-related problems.

Linear Transformation

Linear transformation is a mathematical concept used to convert values from one scale to another. In the context of temperature conversion, it involves establishing a linear relationship between the two scales based on known reference points. This allows for the calculation of equivalent temperatures in different units.

Absolute Zero and Kelvin Scale

The Kelvin scale is an absolute temperature scale starting at absolute zero, the theoretical point where molecular motion ceases. It is crucial for scientific calculations, as it provides a direct measure of thermal energy. The relationship between Celsius and Kelvin is linear, with 0 K equivalent to -273.15°C, which is important for converting temperatures accurately.