Problem 2 - Arihant AIEEE Physics

Question

What is the change in the temperature on Fahrenheit scale and on Kelvin scale, if a iron piece is heated from \(30^{\circ}\) to \(90^{\circ} \mathrm{C} ?\) (a) \(108^{\circ} \mathrm{F}, 60 \mathrm{~K}\) (b) \(100^{\circ} \mathrm{F}, 55 \mathrm{~K}\) (c) \(100^{\circ} \mathrm{F}, 65 \mathrm{~K}\) (d) \(60^{\circ} \mathrm{F}, 108 \mathrm{~K}\)

Step-by-Step Solution

Answer

The temperature change is 108°F and 60 K. Solution: (a).

Step 1: Determine the Change in Celsius

Calculate the change in temperature in Celsius. The initial temperature is \(30^{\circ}\)C and the final temperature is \(90^{\circ}\)C.\[\Delta T_{C} = 90^{\circ}C - 30^{\circ}C = 60^{\circ}C\]

Step 2: Convert Celsius Change to Fahrenheit Change

Use the conversion formula from Celsius to Fahrenheit:\[\Delta T_{F} = \Delta T_{C} \times \frac{9}{5}\]Substitute \(\Delta T_{C} = 60^{\circ}C\):\[\Delta T_{F} = 60 \times \frac{9}{5} = 108^{\circ}F\]

Step 3: Convert Celsius Change to Kelvin Change

Remember that a change of 1 degree Celsius corresponds to an equal change of 1 Kelvin, since both scales have the same incremental unit size:\[\Delta T_{K} = 60 \text{ K}\]

Step 4: Select the Correct Option

From the results:- Change in Fahrenheit: \(108^{\circ}F\)- Change in Kelvin: \(60 \text{ K}\)The correct option is (a) \(108^{\circ} \text{F}, 60 \text{ K}\).

Key Concepts

Celsius to Fahrenheit

Converting temperatures from Celsius to Fahrenheit is a common task in science, especially when dealing with temperature measurements in different systems.
The most important thing to remember is the conversion formula:

• Start with the change in temperature in Celsius, which we'll call \(\Delta T_{C}\).
• Use the conversion formula: \(\Delta T_{F} = \Delta T_{C} \times \frac{9}{5}\).

This formula helps us understand the relationship between the two scales. It reflects that each degree Celsius is equivalent to \(\frac{9}{5}\) degrees Fahrenheit. Therefore, a change of 60 degrees Celsius is calculated as \(60 \times \frac{9}{5} = 108^{\circ}F\). The formula is essential for accurately translating temperature changes or measurements between these two systems.

Fahrenheit Scale

The Fahrenheit scale is often used in the United States for everyday temperature readings, such as weather forecasts and home heating.
It has a unique structure:

• Water freezes at \(32^{\circ}F\) and boils at \(212^{\circ}F\) under standard atmospheric conditions.
• The interval between the freezing and boiling point of water is divided into 180 equal parts.

This scale was proposed by Daniel Gabriel Fahrenheit in 1724. It may seem a bit awkward compared to the Celsius scale, but it's rooted in a historical approach to measuring temperature based on the human experience of weather and climate.

Kelvin Scale

The Kelvin scale is the fundamental unit of temperature in the scientific community and is an absolute scale used worldwide.
It is directly tied to the Celsius scale in a straightforward way:

• 0 K is absolute zero, the point at which all thermal motion ceases, corresponding to \(-273.15^{\circ}C\).
• A change of 1 Kelvin is equal to a change of 1 degree Celsius.

This scale is essential for scientific calculations because it starts at absolute zero, the lowest possible temperature. The Kelvin scale allows scientists to perform calculations without the biases introduced by the freezing and boiling points of water, providing a more universal system.

Temperature Difference

When calculating temperature differences, especially in thermodynamic processes, it is vital to maintain consistency in the units used.
Often, scientists work with differences rather than absolute values. This simplification leads to more straightforward calculations, such as in our exercise, where the change between \(30^{\circ}C\) and \(90^{\circ}C\) was efficiently translated across different scales:

• 60 degrees Celsius translates directly to 60 Kelvin since the two scales have the same incremental unit size.
• To find the difference in Fahrenheit, use \(\Delta T_{F} = \Delta T_{C} \times \frac{9}{5}\), giving \(108^{\circ}F\).

Understanding these calculations helps in fields like chemistry and physics, where precise temperature control is crucial.

Unit Conversion in Thermodynamics

Thermodynamics often requires accurate temperature conversions.
This accuracy is crucial for experiments and calculations, enabling comparisons and predictions. Here are some key points:

• Start with consistent units to ensure valid results.
• Converting between Celsius, Kelvin, and Fahrenheit can affect experimental outcomes if not done correctly.

Science relies on these conversions for reporting results, calculating energy exchanges, and predicting reaction behavior in diverse conditions. Therefore, understanding and applying these conversion techniques is essential for any aspiring scientist.