Problem 1057 - JEE Physics

Question

Maximum density of \(\mathrm{H}_{2} \mathrm{O}\) is at the temperature. (A) \(32^{\circ} \mathrm{F}\) (B) \(39.2^{\circ} \mathrm{F}\) (C) \(42^{\circ} \mathrm{F}\) (D) \(4^{\circ} \mathrm{F}\)

Step-by-Step Solution

Answer

The correct answer is (B) \(39.2^{\circ} \mathrm{F}\), as it is closest to the maximum density of water at 4°C.

Step 1: Recall basic fact about water's density

Water achieves its maximum density at 4°C. As the temperature increases or decreases from this point, water's density decreases. Now, we need to convert the given options from Fahrenheit to Celsius and identify the closest one to 4°C.

Step 2: Convert the given options to Celsius

To convert Fahrenheit to Celsius, we need to use the formula: Celsius = (Fahrenheit - 32) × 5/9. Now let's convert the given options. (A) \(32^{\circ} F\) C = (32 - 32) × 5/9 C = 0°C (B) \(39.2^{\circ} F\) C = (39.2 - 32) × 5/9 C = 7.2 × 5/9 C ≈ 4°C (C) \(42^{\circ} F\) C = (42 - 32) × 5/9 C = 10 × 5/9 C ≈ 5.56°C (D) \(4^{\circ} F\) C = (4 - 32) × 5/9 C = -28 × 5/9 C ≈ -15.56°C

Step 3: Identify the correct option

As we can see from the conversions above, option (B) 39.2°F is the closest to 4°C, which is the temperature at which water's density is maximum. So, the correct answer is: (B) \(39.2^{\circ} \mathrm{F}\)

Key Concepts

Fahrenheit to Celsius Conversion

When working with temperature systems, converting between Fahrenheit and Celsius is essential. One common formula to convert temperatures from Fahrenheit to Celsius is:

• \( \text{Celsius} = (\text{Fahrenheit} - 32) \times \frac{5}{9} \)

Knowing this allows you to convert any given temperature in Fahrenheit to Celsius with ease.
For example, if we have a temperature of \(39.2^{\circ} F\), and we want to find out its equivalent in Celsius, substitute the values into the formula:

• \( \text{Celsius} = (39.2 - 32) \times \frac{5}{9} \)

Which simplifies to:

• \( \approx 4^{\circ} C \)

This conversion is frequently used in scientific contexts where Celsius is preferred. Understanding how to seamlessly switch between these temperature scales is invaluable for study and problem-solving.

Maximum Density

Water has a unique characteristic in which its density is maximized at a specific temperature. This is unusual because most substances become denser as they cool. However, water reaches its maximum density at \( 4^{\circ} C \), which is approximately \( 39.2^{\circ} F \).
At this temperature:

• Water molecules are optimally packed due to the hydrogen bonding.
• Beyond this point, as water cools and eventually freezes, it begins to expand and become less dense.

This behavior explains why ice floats on liquid water, significantly affecting aquatic life and environmental conditions.

Water Properties

Water is a remarkable substance due to its unique set of properties. It's vital for all forms of life and has characteristics that make it stand out:

• High heat capacity: Water can absorb a lot of heat before it begins to get hot, which helps regulate and stabilize temperatures.
• Polarity: Water molecules have a positive and a negative end, making it a great solvent for many substances, which is why it's often termed the 'universal solvent'.
• Cohesion and adhesion: This allows for the capillary action, which plants use to draw water from the soil, and creates surface tension that supports lightweight objects.
• Expands upon freezing: Unlike most substances, water expands and becomes less dense as it solidifies.

These factors contribute to its ability to support life in diverse environments, underpinning the complex systems within natural ecosystems.

Temperature and Density Relationship

The relationship between temperature and density in water exhibits some fascinating behaviors. Generally, as substances cool, they contract and become denser. However, water is exceptional because:

• From \(0^{\circ} C\) to \(4^{\circ} C\), water behaves normally as it cools and becomes denser.
• At \(4^{\circ} C\), it reaches its peak density before starting to expand as it becomes less dense with further cooling.
• This anomalous expansion continues as it reaches \(0^{\circ} C\) and freezes, leading to ice being less dense than liquid water.

This peculiar trait is critical for the natural world. It helps ensure that bodies of water freeze from the top down, allowing aquatic life to survive even in icy conditions. This understanding of temperature and density is fundamental in fields like environmental science and engineering.