Problem 54 - A Complete Resource Book in Chemistry for JEE Main

Question

15 litre atmosphere is equal to (a) \(1.515 \times 10^{8} \mathrm{erg}\) (b) \(15.15 \times 10^{9} \mathrm{erg}\) (c) \(1.515 \times 10^{10} \mathrm{erg}\) (d) \(15.15 \times 10^{12} \mathrm{erg}\)

Step-by-Step Solution

Answer

The correct answer is (a) \(1.515 \times 10^{8} \mathrm{erg}\).

Step 1: Understand Units

To solve this problem, we need to convert liters-atmosphere to ergs. One litre-atmosphere is a measure of energy equivalent to the volume of 1 litre under 1 atmosphere of pressure.

Step 2: Conversion Factor

The conversion factor between litre-atmosphere and ergs is that 1 litre-atmosphere equals \(101325 \times 10^2\) ergs, because 1 Joule is equal to \(10^7\) ergs and 1 litre-atmosphere equals 101.325 Joules.

Step 3: Calculate Energy in Ergs

Calculate the energy in ergs for 15 litre-atmosphere using the conversion factor: \(15 \times 101325 \times 10^2 = 15 \times 10132500 = 151987500 \) ergs.

Step 4: Compare to Given Options

The calculated energy in ergs, 151987500, can be rewritten as \(1.519875 \times 10^8\) ergs. Comparing with the options, the closest answer considering significant digits is \(1.515 \times 10^8\).

Key Concepts

Litre-Atmosphere to Ergs

Converting energy from one unit to another involves understanding the relationship between the units. In this case, we are working with litre-atmosphere to ergs. Litre-atmosphere is a unit of energy used in chemistry and physics to measure the work done by a quantity of gas expanding under one atmosphere of pressure. Ergs, on the other hand, are a unit of energy used primarily in the centimetre-gram-second (CGS) system. To convert litre-atmosphere to ergs, use the conversion:

• 1 litre-atmosphere = 101.325 Joules
• 1 Joule = 10⁷ ergs

This means you multiply the energy in litre-atmosphere by 101.325 and then by 10⁷ to find the energy in ergs.

Energy Conversion

Energy conversion is the process of changing energy from one form to another. In our example, we are converting energy measured in litre-atmosphere to ergs. The formula for the conversion is simple: For 1 litre-atmosphere, it is equivalent to 101.325 Joules. Each Joule can further be converted into ergs by using the factor of 10⁷ because 1 Joule equals 10⁷ ergs. Here's how to perform the conversion step-by-step:

• Calculate the Joules first: 15 litre-atmosphere × 101.325 = 1519.875 Joules
• Convert these Joules to ergs: 1519.875 × 10⁷ = 1.519875 × 10¹⁰ ergs

This allows us to understand exactly how much energy is being represented in a different unit.

Significant Figures

Significant figures are the digits in a number that contribute to its precision. When it comes to measurements and scientific calculations, it's important to maintain consistency in significant figures to ensure accuracy. In our example of converting 15 litre-atmosphere to ergs, we determine significant figures by:

• Identifying the number of digits that are known with certainty plus one digit that is estimated.
• The original number 151987500 ergs can be expressed in scientific notation as 1.519875 × 10⁸ ergs.
• When rounding off for significant figures, such as given options or requirements, we use 1.515 × 10⁸ ergs as our solution, reflecting the consistent precision needed for accurate scientific communication.

By adhering to these principles, we can represent measurements correctly and maintain the reliability of our results.