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 closest option is (c) \(1.515 \times 10^{10} \, \text{erg}\).

Step 1: Understand the units

We are given a volume in litre-atmospheres (L atm) and asked to convert it to ergs. One litre atmosphere is a measure of energy. We need to use the conversion factor: \(1 \, \text{L atm} = 101.325 \, \text{Joules}\). Furthermore, 1 Joule = \(10^7\) erg.

Step 2: Convert litre atmosphere to Joules

First, calculate how many Joules 15 L atm are by multiplying the volume by the conversion factor for L atm to Joules. \[15 \, \text{L atm} = 15 \times 101.325 \, \text{J} = 1519.875 \, \text{J}.\]

Step 3: Convert Joules to ergs

Convert the amount in Joules to ergs using the conversion factor \(1 \, \text{J} = 10^7 \, \text{ergs}\).\[1519.875 \, \text{J} = 1519.875 \times 10^7 \, \text{ergs} = 1.519875 \times 10^{10} \, \text{ergs}.\]

Step 4: Round off the answer

Round 1.519875 \times 10^{10} ergs to three significant figures to match the precision of given options:\[1.520 \times 10^{10} \, \text{ergs} \]

Step 5: Compare with the options

The rounded value 1.520 \times 10^{10} \, \text{erg} does not precisely match any options. However, option (c), \(1.515 \times 10^{10} \, \text{erg}\), is the closest approximation, considering possible slight rounding differences during the conversion process.

Key Concepts

Energy Conversion

Energy conversion is the process of transforming one form of energy into another. This is crucial across various fields, from engineering to everyday activities. A simple example is converting chemical energy in food into kinetic energy for movement.
In this particular exercise, we are dealing with the energy contained in a unit of litre atmosphere (L atm) and converting it into ergs. Knowing the correct conversion factors is key. For instance, when dealing with the conversion of litre atmosphere to joules, the conversion factor is:

• 1 L atm = 101.325 Joules.

This translation from one unit to another often involves multiplying by known constants. Ensuring accuracy in these figures is vital for precise scientific and engineering work.
Conversions are necessary as different fields or regions might use different units, and energy conversion allows for a common ground for calculations and understanding.

Litre Atmosphere to Joules

Converting litre atmosphere to joules involves understanding the definition of a litre atmosphere. It is typically used in thermodynamics as a unit of energy, representing the work done by or against a pressure of one atmosphere acting on a volume of one litre. To convert, you use the conversion factor, where 1 L atm is equal to 101.325 Joules.
For instance, if you have 15 L atm, you would multiply by this conversion factor:

• 15 L atm = 15 x 101.325 J = 1519.875 Joules.

Always remember to carry through units in your calculations to ensure accuracy.
This step of the conversion is essential as it translates energy into a more universally understood unit, joules, which is the standard unit of energy in the International System of Units (SI). Joules are more commonly used in scientific discussions, making this conversion a fundamental part of energy-related calculations.

Joules to Ergs

After converting litre atmospheres to joules, the next step is often to convert those joules into another unit such as ergs, especially in cases involving smaller-scale energy measurements. The erg is a unit of energy or work in the centimetre-gram-second (CGS) system of units. One erg is a considerably smaller unit than a joule.
The conversion factor here is:

• 1 Joule = 10^7 ergs.

Therefore, to convert from joules to ergs, you simply multiply the number of joules by 10 million. For example, 1519.875 joules can be converted to ergs as follows:

• 1519.875 J = 1519.875 x 10^7 ergs = 1.519875 x 10^{10} ergs.

Understanding how to convert between these units enables more versatile and precise communication and calculation in scientific contexts where different systems of measurement might be applied.