Problem 19 - Master Resource Book in JEE Main Physics

Question

The value of universal gas constant is \(R=8.3 \mathrm{~J} / \mathrm{k}-\mathrm{mol}\). The value of \(R\) in atmosphere litre per kelvin per mol (a) \(8.12\) (b) \(0.00812\) (c) \(81.2\) (d) \(0.0812\)

Step-by-Step Solution

Answer

(d) 0.0812

Step 1: Identify Units Transformation

We need to convert the given gas constant from Joules per Kelvin per mole ( \(R = 8.3 \frac{J}{K \cdot mol}\)) to atmosphere liters per Kelvin per mole. This requires converting Joules to atmosphere-liters using the relevant conversion factor.

Step 2: Determine Conversion Factor

1 Joule is equivalent to 0.00986923 liters-atmosphere (since 1 atm liter = 101.3 J). Use this conversion factor to transform the units of the gas constant.

Step 3: Apply the Conversion

Multiply the given value of the gas constant by the conversion factor to change the unit from Joules to atmosphere-liter. \[R = 8.3 \frac{J}{K \cdot mol} \times 0.00986923 \frac{L \cdot atm}{J}\]

Step 4: Calculate Result

Calculate the value of the gas constant using the conversion factor: \[8.3 \times 0.00986923 \approx 0.081921759\] Thus, rounding appropriately, this becomes \(0.0812\) in atmosphere liters per Kelvin per mole.

Step 5: Select the Correct Answer

Compare the calculated value with the given options and determine the correct choice. The correct answer is option (d), which is 0.0812.

Key Concepts

Units Transformation

When dealing with scientific equations, often you'll need to change units to make them compatible. This process is called **units transformation**. It's like translating from one language to another while keeping the meaning intact.

The exercise you're looking at deals with transforming units of the **universal gas constant**.

The original value is given in Joules per Kelvin per mole \(R = 8.3 \frac{J}{K \, mol} \) and the task is to convert it into another set of units: atmosphere liters per Kelvin per mole.

This kind of conversion is important because sometimes we use different measurement systems in practice and science. Understanding how to convert between them is crucial for solving physics and chemistry problems correctly.

Conversion Factor

When you need to transform units, you'll often use something called a **conversion factor**. It's a multiplier that helps you switch from one unit to another.

For the universal gas constant, converting from Joules to atmosphere-liters uses a specific conversion factor. The key to this calculation is knowing that **1 Joule is equal to 0.00986923 liters-atmosphere**.

To convert the gas constant's units, you multiply the original value by this conversion factor:\[R = 8.3 \, \frac{J}{K \, mol} \times 0.00986923 \, \frac{L \, atm}{J}\]

This multiplication effectively translates the energy unit from Joules to atmosphere-liters, avoiding any errors that might occur from a direct calculation without proper conversion.

Atmosphere Liters

In scientific calculations, particularly in chemistry and physics, you will encounter different units for measuring pressure and volume. **Atmosphere liters** is a combined unit. It pairs the unit for pressure—"atmosphere"—and the unit for volume—"liters."

One atmosphere is approximately the pressure exerted by the Earth's atmosphere at sea level. Meanwhile, a liter is a common unit of volume, often used to measure liquids and gases.

When used in combination, these units can express work or energy changes in a system, such as in gas physics using the gas laws. This makes learning to convert existing units to and from atmosphere-liters useful, especially when dealing with the universal gas constant in this context. Understanding these units equips you to solve more complex problems efficiently and accurately.