Problem 40 - Advanced Problems in Physical Chemistry for Competitive Examinations

Question

Number of gas molecules present in \(1 \mathrm{ml}\) of gas at \(0{ }^{\circ} \mathrm{C}\) and \(1 \mathrm{~atm}\) is called Loschmidt number. Its value is about (a) \(2.7 \times 10^{19}\) (b) \(6 \times 10^{23}\) (c) \(2.7 \times 10^{22}\) (d) \(1.3 \times 10^{28}\)

Step-by-Step Solution

Answer

The Loschmidt number is approximately \(2.7 \times 10^{19}\) molecules.

Step 1: Understanding the Concept of Loschmidt Number

The Loschmidt number is defined as the number of molecules of an ideal gas present in a volume of one cubic centimeter at standard temperature and pressure (STP), which is 0°C and 1 atm. To solve for the Loschmidt number, we use Avogadro's number and the ideal gas law.

Step 2: Using Avogadro's Law to Determine the Number of Molecules

According to Avogadro's Law, one mole of any gas occupies 22.4 liters at STP. We know that Avogadro's number is approximately \(6 \times 10^{23}\), which is the number of molecules in one mole of a substance. To find out how many molecules are present in 1 ml, we must convert 1 ml to liters and then use a proportion to find the number of molecules.

Step 3: Convert Volume from Milliliters to Liters

First, convert the volume from milliliters to liters, since we know the volume occupied by a mole of gas is expressed in liters. \(1 \text{ ml} = 1 \times 10^{-3} \text{ liters}\).

Step 4: Calculate the Proportion of Molecules in 1 ml

Calculate the proportion of molecules in 1 ml using the volume of one mole at STP (22.4 liters): \(\frac{6 \times 10^{23} \text{ molecules}}{22.4 \text{ L}} = \frac{x \text{ molecules}}{1 \times 10^{-3} \text{ L}}\). Solve for x to find the Loschmidt number.

Step 5: Solving for the Loschmidt Number

Cross-multiply and divide to find the value of x: \(x = \frac{6 \times 10^{23} \text{ molecules} \times 1 \times 10^{-3} \text{ L}}{22.4 \text{ L}} \ x \) \ \

Key Concepts

Avogadro's Law

Let's delve into Avogadro's Law, a principle that serves as a cornerstone in chemistry and physics. This law states that equal volumes of gases, at the same temperature and pressure, contain an equal number of molecules. To put it simply, it's not the type of gas that matters, but the conditions it's under that determine the number of gas molecules present.

For instance, under standard conditions of temperature and pressure, known as STP (0°C and 1 atm), one mole of any ideal gas occupies 22.4 liters. This is essential to remember because it helps us understand how to relate volume to the number of molecules. If we want to determine the number of molecules in a different volume, we can use a proportion based on the fact that 22.4 liters contain approximately Avogadro's number of molecules, which is about 6 x 10^23 molecules for any gas.

Ideal Gas Law

The ideal gas law fuses several gas laws into one universal equation, tying the state variables—pressure (P), volume (V), temperature (T), and number of moles (n)—together with the gas constant (R). Expressed as PV = nRT, it's a powerful relationship that allows us to calculate one property if we know the others.

The simplicity of this law is that it applies to hypothetical ideal gases, which are assumed to have no intermolecular forces and occupy no volume. Though real gases don't perfectly obey this law, it's very accurate under many conditions, especially at high temperatures and low pressures. In our context, using this law in conjunction with Avogadro's law lets us find the Loschmidt number, which then translates to a concrete count of gas molecules in a certain volume at STP.

STP (Standard Temperature and Pressure)

When working with gases, it's crucial to reference STP—Standard Temperature and Pressure. Defined as 0°C (273.15 K) for temperature and 1 atm for pressure, it provides a common comparison for discussing properties of gases.

Under STP, finding values such as density, volume, and the number of molecules becomes manageable as the variables are fixed and known. When calculating the Loschmidt number, we specifically use the conditions of STP to figure out the number of molecules in 1 ml of a gas. Using STP allows us to simplify our calculations and ensures that we can accurately apply Avogadro's Law and the ideal gas law to find reliable results.