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) \(1.3 \times 10^{23}\)

Short answer

Answer: (c) 2.7 x 10^22

Step-by-step solution

Convert temperature and volume to SI units

Given temperature \(T = 0^{\circ}\mathrm{C}\), we need to convert it to Kelvin, using the formula \(K = \mathrm{ C} + 273.15\). \( T = 0 + 273.15 = 273.15 \mathrm{K}\). The volume is given in milliliters (ml), we need to convert it to cubic meters (m³): \(V = 1 \mathrm{ml} = 1 \times 10^{-6} \mathrm{m^3}\).

Use the Ideal Gas Law formula

We will use the Ideal Gas Law formula, \(PV=nRT\), where P is pressure, V is volume, n is the number of moles, R is the gas constant, and T is temperature. Given Pressure, \(P = 1 \mathrm{atm}\), which is equal to \(101325 \mathrm{Pa}\) in SI units. The gas constant, \(R = 8.314 \mathrm{J/(mol.K)}\). Now we can solve for n (Number of moles): \( n = \frac{PV}{RT} = \frac{101325 \mathrm{Pa} \cdot 1 \times 10^{-6} \mathrm{m^3}}{8.314 \mathrm{J/(mol.K)} \cdot 273.15 \mathrm{K}}\)

Solve for n

Solving the equation in Step 2 for n, we get \(n = \frac{101325 \cdot 1 \times 10^{-6}}{8.314 \cdot 273.15} = 0.044636 \mathrm{moles}\)

Calculate the number of molecules using Avogadro's number

Now we need to multiply the number of moles by Avogadro's number (\(6.022 \times 10^{23} \mathrm{molecules/mol}\)) to find out the number of molecules in 1 ml of gas: Number of molecules = \(n \cdot N_\mathrm{A} = 0.044636 \mathrm{moles} \cdot 6.022 \times 10^{23} \mathrm{molecules/mol} = 2.686 \times 10^{22} \mathrm{molecules}\) Comparing this result to the provided options, we find that the correct answer is (c) \(2.7 \times 10^{22}\).

Key concepts

Avogadro's Number

Avogadro's number, typically denoted as \(N_A\), is a fundamental constant in chemistry and physics that represents the number of atoms, ions, or molecules in one mole of a substance. A mole is one of the seven basic SI units for amount of substance, and \(N_A\) is approximately \(6.022 \times 10^{23}\) entities per mole.

This colossal number allows scientists to work with the massive quantities of atoms and molecules necessary for chemical reactions in a manageable way, translating between macroscopic and microscopic scales. In the context of the Loschmidt number exercise, Avogadro's number is used to convert moles of a gas into the actual number of molecules, linking the macroscopic properties of a mole of gas to the microscopic detail of individual particles.

Ideal Gas Law

The Ideal Gas Law is a fundamental equation in thermodynamics and physical chemistry, relating volume, pressure, temperature, and the number of moles of a gas. The law is provided by the equation \(PV=nRT\), where \(P\) represents pressure, \(V\) is volume, \(n\) stands for the number of moles, \(R\) is the ideal gas constant, and \(T\) is the temperature in Kelvin.

The law assumes the ideal scenario where there are no intermolecular forces and the molecules occupy no volume. Although real gases do not perfectly follow the Ideal Gas Law, it serves as a good approximation under many conditions. In the textbook exercise, this law allows us to calculate the number of moles of gas in a given volume at known pressure and temperature, which is a crucial step in determining the Loschmidt number.

SI Units Conversion

SI units conversion is imperative in science and engineering to ensure consistency and accuracy in measurements and calculations. The International System of Units (SI) provides a standard for seven base units, from which all other units are derived. In particular, it's common to convert temperature from Celsius to Kelvin for thermodynamic equations, as seen with the Ideal Gas Law, and to convert volumes from milliliters to cubic meters to adhere to the SI standard for volume measurement.

For instance, converting temperature to Kelvin involves adding 273.15 to the Celsius value, and converting volume to cubic meters often requires moving the decimal point six places to the left, as there are \(1 \times 10^{6}\) milliliters in a cubic meter. Accurate conversion is critical, as discrepancies can significantly affect the outcome, particularly when dealing with calculations at the molecular level, such as finding the Loschmidt number.