Question

A cylindrical container of length \(56.0 \mathrm{~cm}\) and diameter \(12.5 \mathrm{~cm}\) holds \(0.350\) moles of nitrogen gas at a pressure of \(2.05\) atm. Find the rms speed of the nitrogen molecules.

Short answer

The root mean square speed of the nitrogen molecules is approximately 686 m/s.

Step-by-step solution

Gather information and convert units

Firstly, note down the given information. We have the number of moles of nitrogen gas \(n = 0.350\) moles, pressure \(P = 2.05\) atm and the gas constant \(R = 0.0821\) L atm K⁻¹mol⁻¹. The temperature will need to be calculated using the formula \(P = nRT/V\) where \(V\) is the volume of the gas calculated from the dimensions of the container given in the problem. The volume of a cylinder is given by \(V = \pi r²h\), where \(r\) is the radius, and \(h\) is the height. Substituting \(r = 6.25\) cm and \(h = 56.0\) cm (making sure to convert cm to L by multiplying with \(1.0 \times 10^{-3}\) L/cm), we find \(V \approx 0.686\) L.

Calculate the temperature

Rearrange the formula for \(P = nRT/V\) to find \(T = PV/nR\). Substituting \(P = 2.05\) atm, \(V = 0.686\) L, \(n = 0.350\) moles, and \(R = 0.0821\) L atm K⁻¹mol⁻¹, we find \(T \approx 151\) K.

Calculate the root mean square (rms) speed

The rms speed is given by \(v_{rms} = \sqrt{(3RT/M)}\), where \(M\) is the molar mass. The molar mass of nitrogen is \(M = 28.0\) g/mol = \(28.0 \times 10^{-3}\) kg/mol, and taking \(R = 8.314\) J K⁻¹ mol⁻¹ as the universal gas constant in these units, substituting \(T = 151\) K and \(M = 28.0 \times 10^{-3}\) kg/mol we find \(v_{rms} \approx 686\) m/s.

Key concepts

Ideal Gas Law

The Ideal Gas Law is a fundamental equation in chemistry and physics used to describe the behavior of an ideal gas. It is expressed as \( PV = nRT \). Here's what each symbol represents:

• \( P \) stands for pressure.
• \( V \) is the volume.
• \( n \) represents the number of moles of the gas.
• \( R \) is the gas constant.
• \( T \) is the temperature in Kelvin.

This equation combines several principles, such as Boyle's Law, Charles's Law, and Avogadro's Law. To find the temperature, for example, one can rearrange the equation to \( T = \frac{PV}{nR} \). This makes calculating unknown variables straightforward, given the other quantities. The Ideal Gas Law assumes that molecules don't interact except during elastic collisions. This assumption works well under many conditions but might not perfectly describe real gases at very high pressures or low temperatures.

Molar Mass

Molar mass is a measure of the mass of a given substance divided by the amount of substance, usually measured in moles. It's expressed in units of grams per mole (g/mol). Each element on the periodic table has its molar mass listed, reflecting the average mass of atoms in a mole.For nitrogen gas, represented as \( N_{2} \), the molar mass is \( 28.0 \) g/mol. This accounts for two nitrogen atoms in each molecule. To convert from grams to kilograms, which is sometimes necessary in physics, we can multiply the molar mass by \( 10^{-3} \).Understanding molar mass is critical when dealing with equations like the root mean square speed, where it must be converted to kilograms per mole (kg/mol) for compatibility with SI units used in calculations. Using accurate molar mass values ensures precision in various scientific applications, including the calculation of speeds and kinetic energies.

Gas Constant

The gas constant, denoted as \( R \), is a key factor in the Ideal Gas Law equation. It is a universal constant that can take different forms depending on the units required by the situation. Two common values are:

• \( 0.0821 \) L atm K\(^{-1}\) mol\(^{-1}\) - Used when pressure is in atmospheres and volume in liters.
• \( 8.314 \) J K\(^{-1}\) mol\(^{-1}\) - Used in kinetic theory calculations involving energy and speed.

By bridging pressure, volume, temperature, and amount of gas, the gas constant supports the calculation of unknowns using the Ideal Gas Law. It underpins calculations across chemical and physical applications, making it a cornerstone of molecular science.

Cylindrical Volume Formula

The volume of a cylinder is calculated using the formula \( V = \pi r^2 h \), where \( r \) is the radius and \( h \) is the height. This basic geometry helps in understanding how much space a gas occupies within a cylindrical container.For instance, given a cylinder with a diameter of \( 12.5 \) cm, the radius \( r \) becomes \( 6.25 \) cm. The height provided is \( 56.0 \) cm. To use these dimensions to find the volume in liters, it’s crucial to convert from cubic centimeters (cm\(^3\)) to liters (L): 1 L = 1000 cm\(^3\).Plug these values into the formula to find the volume in cm\(^3\), and then convert to liters by multiplying by \( 1.0 \times 10^{-3} \):\[ V = \pi \times (6.25)^2 \times 56.0 \times 10^{-3} \approx 0.686 \text{ L} \]Understanding this formula allows for accurate determination of volume, necessary for applying other principles like the Ideal Gas Law.