Question

What is the average kinetic energy and rms speed of \(\mathrm{N}_{2}\) molecules at STP? Compare these values with those of \(\mathrm{H}_{2}\) molecules at STP. [Use \(R=8.314 \mathrm{~J} /(\mathrm{mol} \cdot \mathrm{K})\) and express \(\mathscr{M}\) in \(\mathrm{kg} / \mathrm{mol} .]\)

Short answer

Average kinetic energies: 3406.84 J/mol for both \(\mathrm{N}_{2}\) and \(\mathrm{H}_{2}\). RMS speeds: 493.1 m/s for \(\mathrm{N}_{2}\) and 1920 m/s for \(\mathrm{H}_{2}\).

Step-by-step solution

- Identify given values

Identify the given values needed for the calculations. For \(\mathrm{N}_{2}\): \mathscr{M} = 0.028 \,kg/mol, and for \(\mathrm{H}_{2}\): \mathscr{M} = 0.002 \,kg/mol. Use the gas constant R = 8.314 \,J/(mol \cdot \, K)

- Calculate the temperature at STP

The temperature at STP (Standard Temperature and Pressure) is 0°C, which is 273.15 K.

- Use the formula for average kinetic energy

The average kinetic energy of a gas molecule is given by \[ \frac{3}{2} k_B T \], where \ k_B \ is Boltzmann's constant (1.38 \times 10^{-23} \ J/K) and \ T \ is the temperature. However, use the mole-based formula \[ KE = \frac{3}{2} RT \] where R is the gas constant and T is the temperature.

- Compute the average kinetic energy for \(\mathrm{N}_{2}\)

Plug in the values: \[ KE_{N_2} = \frac{3}{2} \, (8.314 \, J/(mol \cdot \, K)) \, (273.15 \, K) = 3406.84 \, J/mol \]

- Compute the average kinetic energy for \(\mathrm{H}_{2}\)

Plug in the values: \[ KE_{H_2} = \frac{3}{2} \, (8.314 \, J/(mol \cdot \, K)) \, (273.15 \, K) = 3406.84 \, J/mol \]

- Use the formula for RMS speed

The root mean square (RMS) speed of molecules is given by the formula \[ v_{rms} = \sqrt{\frac{3RT}{\mathscr{M}}} \]

- Compute RMS speed for \(\mathrm{N}_{2}\)

Plug in the values: \[ v_{rms_{N2}} = \sqrt{\frac{3 \, \times 8.314 \, J/(mol \cdot \, K) \, \times 273.15 \, K}{0.028 \, kg/mol}} = 493.1 \, m/s \]

- Compute RMS speed for \(\mathrm{H}_{2}\)

Plug in the values: \[ v_{rms_{H2}} = \sqrt{\frac{3 \, \times 8.314 \, J/(mol \cdot \, K) \, \times 273.15 \, K}{0.002 \, kg/mol}} = 1920 \, m/s \]

- Compare values

Note that while the average kinetic energies of \(\mathrm{N}_{2}\) and \(\mathrm{H}_{2}\) are the same at STP, the RMS speeds differ due to their masses, with \(\mathrm{H}_{2}\) having a higher RMS speed.

Key concepts

average kinetic energy

The average kinetic energy of gas molecules is a measure of the energy contained in their random motion. For a single molecule, it's given by the formula \[ \text{KE} = \frac{3}{2} k_B T \] where \( k_B \) is Boltzmann's constant and \( T \) is the temperature in Kelvin. However, when dealing with a mole of gas molecules, we use the formula \[ \text{KE} = \frac{3}{2} RT \] where \( R \) is the gas constant. This formula tells us how the average kinetic energy depends solely on temperature, not on the type of gas. At Standard Temperature and Pressure (STP), the computed average kinetic energy for both \( N_2 \) and \( H_2 \) is the same, 3406.84 J/mol, showing that kinetic energy is independent of the gas's identity.

RMS speed

Root Mean Square (RMS) speed gives a measure of the speed of gas molecules in a sample. It's defined by the equation \[ v_{\text{rms}} = \sqrt{\frac{3RT}{ \mathscr{M} }} \] Here, \( R \) is the gas constant, \( T \) is the temperature in Kelvin, and \( \mathscr{M} \) is the molar mass of the gas in kg/mol. RMS speed accounts for the different speeds of molecules due to their masses. For instance, at STP, \( N_2 \) molecules have an RMS speed of 493.1 m/s owing to their larger molar mass of 0.028 kg/mol, whereas \( H_2 \) molecules, which are lighter with a molar mass of 0.002 kg/mol, zip around faster with an RMS speed of 1920 m/s.

Standard Temperature and Pressure (STP)

Standard Temperature and Pressure (STP) is a set of conditions used in experiments to allow for consistent and comparable data. At STP, the temperature is defined as 0°C (273.15 K), and the pressure is 1 atm (101.325 kPa). These conditions are considered the standard for conducting and comparing gas experiments. By calculating values at STP, we can easily compare the behaviors of different gases without worrying about varying experimental conditions.

molar mass

Molar mass is a measure of the mass of one mole of a substance. It's expressed in kg/mol for these calculations. Molar mass significantly influences RMS speed because the heavier the gas molecules, the slower they move at a given temperature. For example, nitrogen (\( N_2 \)) has a molar mass of 0.028 kg/mol, while hydrogen (\( H_2 \)) has a much lower molar mass of 0.002 kg/mol, explaining why \( H_2 \) molecules travel faster than \( N_2 \) molecules at the same temperature.

gas constant

The gas constant, denoted as \( R \), is a fundamental constant that appears in various gas-related equations, such as the Ideal Gas Law and kinetic theory formulas. Its value is 8.314 J/(mol·K). This constant helps relate various properties of gases, such as volume, temperature, pressure, and energy. In our formulas for average kinetic energy and RMS speed, \( R \) assists in connecting the temperature and energy or speed of gas molecules, allowing us to perform calculations that demonstrate the behavior of gases under different conditions.