Question

1. calculate the root mean square velocity and kinetic energy of f₂, cl₂, and br₂ at 298 k. rank these three halogens with respect to their rate of effusion.

missed this? read section 6.8; watch kcv 6.8, iwe 6.14

Explanation:

Step1: Recall Root Mean Square Velocity Formula

The root mean square (rms) velocity formula is \( v_{\text{rms}} = \sqrt{\frac{3RT}{M}} \), where \( R \) is the gas constant (\( 8.314 \, \text{J/(mol·K)} \)), \( T \) is temperature in Kelvin, and \( M \) is molar mass in kg/mol. Kinetic energy (\( KE \)) for a gas is \( KE = \frac{3}{2}RT \) per mole (or \( \frac{1}{2}mv^2 \) for particles), and rate of effusion is related to molar mass by Graham's law: \( \text{Rate} \propto \frac{1}{\sqrt{M}} \).

Step2: Find Molar Masses

• \( \text{F}_2 \): Molar mass \( M_{\text{F}_2} = 2 \times 19.00 = 38.00 \, \text{g/mol} = 0.038 \, \text{kg/mol} \)
• \( \text{Cl}_2 \): \( M_{\text{Cl}_2} = 2 \times 35.45 = 70.90 \, \text{g/mol} = 0.0709 \, \text{kg/mol} \)
• \( \text{Br}_2 \): \( M_{\text{Br}_2} = 2 \times 79.90 = 159.80 \, \text{g/mol} = 0.1598 \, \text{kg/mol} \)

Step3: Analyze \( v_{\text{rms}} \)

Since \( v_{\text{rms}} \propto \frac{1}{\sqrt{M}} \) (same \( T \) and \( R \)), lower molar mass means higher \( v_{\text{rms}} \). So order: \( \text{F}_2 > \text{Cl}_2 > \text{Br}_2 \).

Step4: Analyze Kinetic Energy

Kinetic energy per mole depends only on temperature (\( KE = \frac{3}{2}RT \)). At same \( T \), all have equal \( KE \) per mole. For average kinetic energy per particle, \( KE = \frac{1}{2}mv^2 \), but since \( KE_{\text{per mole}} \) is same, and \( m \) (molar mass) is different, but per mole, \( KE \) is equal. So \( \text{F}_2 = \text{Cl}_2 = \text{Br}_2 \) for \( KE \) per mole.

Step5: Analyze Rate of Effusion (Graham's Law)

Rate of effusion \( \propto \frac{1}{\sqrt{M}} \), so same as \( v_{\text{rms}} \) order: \( \text{F}_2 > \text{Cl}_2 > \text{Br}_2 \).

Answer:

Root Mean Square Velocity:

Order (highest to lowest): \( \boldsymbol{\text{F}_2 > \text{Cl}_2 > \text{Br}_2} \) (inversely proportional to \( \sqrt{\text{molar mass}} \)).

Kinetic Energy (per mole):

All three have equal kinetic energy at 298 K (depends only on temperature).

Rate of Effusion:

Order (highest to lowest): \( \boldsymbol{\text{F}_2 > \text{Cl}_2 > \text{Br}_2} \) (Graham’s law: rate \( \propto \frac{1}{\sqrt{M}} \)).