Question

calculate the root mean square velocity and kinetic energy of $f_{2}$, $cl_{2}$, and $br_{2}$ at 298 k. rank these three halogens with respect to their rates of effusion.

Explanation:

Step1: Calculate molar masses

Molar mass of \(F_2\): \(M_{F_2}=38.00\space g/mol = 0.03800\space kg/mol\)
Molar mass of \(Cl_2\): \(M_{Cl_2}=70.90\space g/mol= 0.07090\space kg/mol\)
Molar mass of \(Br_2\): \(M_{Br_2}=159.80\space g/mol = 0.15980\space kg/mol\)

Step2: Calculate root - mean - square velocity

The formula for root - mean - square velocity is \(v_{rms}=\sqrt{\frac{3RT}{M}}\), where \(R = 8.314\space J/(mol\cdot K)\) and \(T = 298\space K\)

For \(F_2\):

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For \(Cl_2\):

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For \(Br_2\):

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Step3: Calculate kinetic energy

The formula for kinetic energy per mole is \(E_{k}=\frac{3}{2}RT\)

Since \(R = 8.314\space J/(mol\cdot K)\) and \(T = 298\space K\)

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Kinetic energy per mole is the same for \(F_2\), \(Cl_2\) and \(Br_2\) at the same temperature (\(E_{k,F_2}=E_{k,Cl_2}=E_{k,Br_2} = 3.72\times 10^{3}\space J/mol\))

Step4: Rank the rates of effusion

According to Graham's law of effusion, \(r\propto\frac{1}{\sqrt{M}}\). Since \(M_{F_2}r_{Cl_2}>r_{Br_2}\)

Answer:

• Root - mean - square velocities: \(v_{rms,F_2}\approx442\space m/s\), \(v_{rms,Cl_2}\approx324\space m/s\), \(v_{rms,Br_2}\approx216\space m/s\)
• Kinetic energies: \(E_{k,F_2}=E_{k,Cl_2}=E_{k,Br_2} = 3.72\times 10^{3}\space J/mol\)
• Rate of effusion ranking: \(F_2>Cl_2>Br_2\)