QUARTER FINAL STAGE 2026
St. John's School: 43 points
Saviour SHS: 33 points
Mfantsiman Girls' SHS: 2 points
QUESTION
Calculate the RMS speed of a molecule in an ideal gas of density $1.2\text{ kg/m}^3$ and pressure $3.6\times10^{4}\text{ Pa}$.
ANSWER: $3.0\times10^{2}\text{ m/s}$
SOLUTION:
From kinetic theory,
$P=\dfrac{1}{3}\rho v_{rms}^2$,
$v_{rms}=\sqrt{\dfrac{3P}{\rho}}$
$v_{rms}=\sqrt{\dfrac{3\times3.6\times10^{4}}{1.2}}$
$v_{rms}=\sqrt{9\times10^{4}}$
$v_{rms}=3.0\times10^{2}\text{ m/s}$
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PRACTICE QUESTIONS
1. QUESTION: Calculate the RMS speed of a molecule in an ideal gas of density $1.2\text{ kg/m}^3$ and pressure $4.84\times10^{3}\text{ Pa}$.
ANSWER: 110 m/s or $1.1\times10^{2}$ m/s
SOLUTION:
$P = \dfrac{1}{3}\rho v_{rms}^2$
$v_{rms}^2 = \dfrac{3P}{\rho}$
$v_{rms}^2 = \dfrac{3 \times 4.84\times10^{3}}{1.2}$
$v_{rms}^2 = \dfrac{3 \times 484\times10^{1}}{12\times10^{-1}}$
$v_{rms}^2 = \dfrac{3 \times 121}{3} \times10^{2}$
$v_{rms}^2 = 1.21\times10^{4}\text{ m}^2/\text{s}^2$
$v_{rms} = \sqrt{1.21\times10^{4}}$
$v_{rms} = 1.1\times10^{2}\text{ m/s}$
2. QUESTION: Calculate the RMS speed of a molecule in an ideal gas of density $1.2\text{ kg/m}^3$ and pressure $6.76\times10^{3}\text{ Pa}$.
ANSWER: 130 m/s or $1.3\times10^{2}$ m/s
SOLUTION:
$P = \dfrac{1}{3}\rho v_{rms}^2$
$v_{rms}^2 = \dfrac{3P}{\rho}$
$v_{rms}^2 = \dfrac{3 \times 6.76\times10^{3}}{1.2}$
$v_{rms}^2 = \dfrac{3 \times 676\times10^{1}}{12\times10^{-1}}$
$v_{rms}^2 = \dfrac{3 \times 169}{3} \times10^{2}$
$v_{rms}^2 = 1.69\times10^{4}\text{ m}^2/\text{s}^2$
$v_{rms} = \sqrt{1.69\times10^{4}}$
$v_{rms} = 1.3\times10^{2}\text{ m/s}$
3. QUESTION: Calculate the RMS speed of a molecule in an ideal gas of density $1.2\text{ kg/m}^3$ and pressure $7.84\times10^{3}\text{ Pa}$.
ANSWER: 140 m/s or $1.4\times10^{2}$ m/s
SOLUTION:
$P = \dfrac{1}{3}\rho v_{rms}^2$
$v_{rms}^2 = \dfrac{3P}{\rho}$
$v_{rms}^2 = \dfrac{3 \times 7.84\times10^{3}}{1.2}$
$v_{rms}^2 = \dfrac{3 \times 784\times10^{1}}{12\times10^{-1}}$
$v_{rms}^2 = \dfrac{3 \times 196}{3} \times10^{2}$
$v_{rms}^2 = 1.96\times10^{4}\text{ m}^2/\text{s}^2$
$v_{rms} = \sqrt{1.96\times10^{4}}$
$v_{rms} = 1.4\times10^{2}\text{ m/s}$
4. QUESTION: Calculate the RMS speed of a molecule in an ideal gas of density $1.2\text{ kg/m}^3$ and pressure $1.024\times10^{4}\text{ Pa}$.
ANSWER: 160 m/s or $1.6\times10^{2}$ m/s
SOLUTION:
$P = \dfrac{1}{3}\rho v_{rms}^2$
$v_{rms}^2 = \dfrac{3P}{\rho}$
$v_{rms}^2 = \dfrac{3 \times 1.024\times10^{4}}{1.2}$
$v_{rms}^2 = \dfrac{3 \times 1024\times10^{1}}{12\times10^{-1}}$
$v_{rms}^2 = \dfrac{3 \times 256}{3} \times10^{2}$
$v_{rms}^2 = 2.56\times10^{4}\text{ m}^2/\text{s}^2$
$v_{rms} = \sqrt{2.56\times10^{4}}$
$v_{rms} = 1.6\times10^{2}\text{ m/s}$
5. QUESTION: Calculate the RMS speed of a molecule in an ideal gas of density $1.2\text{ kg/m}^3$ and pressure $1.156\times10^{4}\text{ Pa}$.
ANSWER: 170 m/s or $1.7\times10^{2}$ m/s
SOLUTION:
$P = \dfrac{1}{3}\rho v_{rms}^2$
$v_{rms}^2 = \dfrac{3P}{\rho}$
$v_{rms}^2 = \dfrac{3 \times 1.156\times10^{4}}{1.2}$
$v_{rms}^2 = \dfrac{3 \times 1156\times10^{1}}{12\times10^{-1}}$
$v_{rms}^2 = \dfrac{3 \times 289}{3} \times10^{2}$
$v_{rms}^2 = 2.89\times10^{4}\text{ m}^2/\text{s}^2$
$v_{rms} = \sqrt{2.89\times10^{4}}$
$v_{rms} = 1.7\times10^{2}\text{ m/s}$
6. QUESTION: Calculate the RMS speed of a molecule in an ideal gas of density $1.2\text{ kg/m}^3$ and pressure $1.296\times10^{4}\text{ Pa}$.
ANSWER: 180 m/s or $1.8\times10^{2}$ m/s
SOLUTION:
$P = \dfrac{1}{3}\rho v_{rms}^2$
$v_{rms}^2 = \dfrac{3P}{\rho}$
$v_{rms}^2 = \dfrac{3 \times 1.296\times10^{4}}{1.2}$
$v_{rms}^2 = \dfrac{3 \times 1296\times10^{1}}{12\times10^{-1}}$
$v_{rms}^2 = 3 \times 108 \times10^{2}$
$v_{rms}^2 = 3.24\times10^{4}\text{ m}^2/\text{s}^2$
$v_{rms} = \sqrt{3.24\times10^{4}}$
$v_{rms} = 1.8\times10^{2}\text{ m/s}$
7. QUESTION: Calculate the RMS speed of a molecule in an ideal gas of density $1.2\text{ kg/m}^3$ and pressure $1.444\times10^{4}\text{ Pa}$.
ANSWER: 190 m/s or $1.9\times10^{2}$ m/s
SOLUTION:
$P = \dfrac{1}{3}\rho v_{rms}^2$
$v_{rms}^2 = \dfrac{3P}{\rho}$
$v_{rms}^2 = \dfrac{3 \times 1.444\times10^{4}}{1.2}$
$v_{rms}^2 = \dfrac{3 \times 1444\times10^{1}}{12\times10^{-1}}$
$v_{rms}^2 = \dfrac{3 \times 361}{3} \times10^{2}$
$v_{rms}^2 = 3.61\times10^{4}\text{ m}^2/\text{s}^2$
$v_{rms} = \sqrt{3.61\times10^{4}}$
$v_{rms} = 1.9\times10^{2}\text{ m/s}$
8. QUESTION: Calculate the RMS speed of a molecule in an ideal gas of density $1.2\text{ kg/m}^3$ and pressure $1.764\times10^{4}\text{ Pa}$.
ANSWER: 210 m/s or $2.1\times10^{2}$ m/s
SOLUTION:
$P = \dfrac{1}{3}\rho v_{rms}^2$
$v_{rms}^2 = \dfrac{3P}{\rho}$
$v_{rms}^2 = \dfrac{3 \times 1.764\times10^{4}}{1.2}$
$v_{rms}^2 = \dfrac{3 \times 1764\times10^{1}}{12\times10^{-1}}$
$v_{rms}^2 = 3 \times 147 \times10^{2}$
$v_{rms}^2 = 4.41\times10^{4}\text{ m}^2/\text{s}^2$
$v_{rms} = \sqrt{4.41\times10^{4}}$
$v_{rms} = 2.1\times10^{2}\text{ m/s}$