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A cylindrical container with a height of 0.2 m and a radius of 0.1 m is filled
with an ideal gas at a temperature of 800 K. The gas contains $8.0 \times 10^{23}$ gas
molecules. Each molecule has a mass of $4.0 \times 10^{-26}$ kg. Calculate the average
kinetic energy of the gas molecules, the root mean square speed of the gas
molecules, and the pressure exerted by the gas on the walls of the container.
All the given assumptions apply.
Option: 1
$5. 43\times10^5N/m^2$

Option: 2
$7. 21\times10^3N/m^2$

Option: 3
$4. 56\times10^6N/m^2$

Option: 4
$5. 96\times10^4N/m^2$

Answers (1)

best_answer

Given data:
Height of the cylindrical container (h) = 0.2 m
Radius of the cylindrical container (r) = 0.1 m
Temperature (T) = 800 K
Number of molecules (N) $=8.0\times10^{23}$

Mass of each molecule (m) $=4.0\times10^{-26}kg$

Boltzmann constant (k) $=1.38\times10^{-23}J/K$

Step 1: Calculate the volume (V ) of the cylindrical container:

$V= \pi r^2 h$

Substitute the given values and solve for V :

$V= \pi\times(0.1)^2\times0.2$

$V= 0.00628 m^3$

Step 2: Calculate the average kinetic energy (KE) of the gas molecules. The average kinetic energy is given by:

$KE =\frac{3}{2}NkT$

Substitute the given values and solve for KE:

$KE =\frac{3}{2}\times8.0\times10^{23}\times1.38\times10^{-23}\times800$

$KE\approx 2.76\times10^{-15}J$

Step 3: Calculate the root mean square (rms) speed of the gas molecules. The rms speed is given by the formula:

$V_{rms}=\sqrt{\frac{3kT}{m}}$

Substitute the given values and solve for $V_{rms}$ :

$V_{rms} = \sqrt{\frac{3\times1.38\times10^{-23}\times800}{4.0\times10^{-26}}}$

$V_{rms} \approx 1700.82m/s$

Step 4: Calculate the pressure (P) exerted by the gas on the walls of the container. The pressure can be calculated using the formula:

$P= \frac{Nmv^2_{rms}}{3V}$

Substitute the given values and the calculated vrms, and solve for P:

$P=\frac{8.0\times10^{23}\times4.0\times10^{-26}\times(1700.82)^2}{3\times0.00628}$

$P \approx 5.43\times 10^5N/m^2$

Therefore, the correct option is 1.

Posted by

Rakesh

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