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A cubic container with sides of length 0.1 m is filled with an ideal gas at a temperature of 600 K. The gas contains $5.0\times10^{24}$ gas molecules. Each molecule has a mass of $2.0 \times 10^{-26} kg$ . Determine the root mean square speed of the gas
molecules, the total kinetic energy of the gas, and the pressure exerted by the gas on the walls of the container. All the given assumptions apply.
Option: 1
$2.30 \times 10^{9}N/m^2$

Option: 2
$1.39 \times 10^{4}N/m^2$

Option: 3
$2.53 \times 10^{3}N/m^2$

Option: 4
$2.38 \times 10^{5}N/m^2$

Answers (1)

best_answer

Given data:
Length of each side of the cubic container (L) = 0.1 m
Temperature (T) = 600 K
Number of molecules (N) $=5.0 \times 10^{24}$

Mass of each molecule (m) $=2.0 \times 10^{26}kg$

Boltzmann constant (k) $=1.38 \times 10^{-23} J/K$

Step 1: 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 $v_{rms}$

$v_{rms} = \sqrt {\frac{3\times 1.38\times 10 ^{-23}\times 600}{2.0\times 10^{-26}}}$

$\approx1592.69m/s$

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

$KE = \frac{3}{2}\times .5.0\times 10^{24}\times1.38\times10^{-23}\times 600$

$\approx 2.07 \times 10^{-15}J$

Step 3: 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 $v_{rms}$ , and solve for P:

$P=\frac{5.0\times10^{24}\times 2.0\times10^{-26}\times(1592.69)^2}{3\times(0.1)^3}$

$\approx 2.38\times 10^5N/m^2$

Therefore, the correct option is 4.

Posted by

sudhir kumar

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