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  • A cubical container with sides of 0.2 m is filled with an ideal gas at a temperature of 400 K. The gas consists of 5.0 × <img alt="10^{23}" src="https://entrancecorner.oncodecogs.com/gif

A cubical container with sides of 0.2 m is filled with an ideal gas at a temperature of 400 K. The gas consists of 5.0 × 10^{23} gas molecules. Each molecule has a mass of 2.0 × 10^{-26} kg. Calculate the average force exerted by the gas molecules on the walls of the container, considering perfectly elastic collisions.

Option: 1

9.24 × 10^7 N


Option: 2

5.47 × 10^5 N


Option: 3

5.03 × 10^5 N
 


Option: 4

3.25 × 10^6 N


Answers (1)

best_answer

Given data:

                Side length of the container (L) = 0.2 m
                                        Temperature (T) = 400 K
                          Number of molecules (N) = 5.0 × 1023
                       Mass of each molecule (m) = 2.0 × 10−26 kg
                              Boltzmann constant (k) = 1.38 × 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_{\mathrm{rms}}=\sqrt{\frac{3 k T}{m}}

Substitute the given values and solve for v_{rms}:

\begin{aligned} v_{\mathrm{rms}} & =\sqrt{\frac{3 \times 1.38 \times 10^{-23} \times 400}{2.0 \times 10^{-26}}} \\ & \approx 1005.15 \mathrm{~m} / \mathrm{s} \end{aligned}

Step 2: Calculate the average force exerted by the gas molecules on the walls of the container. The average force can be calculated using the formula:

F=\frac{2 m N v_{\mathrm{rms}}^2}{L}

Substitute the given values and the calculated v_{rms}, and solve for F:

\begin{aligned} F & =\frac{2 \times 5.0 \times 10^{23} \times 2.0 \times 10^{-26} \times(1005.15)^2}{0.2} \\ & \approx 5.03 \times 10^5 \mathrm{~N} \end{aligned}

Therefore, the correct option is 3.

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Nehul

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