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  • A container holds an ideal gas at a temperature of 300 K. The gas consists of 

A container holds an ideal gas at a temperature of 300 K. The gas consists of 1.5\times 10^{24} gas molecules. Each molecule has a mass of4.0\times 10^{-26} kg. Calculate the RMS velocity of the gas molecules in all three directions (x, y, and z) and verify that they are the same.

Option: 1

1465.25 m/s


Option: 2

1564.98 m/s


Option: 3

1693.41 m/s


Option: 4

1451.36 m/s


Answers (1)

best_answer

Given data:

Temperature (T) = 300 K

\text { Number of molecules }(N)=1.5 \times 10^{24}

\text { Mass of each molecule }(m)=4.0 \times 10^{-26} \mathrm{~kg}

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

Step 1: Calculate the RMS velocity in one direction (vrms) using the formula

v_{\mathrm{rms}}=\sqrt{\frac{3 k T}{m}}

Substitute the given values and solve for vrms:

\begin{gathered} v_{\mathrm{rms}}=\sqrt{\frac{3 \times 1.38 \times 10^{-23} \times 300}{4.0 \times 10^{-26}}} \\ v_{\mathrm{rms}} \approx 1564.98 \mathrm{~m} / \mathrm{s} \end{gathered}

Step 2: Calculate the RMS velocity in all three directions. Since the RMS velocity is the same in all directions, the RMS velocity in each direction \left(v_{\mathrm{rms}}^x\right. \text {, }\left.v_{\mathrm{rms}}^y, v_{\mathrm{rms}}^z\right) \text { will be equal to } v_{\mathrm{rms}} \text { calculated in step } 1 .

v_{\mathrm{rms}}^x=v_{\mathrm{rms}}^y=v_{\mathrm{rms}}^z=1564.98 \mathrm{~m} / \mathrm{s}

Therefore, the correct option is (2).

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