Question

Find the rms speed of hydrogen molecules in a sample of hydrogen gas at 300 K. Find the temperature at which the rms speed is double the speed calculated in the previous part.

Answer 1

The temperature at which the rms speed is double the speed calculated in the previous part is about 1200 K

Explanation:

Step 1:

Given data in the question  

Hydrogen Molar mass, Mo = 2 g/mol=0.002 kg /mol

We know that

$C=\sqrt{\frac{3 R T}{M_{0}}}$

$C=\sqrt{\frac{3 \times 8.3 \times 300}{0.002}}$

$C=\sqrt{\frac{900 \times 8.3}{0.002}}$

$C=\sqrt{\frac{7470}{0.002}}$

$C=\sqrt{3735000}$

C = 1932.61

Step 2:

In the second case, let the temperature required be T.

Use the same formula, and we get

$C=\sqrt{\frac{3 R T}{M_{0}}}$

$\begin{aligned}&\sqrt{\frac{3 \times 8.3 T}{0.002}}=2 \times 1932.61\\&\sqrt{\frac{3 \times 8.3 T}{0.002}}=3865.22\\&(\sqrt{\frac{3 \times 8.3 T}{0.002}})^{2}=3865.22^{2}\end{aligned}$

$\begin{aligned}&3 \times 8.3 T=0.002 \times 14939925.65\\&3 \times 8.3 T=29879.85\\&T=\frac{29879.85}{3 \times 8.3}\\&T=\frac{29879.85}{24.9}\\&T=1199.9 \approx 1200 k\end{aligned}$

  = 1200 K

Answer 2

The temperature of the hydrogen is 1200 K when the rms speed value of 1932.61 is double the speed

Explanation:

Given data in the question  

Molar mass of Hydrogen , Mo = 2 g/mol  =0.002 kg /mol

We know that

$C=\sqrt{\frac{3 R T}{M_{0}}}$

$C=\sqrt{\frac{3 \times 8.3 \times 300}{0.002}}$

$C=\sqrt{\frac{900 \times 8.3}{0.002}}$

$C=\sqrt{\frac{7470}{0.002}}$

C =1932.61

In the second case, let the temperature required be T.

Use the same formula, and we get

$C=\sqrt{\frac{3 R T}{M_{0}}}$

$\sqrt{\frac{3 \times 8.3 T}{0.002}}=2 \times 1932.61$

$\sqrt{\frac{3 \times 8.3 T}{0.002}}=3865.22$

Squaring on both sides of the above equation

$(\sqrt{\frac{3 \times 8.3 T}{0.002}})^{2}=3865.22^{2}$

$\frac{3 \times 8.3 T}{0.002}=14939925.65$

$3 \times 8.3 T=0.002 \times 14939925.65$

$3 \times 8.3 T=29879.85$

$T=\frac{29879.85}{3 \times 8.3}$

$T=\frac{29879,85}{24.9}$

$T=1199.9 \approx 1200 K$

1200 K is the temperature of the hydrogen when the rms speed of 1932.61 is double the speed and the temperature of the sample gas is 300 K