The root mean square (RMS) speed of a gas is given by $v_{rms} = \sqrt{\frac{3RT}{M}}$, where $T$ is the absolute temperature and $M$ is the molar mass.

1. Condition: $v_{rms}(\text{Ne}) = v_{rms}(\text{H}_2)$ $\sqrt{\frac{3RT_{\text{Ne}}}{M_{\text{Ne}}}} = \sqrt{\frac{3RT_{\text{H}_2}}{M_{\text{H}_2}}}$

2. Simplify: Squaring both sides and canceling constants ($3R$): $\frac{T_{\text{Ne}}}{M_{\text{Ne}}} = \frac{T_{\text{H}_2}}{M_{\text{H}_2}}$ $T_{\text{Ne}} = T_{\text{H}_2} \times \frac{M_{\text{Ne}}}{M_{\text{H}_2}}$

3. Substitute Values: $T_{\text{H}_2} = 15^\circ\text{C} = 15 + 273 = 288 \text{ K}$ $M_{\text{Ne}} = 20.2 \text{ u}$

• $M_{\text{H}_2} = 2 \text{ u}$ $T_{\text{Ne}} = 288 \times \frac{20.2}{2} = 288 \times 10.1$ $T_{\text{Ne}} \approx 2908.8 \text{ K}$

4. Scientific Notation: $2908.8 \text{ K} \approx 2.9 \times 10^3 \text{ K}$