3. (15 points) Determine the mass $m$ of helium needed to fill an empty balloon of mass $m_{1}=10$ g so that the balloon will rise. Assume the temperature and pressure of the gas in the balloon are equal to the temperature and pressure of the air. The molar mass of helium $M_{\mathrm{r}}=4$ g/mol, the molar mass of air $M_{\mathrm{B}}=29$ g/mol.
3. For the balloon to take off, the condition is $F_{\text {Arch }}=\left(m+m_{1}\right) g$

$F_{\text {Arch }}=\rho_{0} g V$

From the Clapeyron-Mendeleev equation, the density of air is

$\rho_{0}=\frac{P_{0} M_{\mathrm{B}}}{R T_{0}}, \quad$ and the volume of the balloon $V=\frac{m}{M_{\mathrm{r}}} \frac{R T_{0}}{P_{0}}$

Then $m \frac{M_{\mathrm{B}}}{M_{\mathrm{r}}}=\left(m+m_{1}\right)$

Finally, $m=\frac{m_{1} M_{\mathrm{r}}}{M_{\mathrm{B}}-M_{\mathrm{r}}}=1.6$ g.

(15 points)