解:$①$设$13.6 \mathrm {mL}_{5}\%$过氧化氢溶液可制得氧气的质量为$x,$可得:
$2\mathrm {H}_2\mathrm {O}_2\xlongequal{\mathrm {MnO}_2}2\mathrm {H}_2\mathrm {O}+\mathrm {O}_2 $
68 32
$13.6 \mathrm {mL}×1 \mathrm {g}·\mathrm {mL}^{-1}×5\% $ x
$\frac {68}{32}=\frac {13.6 \mathrm {mL}×1 \mathrm {g}·\mathrm {mL}^{-1}×5\%}x,$$x=0.32 \mathrm {g}$
答:$13.6 \mathrm {mL}_{5}\%$过氧化氢溶液可制得氧气的质量为$0.32 \mathrm {g}。$
$②$设制得0.32 g氧气需要过碳酸钠的质量为$y。$由题意可得反应的关系式为
$2\mathrm {Na}_2\mathrm {CO}_3·3\mathrm {H}_2\mathrm {O}_2∼3\mathrm {H}_2\mathrm {O}_2∼\frac 32\mathrm {O}_2$
314 48
y 0.32 g
$\frac {314}{48}=\frac {y}{0.32 \mathrm {g}},$$y≈2.09 \mathrm {g}$
答:制得0.32 g氧气需要过碳酸钠的质量为$2.09 \mathrm {g}。$