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Although not a chemical method, using flotation in bromine to separate your mixture will work at room temperature. Unlike the chemical methods given in other answers, this physical method of separation does not destroy or otherwise degrade the components of your mixture, so you can recover them.

The density of bromine at 20 $^\circ$C is 3.1 g$\cdot$cm$^{-3}$; the densities of $\ce{Al2O3}$ and $\ce{SiO2}$ at the same temperature are 3.95 g$\cdot$cm$^{-3}$ and 2.65 g$\cdot$cm$^{-3}$, respectively. The $\ce{SiO2}$ will float and the $\ce{Al2O3}$ will sink.

As noted in the comment thread above, this method will work unless the particle size is too small.

And as noted in the comments below, sodium polytungstate solution (with a density of 3.1 g$\cdot$cm$^{-3}$) is an alternative to bromine.

Because there are likely a large number of substances with a density in the range required to separate your mixture, the suggestions here are representative of a small number of choices.

Although not a chemical method, using flotation in bromine to separate your mixture will work at room temperature. Unlike the chemical methods given in other answers, this physical method of separation does not destroy or otherwise degrade the components of your mixture, so you can recover them.

The density of bromine at 20 $^\circ$C is 3.1 g$\cdot$cm$^{-3}$; the densities of $\ce{Al2O3}$ and $\ce{SiO2}$ at the same temperature are 3.95 g$\cdot$cm$^{-3}$ and 2.65 g$\cdot$cm$^{-3}$, respectively. The $\ce{SiO2}$ will float and the $\ce{Al2O3}$ will sink.

As noted in the comment thread above, this method will work unless the particle size is too small.

And as noted in the comments below, sodium polytungstate solution (with a density of 3.1 g$\cdot$cm$^{-3}$) is an alternative to bromine.

Because there are likely a large number of substances with a density in the range required to separate your mixture, the suggestions here represent a small number of those choices.

Although not a chemical method, using flotation in bromine to separate your mixture will work at room temperature. Unlike the chemical methods given in other answers, this physical method of separation does not destroy or otherwise degrade the components of your mixture, so you can recover them.

The density of bromine at 20 $^\circ$C is 3.1 g$\cdot$cm$^{-3}$; the densities of $\ce{Al2O3}$ and $\ce{SiO2}$ at the same temperature are 3.95 g$\cdot$cm$^{-3}$ and 2.65 g$\cdot$cm$^{-3}$, respectively. The $\ce{SiO2}$ will float and the $\ce{Al2O3}$ will sink.

As noted in the comment thread above, this method will work unless the particle size is too small.

Although not a chemical method, using flotation in bromine to separate your mixture will work at room temperature. Unlike the chemical methods given in other answers, this physical method of separation does not destroy or otherwise degrade the components of your mixture, so you can recover them.

The density of bromine at 20 $^\circ$C is 3.1 g$\cdot$cm$^{-3}$; the densities of $\ce{Al2O3}$ and $\ce{SiO2}$ at the same temperature are 3.95 g$\cdot$cm$^{-3}$ and 2.65 g$\cdot$cm$^{-3}$, respectively. The $\ce{SiO2}$ will float and the $\ce{Al2O3}$ will sink.

As noted in the comment thread above, this method will work unless the particle size is too small.

And as noted in the comments below, sodium polytungstate solution (with a density of 3.1 g$\cdot$cm$^{-3}$) is an alternative to bromine.

Because there are likely a large number of substances with a density in the range required to separate your mixture, the suggestions here are representative of a small number of choices.

Although not a chemical method, using flotation in bromine to separate your mixture will work at room temperature. Unlike the chemical methods given in other answers, this physical method of separation does not destroy or otherwise degrade the components of your mixture, so you can recover them.

The density of bromine at 20$^\circ$ is 3.1 g$\cdot$cm$^{-3}$; the densities of $\ce{Al2O3}$ and $\ce{SiO2}$ at the same temperature are 3.95 g$\cdot$cm$^{-3}$ and 2.65 g$\cdot$cm$^{-3}$, respectively. The $\ce{SiO2}$ will float and the $\ce{Al2O3}$ will sink.

As noted in the comment thread above, this method will work unless the particle size is too small.

Although not a chemical method, using flotation in bromine to separate your mixture will work at room temperature. Unlike the chemical methods given in other answers, this physical method of separation does not destroy or otherwise degrade the components of your mixture, so you can recover them.

The density of bromine at 20  $^\circ$C is 3.1 g$\cdot$cm$^{-3}$; the densities of $\ce{Al2O3}$ and $\ce{SiO2}$ at the same temperature are 3.95 g$\cdot$cm$^{-3}$ and 2.65 g$\cdot$cm$^{-3}$, respectively. The $\ce{SiO2}$ will float and the $\ce{Al2O3}$ will sink.

As noted in the comment thread above, this method will work unless the particle size is too small.