Is it permissible to pay Zakat by renting an apartment to a needy Azhari student, knowing that the rental value exceeds the amount of Zakat due?

It is not permissible to pay Zakat in a way that benefits the payer, such as using it to settle his debt or to purchase something, because Zakat is a right of Allah and of the deserving. Therefore, the questioner must pay Zakat to the needy students from other funds after collecting them.