Can he give his zakaah to his relative who is studying in university and cannot afford the expenses?

It is permissible to give zakaah to relatives who are entitled to it; that is better than giving it to people who are not your relatives, because charity that is given to a relative is both charity and upholding ties of kinship. The Prophet (blessings and peace of Allah be upon him) said: “Charity given to a poor person is charity, but that which is given to a relative is two things, charity and upholding ties of kinship.”

Narrated by an-Nasaa’i, 2581; at-Tirmidhi, 658; classed as saheeh by al-Albaani in Saheeh an-Nasaa’i, 2420. 

But that is subject to two conditions: 

1.

That the relative who receives the zakaah should be one who is entitled to it (poor or needy), even if he has a job, so long as his earnings are not sufficient to meet his needs.

Moreover, it makes no difference whether the university is near or far; rather what matters is whether the man is poor or needy. 

2.

That these relatives should not be among those on whose maintenance it is obligatory for the zakaah-giver to spend. If spending on their maintenance is obligatory for him, then it is not permissible for him to give them anything from the zakaah on his wealth. 

Ash-Shaafa‘i said in al-Umm (2/87): He should not give (the zakaah of his wealth) to his father, mother, grandfather or grandmother. End quote. 

Ibn Qudaamah said in al-Mughni (2/509): Obligatory charity (i.e., zakaah) should not be given to parents or grandparents, no matter how far the line of ascent reaches, or to children or grandchildren, no matter how far the line of descent reaches. 

If these two conditions apply with regard to this relative of yours, it is permissible to give him enough to meet his needs from the zakaah of your wealth. One of those needs is to complete his studies, so long as it is a permissible field of studies that is beneficial in religious or worldly terms. Al-Mirdaawi said in al-Insaaf (3/218): Shaykh Taqiy ad-Deen favoured the view that it is permissible to accept zakaah in order to buy books that he needs, books of knowledge that are essential to his religious and worldly interests. End quote. And this is the correct view. End quote. 

And Allah knows best.